lifting.v 12.8 KB
Newer Older
1
From iris.algebra Require Import auth gmap.
2
From iris.base_logic Require Export gen_heap.
3 4
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import ectx_lifting total_ectx_lifting.
5
From iris.heap_lang Require Export lang proph_map.
6
From iris.heap_lang Require Import tactics.
7
From iris.proofmode Require Import tactics.
Ralf Jung's avatar
Ralf Jung committed
8
From stdpp Require Import fin_maps.
9
Set Default Proof Using "Type".
Ralf Jung's avatar
Ralf Jung committed
10

11 12
Class heapG Σ := HeapG {
  heapG_invG : invG Σ;
13
  heapG_gen_heapG :> gen_heapG loc val Σ;
14
  heapG_proph_mapG :> proph_mapG proph_id val Σ
15 16 17 18
}.

Instance heapG_irisG `{heapG Σ} : irisG heap_lang Σ := {
  iris_invG := heapG_invG;
19
  state_interp σ κs :=
20
    (gen_heap_ctx σ.(heap)  proph_map_ctx κs σ.(used_proph_id))%I
21 22 23 24
}.

(** Override the notations so that scopes and coercions work out *)
Notation "l ↦{ q } v" := (mapsto (L:=loc) (V:=val) l q v%V)
Robbert Krebbers's avatar
Robbert Krebbers committed
25
  (at level 20, q at level 50, format "l  ↦{ q }  v") : bi_scope.
26
Notation "l ↦ v" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
27
  (mapsto (L:=loc) (V:=val) l 1 v%V) (at level 20) : bi_scope.
28
Notation "l ↦{ q } -" := ( v, l {q} v)%I
Robbert Krebbers's avatar
Robbert Krebbers committed
29 30
  (at level 20, q at level 50, format "l  ↦{ q }  -") : bi_scope.
Notation "l ↦ -" := (l {1} -)%I (at level 20) : bi_scope.
31

32 33 34 35 36 37 38 39 40
(** The tactic [inv_head_step] performs inversion on hypotheses of the shape
[head_step]. The tactic will discharge head-reductions starting from values, and
simplifies hypothesis related to conversions from and to values, and finite map
operations. This tactic is slightly ad-hoc and tuned for proving our lifting
lemmas. *)
Ltac inv_head_step :=
  repeat match goal with
  | _ => progress simplify_map_eq/= (* simplify memory stuff *)
  | H : to_val _ = Some _ |- _ => apply of_to_val in H
41
  | H : head_step ?e _ _ _ _ _ |- _ =>
42 43 44 45 46
     try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
     and can thus better be avoided. *)
     inversion H; subst; clear H
  end.

47 48
Local Hint Extern 0 (head_reducible _ _) => eexists _, _, _, _; simpl.
Local Hint Extern 0 (head_reducible_no_obs _ _) => eexists _, _, _; simpl.
49

50 51 52 53 54
(* [simpl apply] is too stupid, so we need extern hints here. *)
Local Hint Extern 1 (head_step _ _ _ _ _ _) => econstructor.
Local Hint Extern 0 (head_step (CAS _ _ _) _ _ _ _ _) => eapply CasSucS.
Local Hint Extern 0 (head_step (CAS _ _ _) _ _ _ _ _) => eapply CasFailS.
Local Hint Extern 0 (head_step (Alloc _) _ _ _ _ _) => apply alloc_fresh.
55
Local Hint Extern 0 (head_step NewProph _ _ _ _ _) => apply new_proph_id_fresh.
56
Local Hint Resolve to_of_val.
57

Ralf Jung's avatar
fix TWP  
Ralf Jung committed
58
Local Ltac solve_exec_safe := intros; subst; do 3 eexists; econstructor; eauto.
59
Local Ltac solve_exec_puredet := simpl; intros; by inv_head_step.
60
Local Ltac solve_pure_exec :=
61
  subst; intros ?; apply nsteps_once, pure_head_step_pure_step;
62
    constructor; [solve_exec_safe | solve_exec_puredet].
63

64 65 66 67 68
Class AsRecV (v : val) (f x : binder) (erec : expr) :=
  as_recv : v = RecV f x erec.
Instance AsRecV_recv f x e : AsRecV (RecV f x e) f x e := eq_refl.
Instance AsRecV_recv_locked v f x e :
  AsRecV v f x e  AsRecV (locked v) f x e.
69 70
Proof. by unlock. Qed.

