ownp.v 11.1 KB
Newer Older
1 2 3 4 5
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import lifting adequacy.
From iris.program_logic Require ectx_language.
From iris.algebra Require Import auth.
From iris.proofmode Require Import tactics classes.
6
Set Default Proof Using "Type".
7 8 9 10 11 12 13 14 15 16 17 18

Class ownPG' (Λstate : Type) (Σ : gFunctors) := OwnPG {
  ownP_invG : invG Σ;
  ownP_inG :> inG Σ (authR (optionUR (exclR (leibnizC Λstate))));
  ownP_name : gname;
}.
Notation ownPG Λ Σ := (ownPG' (state Λ) Σ).

Instance ownPG_irisG `{ownPG' Λstate Σ} : irisG' Λstate Σ := {
  iris_invG := ownP_invG;
  state_interp σ := own ownP_name ( (Excl' (σ:leibnizC Λstate)))
}.
19
Global Opaque iris_invG.
20 21 22

Definition ownPΣ (Λstate : Type) : gFunctors :=
  #[invΣ;
23
    GFunctor (authUR (optionUR (exclR (leibnizC Λstate))))].
24 25 26 27 28 29 30 31

Class ownPPreG' (Λstate : Type) (Σ : gFunctors) : Set := IrisPreG {
  ownPPre_invG :> invPreG Σ;
  ownPPre_inG :> inG Σ (authR (optionUR (exclR (leibnizC Λstate))))
}.
Notation ownPPreG Λ Σ := (ownPPreG' (state Λ) Σ).

Instance subG_ownPΣ {Λstate Σ} : subG (ownPΣ Λstate) Σ  ownPPreG' Λstate Σ.
32
Proof. solve_inG. Qed.
33 34 35 36 37 38 39 40 41

(** Ownership *)
Definition ownP `{ownPG' Λstate Σ} (σ : Λstate) : iProp Σ :=
  own ownP_name ( (Excl' σ)).
Typeclasses Opaque ownP.
Instance: Params (@ownP) 3.


(* Adequacy *)
42 43 44
Theorem ownP_adequacy Σ `{ownPPreG Λ Σ} p e σ φ :
  ( `{ownPG Λ Σ}, ownP σ  WP e @ p;  {{ v, ⌜φ v }}) 
  adequate p e σ φ.
45 46 47 48 49 50 51 52
Proof.
  intros Hwp. apply (wp_adequacy Σ _).
  iIntros (?) "". iMod (own_alloc ( (Excl' (σ : leibnizC _))   (Excl' σ)))
    as (γσ) "[Hσ Hσf]"; first done.
  iModIntro. iExists (λ σ, own γσ ( (Excl' (σ:leibnizC _)))). iFrame "Hσ".
  iApply (Hwp (OwnPG _ _ _ _ γσ)). by rewrite /ownP.
Qed.

53
Theorem ownP_invariance Σ `{ownPPreG Λ Σ} p e σ1 t2 σ2 φ :
54
  ( `{ownPG Λ Σ},
55
    ownP σ1 ={}= WP e @ p;  {{ _, True }}  |={,}=>  σ', ownP σ'  ⌜φ σ') 
56 57 58
  rtc step ([e], σ1) (t2, σ2) 
  φ σ2.
Proof.
59
  intros Hwp Hsteps. eapply (wp_invariance Σ Λ p e σ1 t2 σ2 _)=> //.
60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77
  iIntros (?) "". iMod (own_alloc ( (Excl' (σ1 : leibnizC _))   (Excl' σ1)))
    as (γσ) "[Hσ Hσf]"; first done.
  iExists (λ σ, own γσ ( (Excl' (σ:leibnizC _)))). iFrame "Hσ".
  iMod (Hwp (OwnPG _ _ _ _ γσ) with "[Hσf]") as "[$ H]"; first by rewrite /ownP.
  iIntros "!> Hσ". iMod "H" as (σ2') "[Hσf %]". rewrite /ownP.
  iDestruct (own_valid_2 with "Hσ Hσf")
    as %[->%Excl_included%leibniz_equiv _]%auth_valid_discrete_2; auto.
Qed.


(** Lifting *)
Section lifting.
  Context `{ownPG Λ Σ}.
  Implicit Types e : expr Λ.
  Implicit Types Φ : val Λ  iProp Σ.

  Lemma ownP_twice σ1 σ2 : ownP σ1  ownP σ2  False.
  Proof. rewrite /ownP -own_op own_valid. by iIntros (?). Qed.
78
  Global Instance ownP_timeless σ : Timeless (@ownP (state Λ) Σ _ σ).
79 80
  Proof. rewrite /ownP; apply _. Qed.

81
  Lemma ownP_lift_step p E Φ e1 :
82 83
    (|={E,}=>  σ1, reducible e1 σ1   ownP σ1 
        e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs - ownP σ2
84 85
            ={,E}= WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
86 87 88 89 90 91
  Proof.
    iIntros "H". destruct (to_val e1) as [v|] eqn:EQe1.
    - apply of_to_val in EQe1 as <-. iApply fupd_wp.
      iMod "H" as (σ1) "[Hred _]"; iDestruct "Hred" as %Hred%reducible_not_val.
      move: Hred; by rewrite to_of_val.
    - iApply wp_lift_step; [done|]; iIntros (σ1) "Hσ".
92
      iMod "H" as (σ1' ?) "[>Hσf H]". rewrite /ownP.
93 94 95 96 97 98
      iDestruct (own_valid_2 with "Hσ Hσf")
        as %[->%Excl_included%leibniz_equiv _]%auth_valid_discrete_2.
      iModIntro; iSplit; [done|]; iNext; iIntros (e2 σ2 efs Hstep).
      iMod (own_update_2 with "Hσ Hσf") as "[Hσ Hσf]".
      { by apply auth_update, option_local_update,
          (exclusive_local_update _ (Excl σ2)). }
99
      iFrame "Hσ". iApply ("H" with "[]"); eauto.
100 101
  Qed.

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
  Lemma ownP_lift_stuck E Φ e :
    (|={E,}=>  σ, ¬ progressive e σ⌝   ownP σ)
     WP e @ E ?{{ Φ }}.
  Proof.
    iIntros "H". destruct (to_val e) as [v|] eqn:EQe.
    - apply of_to_val in EQe as <-; iApply fupd_wp.
      iMod "H" as (σ1) "[#H _]"; iDestruct "H" as %Hstuck; exfalso.
      by apply Hstuck; left; rewrite to_of_val; exists v.
    - iApply wp_lift_stuck; [done|]; iIntros (σ1) "Hσ".
      iMod "H" as (σ1') "(% & >Hσf)"; rewrite /ownP.
      by iDestruct (own_valid_2 with "Hσ Hσf")
        as %[->%Excl_included%leibniz_equiv _]%auth_valid_discrete_2.
  Qed.

  Lemma ownP_lift_pure_step `{Inhabited (state Λ)} p E Φ e1 :
117 118 119
    ( σ1, reducible e1 σ1) 
    ( σ1 e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs  σ1 = σ2) 
    (  e2 efs σ, prim_step e1 σ e2 σ efs 
120 121
      WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
122 123 124 125 126 127 128 129 130 131
  Proof.
    iIntros (Hsafe Hstep) "H". iApply wp_lift_step.
    { eapply reducible_not_val, (Hsafe inhabitant). }
    iIntros (σ1) "Hσ". iMod (fupd_intro_mask' E ) as "Hclose"; first set_solver.
    iModIntro. iSplit; [done|]; iNext; iIntros (e2 σ2 efs ?).
    destruct (Hstep σ1 e2 σ2 efs); auto; subst.
    iMod "Hclose"; iModIntro. iFrame "Hσ". iApply "H"; auto.
  Qed.