71 72 73 74 75 76 77 78 79 80 81 82
Instance pure_recc f x (erec : expr) :
  PureExec True 1 (Rec f x erec) (Val $ RecV f x erec).
Proof. solve_pure_exec. Qed.
Instance pure_pairc (v1 v2 : val) :
  PureExec True 1 (Pair (Val v1) (Val v2)) (Val $ PairV v1 v2).
Proof. solve_pure_exec. Qed.
Instance pure_injlc (v : val) :
  PureExec True 1 (InjL $ Val v) (Val $ InjLV v).
Proof. solve_pure_exec. Qed.
Instance pure_injrc (v : val) :
  PureExec True 1 (InjR $ Val v) (Val $ InjRV v).
Proof. solve_pure_exec. Qed.
83

84 85 86 87 88 89
Instance pure_beta f x (erec : expr) (v1 v2 : val) `{AsRecV v1 f x erec} :
  PureExec True 1 (App (Val v1) (Val v2)) (subst' x v2 (subst' f v1 erec)).
Proof. unfold AsRecV in *. solve_pure_exec. Qed.

Instance pure_unop op v v' :
  PureExec (un_op_eval op v = Some v') 1 (UnOp op (Val v)) (Val v').
90
Proof. solve_pure_exec. Qed.
91

92 93
Instance pure_binop op v1 v2 v' :
  PureExec (bin_op_eval op v1 v2 = Some v') 1 (BinOp op (Val v1) (Val v2)) (Val v').
94
Proof. solve_pure_exec. Qed.
95

96
Instance pure_if_true e1 e2 : PureExec True 1 (If (Val $ LitV $ LitBool true) e1 e2) e1.
97
Proof. solve_pure_exec. Qed.
98

99
Instance pure_if_false e1 e2 : PureExec True 1 (If (Val $ LitV  $ LitBool false) e1 e2) e2.
100
Proof. solve_pure_exec. Qed.
101

102 103
Instance pure_fst v1 v2 :
  PureExec True 1 (Fst (Val $ PairV v1 v2)) (Val v1).
104
Proof. solve_pure_exec. Qed.
105

106 107
Instance pure_snd v1 v2 :
  PureExec True 1 (Snd (Val $ PairV v1 v2)) (Val v2).
108
Proof. solve_pure_exec. Qed.
109

110 111
Instance pure_case_inl v e1 e2 :
  PureExec True 1 (Case (Val $ InjLV v) e1 e2) (App e1 (Val v)).
112
Proof. solve_pure_exec. Qed.
113

114 115
Instance pure_case_inr v e1 e2 :
  PureExec True 1 (Case (Val $ InjRV v) e1 e2) (App e2 (Val v)).
116
Proof. solve_pure_exec. Qed.
117

118 119 120 121 122 123 124
Section lifting.
Context `{heapG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val  iProp Σ.
Implicit Types efs : list expr.
Implicit Types σ : state.

Ralf Jung's avatar
Ralf Jung committed
125
(** Fork: Not using Texan triples to avoid some unnecessary [True] *)
126
Lemma wp_fork s E e Φ :
Ralf Jung's avatar
Ralf Jung committed
127
   WP e @ s;  {{ _, True }} -  Φ (LitV LitUnit) - WP Fork e @ s; E {{ Φ }}.
128
Proof.
Ralf Jung's avatar
Ralf Jung committed
129
  iIntros "He HΦ".
Ralf Jung's avatar
Ralf Jung committed
130
  iApply wp_lift_pure_det_head_step; [by eauto|intros; inv_head_step; by eauto|].
131 132
  iModIntro; iNext; iIntros "!> /= {$He}". by iApply wp_value.
Qed.
133

134
Lemma twp_fork s E e Φ :
Ralf Jung's avatar
Ralf Jung committed
135
  WP e @ s;  [{ _, True }] - Φ (LitV LitUnit) - WP Fork e @ s; E [{ Φ }].
136
Proof.
Ralf Jung's avatar
Ralf Jung committed
137
  iIntros "He HΦ".
138
  iApply twp_lift_pure_det_head_step; [eauto|intros; inv_head_step; eauto|].
139 140 141
  iIntros "!> /= {$He}". by iApply twp_value.
Qed.