  (** Derived lifting lemmas. *)
132
  Lemma ownP_lift_atomic_step {p E Φ} e1 σ1 :
133 134
    reducible e1 σ1 
    ( ownP σ1    e2 σ2 efs, prim_step e1 σ1 e2 σ2 efs - ownP σ2 -
135 136
      default False (to_val e2) Φ  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
137
  Proof.
138
    iIntros (?) "[Hσ H]". iApply ownP_lift_step.
139 140 141 142 143 144 145 146
    iMod (fupd_intro_mask' E ) as "Hclose"; first set_solver. iModIntro.
    iExists σ1. iFrame "Hσ"; iSplit; eauto.
    iNext; iIntros (e2 σ2 efs) "% Hσ".
    iDestruct ("H" $! e2 σ2 efs with "[] [Hσ]") as "[HΦ $]"; [by eauto..|].
    destruct (to_val e2) eqn:?; last by iExFalso.
    iMod "Hclose". iApply wp_value; auto using to_of_val. done.
  Qed.

147
  Lemma ownP_lift_atomic_det_step {p E Φ e1} σ1 v2 σ2 efs :
148 149 150 151
    reducible e1 σ1 
    ( e2' σ2' efs', prim_step e1 σ1 e2' σ2' efs' 
                     σ2 = σ2'  to_val e2' = Some v2  efs = efs') 
     ownP σ1   (ownP σ2 -
152 153
      Φ v2  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
154
  Proof.
155
    iIntros (? Hdet) "[Hσ1 Hσ2]". iApply ownP_lift_atomic_step; try done.
156 157 158 159
    iFrame. iNext. iIntros (e2' σ2' efs') "% Hσ2'".
    edestruct Hdet as (->&Hval&->). done. rewrite Hval. by iApply "Hσ2".
  Qed.

160 161 162 163 164 165 166 167 168 169 170
  Lemma ownP_lift_atomic_det_step_no_fork {p E e1} σ1 v2 σ2 :
    reducible e1 σ1 
    ( e2' σ2' efs', prim_step e1 σ1 e2' σ2' efs' 
      σ2 = σ2'  to_val e2' = Some v2  [] = efs') 
    {{{  ownP σ1 }}} e1 @ p; E {{{ RET v2; ownP σ2 }}}.
  Proof.
    intros. rewrite -(ownP_lift_atomic_det_step σ1 v2 σ2 []); [|done..].
    rewrite big_sepL_nil right_id. by apply uPred.wand_intro_r.
  Qed.

  Lemma ownP_lift_pure_det_step `{Inhabited (state Λ)} {p E Φ} e1 e2 efs :
171 172
    ( σ1, reducible e1 σ1) 
    ( σ1 e2' σ2 efs', prim_step e1 σ1 e2' σ2 efs'  σ1 = σ2  e2 = e2'  efs = efs')
173 174
     (WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p; {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
175
  Proof.
176
    iIntros (? Hpuredet) "?". iApply ownP_lift_pure_step; try done.
177 178
    by intros; eapply Hpuredet. iNext. by iIntros (e' efs' σ (_&->&->)%Hpuredet).
  Qed.
179 180 181 182 183 184 185 186 187

  Lemma ownP_lift_pure_det_step_no_fork `{Inhabited (state Λ)} {p E Φ} e1 e2 :
    to_val e1 = None 
    ( σ1, reducible e1 σ1) 
    ( σ1 e2' σ2 efs', prim_step e1 σ1 e2' σ2 efs'  σ1 = σ2  e2 = e2'  [] = efs') 
     WP e2 @ p; E {{ Φ }}  WP e1 @ p; E {{ Φ }}.
  Proof.
    intros. rewrite -(wp_lift_pure_det_step e1 e2 []) ?big_sepL_nil ?right_id; eauto.
  Qed.
188 189 190 191 192
End lifting.

Section ectx_lifting.
  Import ectx_language.
  Context {expr val ectx state} {Λ : EctxLanguage expr val ectx state}.
193
  Context `{ownPG (ectx_lang expr) Σ} {Hinh : Inhabited state}.
194 195 196 197
  Implicit Types Φ : val  iProp Σ.
  Implicit Types e : expr.
  Hint Resolve head_prim_reducible head_reducible_prim_step.