142
(** Heap *)
143 144
Lemma wp_alloc s E v :
  {{{ True }}} Alloc (Val v) @ s; E {{{ l, RET LitV (LitLoc l); l  v }}}.
145
Proof.
146 147
  iIntros (Φ) "_ HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
  iIntros (σ1 κ κs) "[Hσ Hκs] !>"; iSplit; first by auto.
148
  iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
149 150 151
  iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
  iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
152 153
Lemma twp_alloc s E v :
  [[{ True }]] Alloc (Val v) @ s; E [[{ l, RET LitV (LitLoc l); l  v }]].
154
Proof.
155
  iIntros (Φ) "_ HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
156
  iIntros (σ1 κs) "[Hσ Hκs] !>"; iSplit; first by eauto.
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
157
  iIntros (κ v2 σ2 efs Hstep); inv_head_step.
158
  iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
159
  iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
160
Qed.
161

162
Lemma wp_load s E l q v :
163
  {{{  l {q} v }}} Load (Val $ LitV $ LitLoc l) @ s; E {{{ RET v; l {q} v }}}.
164 165
Proof.
  iIntros (Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
166 167
  iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
  iSplit; first by eauto. iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
168 169
  iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
170
Lemma twp_load s E l q v :
171
  [[{ l {q} v }]] Load (Val $ LitV $ LitLoc l) @ s; E [[{ RET v; l {q} v }]].
172 173
Proof.
  iIntros (Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
174
  iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
175
  iSplit; first by eauto. iIntros (κ v2 σ2 efs Hstep); inv_head_step.
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
176
  iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
177
Qed.
178

179 180 181
Lemma wp_store s E l v' v :
  {{{  l  v' }}} Store (Val $ LitV (LitLoc l)) (Val v) @ s; E
  {{{ RET LitV LitUnit; l  v }}}.
182
Proof.
183
  iIntros (Φ) ">Hl HΦ".
184
  iApply wp_lift_atomic_head_step_no_fork; auto.
185 186
  iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
  iSplit; first by eauto. iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
187
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
188
  iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
189
Qed.
190 191 192
Lemma twp_store s E l v' v :
  [[{ l  v' }]] Store (Val $ LitV $ LitLoc l) (Val v) @ s; E
  [[{ RET LitV LitUnit; l  v }]].
193
Proof.
194
  iIntros (Φ) "Hl HΦ".
195
  iApply twp_lift_atomic_head_step_no_fork; auto.
196
  iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
197
  iSplit; first by eauto. iIntros (κ v2 σ2 efs Hstep); inv_head_step.
198
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
199
  iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
200
Qed.
201

202 203 204
Lemma wp_cas_fail s E l q v' v1 v2 :
  v'  v1  vals_cas_compare_safe v' v1 
  {{{  l {q} v' }}} CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2) @ s; E
205 206
  {{{ RET LitV (LitBool false); l {q} v' }}}.
Proof.
207
  iIntros (?? Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
208 209
  iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
  iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
210 211
  iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
212 213 214
Lemma twp_cas_fail s E l q v' v1 v2 :
  v'  v1  vals_cas_compare_safe v' v1 
  [[{ l {q} v' }]] CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2) @ s; E
215 216
  [[{ RET LitV (LitBool false); l {q} v' }]].
Proof.
217
  iIntros (?? Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
218
  iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
219 220
  iSplit; first by eauto. iIntros (κ v2' σ2 efs Hstep); inv_head_step.
  iModIntro; iSplit=> //. iSplit; first done. iFrame. by iApply "HΦ".
221
Qed.
222

223 224 225
Lemma wp_cas_suc s E l v1 v2 :
  vals_cas_compare_safe v1 v1 
  {{{  l  v1 }}} CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2) @ s; E
226 227
  {{{ RET LitV (LitBool true); l  v2 }}}.
Proof.
228
  iIntros (? Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
229 230
  iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
  iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
231
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
232
  iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
233
Qed.
234 235 236
Lemma twp_cas_suc s E l v1 v2 :
  vals_cas_compare_safe v1 v1 
  [[{ l  v1 }]] CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2) @ s; E
237 238
  [[{ RET LitV (LitBool true); l  v2 }]].
Proof.
239
  iIntros (? Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
240
  iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
241
  iSplit; first by eauto. iIntros (κ v2' σ2 efs Hstep); inv_head_step.
242
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
243
  iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
244
Qed.
245