198
  Lemma ownP_lift_head_step p E Φ e1 :
199 200
    (|={E,}=>  σ1, head_reducible e1 σ1   ownP σ1 
        e2 σ2 efs, head_step e1 σ1 e2 σ2 efs - ownP σ2
201 202
            ={,E}= WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p; {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
203
  Proof.
204
    iIntros "H". iApply (ownP_lift_step p E); try done.
205
    iMod "H" as (σ1 ?) "[Hσ1 Hwp]". iModIntro. iExists σ1.
206
    iSplit; first by eauto. iFrame. iNext. iIntros (e2 σ2 efs) "% ?".
207
    iApply ("Hwp" with "[]"); eauto.
208 209
  Qed.

210
  Lemma ownP_lift_pure_head_step p E Φ e1 :
211 212 213
    ( σ1, head_reducible e1 σ1) 
    ( σ1 e2 σ2 efs, head_step e1 σ1 e2 σ2 efs  σ1 = σ2) 
    (  e2 efs σ, head_step e1 σ e2 σ efs 
214 215
      WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
216
  Proof using Hinh.
217 218 219 220
    iIntros (??) "H". iApply ownP_lift_pure_step; eauto. iNext.
    iIntros (????). iApply "H". eauto.
  Qed.

221
  Lemma ownP_lift_atomic_head_step {p E Φ} e1 σ1 :
222 223 224
    head_reducible e1 σ1 
     ownP σ1   ( e2 σ2 efs,
    head_step e1 σ1 e2 σ2 efs - ownP σ2 -
225 226
      default False (to_val e2) Φ  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
227 228
  Proof.
    iIntros (?) "[? H]". iApply ownP_lift_atomic_step; eauto. iFrame. iNext.
229
    iIntros (???) "% ?". iApply ("H" with "[]"); eauto.
230 231
  Qed.

232
  Lemma ownP_lift_atomic_det_head_step {p E Φ e1} σ1 v2 σ2 efs :
233 234 235
    head_reducible e1 σ1 
    ( e2' σ2' efs', head_step e1 σ1 e2' σ2' efs' 
      σ2 = σ2'  to_val e2' = Some v2  efs = efs') 
236 237
     ownP σ1   (ownP σ2 - Φ v2  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
238 239
  Proof. eauto using ownP_lift_atomic_det_step. Qed.

240
  Lemma ownP_lift_atomic_det_head_step_no_fork {p E e1} σ1 v2 σ2 :
241 242 243
    head_reducible e1 σ1 
    ( e2' σ2' efs', head_step e1 σ1 e2' σ2' efs' 
      σ2 = σ2'  to_val e2' = Some v2  [] = efs') 
244 245
    {{{  ownP σ1 }}} e1 @ p; E {{{ RET v2; ownP σ2 }}}.
  Proof. eauto using ownP_lift_atomic_det_step_no_fork. Qed.
246

247
  Lemma ownP_lift_pure_det_head_step {p E Φ} e1 e2 efs :
248 249
    ( σ1, head_reducible e1 σ1) 
    ( σ1 e2' σ2 efs', head_step e1 σ1 e2' σ2 efs'  σ1 = σ2  e2 = e2'  efs = efs') 
250 251 252
     (WP e2 @ p; E {{ Φ }}  [ list] ef  efs, WP ef @ p;  {{ _, True }})
     WP e1 @ p; E {{ Φ }}.
  Proof using Hinh. eauto using ownP_lift_pure_det_step. Qed.
253

254
  Lemma ownP_lift_pure_det_head_step_no_fork {p E Φ} e1 e2 :
255 256 257
    to_val e1 = None 
    ( σ1, head_reducible e1 σ1) 
    ( σ1 e2' σ2 efs', head_step e1 σ1 e2' σ2 efs'  σ1 = σ2  e2 = e2'  [] = efs') 
258 259
     WP e2 @ p; E {{ Φ }}  WP e1 @ p; E {{ Φ }}.
  Proof using Hinh. eauto using ownP_lift_pure_det_step_no_fork. Qed.
260
End ectx_lifting.