246 247
Lemma wp_faa s E l i1 i2 :
  {{{  l  LitV (LitInt i1) }}} FAA (Val $ LitV $ LitLoc l) (Val $ LitV $ LitInt i2) @ s; E
248 249
  {{{ RET LitV (LitInt i1); l  LitV (LitInt (i1 + i2)) }}}.
Proof.
250
  iIntros (Φ) ">Hl HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
251 252
  iIntros (σ1 κ κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
  iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
253
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
254
  iModIntro. iSplit=>//. iFrame. by iApply "HΦ".
255
Qed.
256 257
Lemma twp_faa s E l i1 i2 :
  [[{ l  LitV (LitInt i1) }]] FAA (Val $ LitV $ LitLoc l) (Val $ LitV $ LitInt i2) @ s; E
258 259
  [[{ RET LitV (LitInt i1); l  LitV (LitInt (i1 + i2)) }]].
Proof.
260
  iIntros (Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
261
  iIntros (σ1 κs) "[Hσ Hκs] !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
262
  iSplit; first by eauto. iIntros (κ e2 σ2 efs Hstep); inv_head_step.
263
  iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
Ralf Jung's avatar
fix TWP  
Ralf Jung committed
264
  iModIntro. iSplit=>//. iSplit; first done. iFrame. by iApply "HΦ".
265
Qed.
266 267 268

(** Lifting lemmas for creating and resolving prophecy variables *)
Lemma wp_new_proph :
269
  {{{ True }}} NewProph {{{ v (p : proph_id), RET (LitV (LitProphecy p)); proph p v }}}.
270 271
Proof.
  iIntros (Φ) "_ HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
272 273 274 275
  iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
  iSplit; first by eauto.
  iNext; iIntros (v2 σ2 efs Hstep). inv_head_step.
  iMod (@proph_map_alloc with "HR") as "[HR Hp]".
276 277 278 279 280 281
  { intro Hin. apply (iffLR (elem_of_subseteq _ _) Hdom) in Hin. done. }
  iModIntro; iSplit=> //. iFrame. iSplitL "HR".
  - iExists _. iSplit; last done.
    iPureIntro. split.
    + apply first_resolve_insert; auto.
    + rewrite dom_insert_L. by apply union_mono_l.
Ralf Jung's avatar
Ralf Jung committed
282
  - iApply "HΦ". done.
283 284 285 286 287
Qed.

Lemma wp_resolve_proph e1 e2 p v w:
  IntoVal e1 (LitV (LitProphecy p)) 
  IntoVal e2 w 
288
  {{{ proph p v }}} ResolveProph e1 e2 {{{ RET (LitV LitUnit); v = Some w }}}.
289 290
Proof.
  iIntros (<- <- Φ) "Hp HΦ". iApply wp_lift_atomic_head_step_no_fork; auto.
291 292 293 294
  iIntros (σ1 κ κs) "[Hσ HR] !>". iDestruct "HR" as (R [Hfr Hdom]) "HR".
  iDestruct (@proph_map_valid with "HR Hp") as %Hlookup.
  iSplit; first by eauto.
  iNext; iIntros (v2 σ2 efs Hstep); inv_head_step. iApply fupd_frame_l.
295
  iSplit=> //. iFrame.
Ralf Jung's avatar
Ralf Jung committed
296
  iMod (@proph_map_remove with "HR Hp") as "Hp". iModIntro.
297 298 299 300 301 302
  iSplitR "HΦ".
  - iExists _. iFrame. iPureIntro. split; first by eapply first_resolve_delete.
    rewrite dom_delete. rewrite <- difference_empty_L. by eapply difference_mono.
  - iApply "HΦ". iPureIntro. by eapply first_resolve_eq.
Qed.

Ralf Jung's avatar
Ralf Jung committed
303
End lifting.