lifting.v 5.95 KB
Newer Older
1 2
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import ownership ectx_lifting. (* for ownP *)
3
From iris.heap_lang Require Export lang.
4
From iris.heap_lang Require Import tactics.
5
From iris.proofmode Require Import weakestpre.
6
Import uPred.
Robbert Krebbers's avatar
Robbert Krebbers committed
7
Local Hint Extern 0 (head_reducible _ _) => do_head_step eauto 2.
8

9
Section lifting.
10
Context {Σ : iFunctor}.
11 12
Implicit Types P Q : iProp heap_lang Σ.
Implicit Types Φ : val  iProp heap_lang Σ.
13
Implicit Types ef : option (expr []).
14

15
(** Bind. This bundles some arguments that wp_ectx_bind leaves as indices. *)
16
Lemma wp_bind {E e} K Φ :
17
  WP e @ E {{ v, WP fill K (of_val v) @ E {{ Φ }} }}  WP fill K e @ E {{ Φ }}.
18
Proof. exact: wp_ectx_bind. Qed.
19

20
Lemma wp_bindi {E e} Ki Φ :
21
  WP e @ E {{ v, WP fill_item Ki (of_val v) @ E {{ Φ }} }} 
22 23 24
     WP fill_item Ki e @ E {{ Φ }}.
Proof. exact: weakestpre.wp_bind. Qed.

25
(** Base axioms for core primitives of the language: Stateful reductions. *)
26
Lemma wp_alloc_pst E σ e v Φ :
27
  to_val e = Some v 
28
  ( ownP σ   ( l, σ !! l = None  ownP (<[l:=v]>σ) - Φ (LitV (LitLoc l))))
29
   WP Alloc e @ E {{ Φ }}.
30
Proof.
Ralf Jung's avatar
Ralf Jung committed
31
  iIntros {?}  "[HP HΦ]".
Ralf Jung's avatar
Ralf Jung committed
32
  (* TODO: This works around ssreflect bug #22. *)
Ralf Jung's avatar
Ralf Jung committed
33
  set (φ (e' : expr []) σ' ef :=  l,
34
    ef = None  e' = Lit (LitLoc l)  σ' = <[l:=v]>σ  σ !! l = None).
Ralf Jung's avatar
Ralf Jung committed
35 36 37
  iApply (wp_lift_atomic_head_step (Alloc e) φ σ); try (by simpl; eauto);
    [by intros; subst φ; inv_head_step; eauto 8|].
  iFrame "HP". iNext. iIntros {v2 σ2 ef} "[% HP]".
38
  (* FIXME: I should not have to refer to "H0". *)
Robbert Krebbers's avatar
Robbert Krebbers committed
39 40
  destruct H0 as (l & -> & [= <-]%of_to_val_flip & -> & ?); simpl.
  iSplit; last done. iApply "HΦ"; by iSplit.
41
Qed.
42

43
Lemma wp_load_pst E σ l v Φ :
44
  σ !! l = Some v 
45
  ( ownP σ   (ownP σ - Φ v))  WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
46
Proof.
47
  intros. rewrite -(wp_lift_atomic_det_head_step σ v σ None) ?right_id //;
Robbert Krebbers's avatar
Robbert Krebbers committed
48
    last (by intros; inv_head_step; eauto using to_of_val); simpl; by eauto.
49
Qed.
50

51
Lemma wp_store_pst E σ l e v v' Φ :
52
  to_val e = Some v  σ !! l = Some v' 
53
  ( ownP σ   (ownP (<[l:=v]>σ) - Φ (LitV LitUnit)))
54
   WP Store (Lit (LitLoc l)) e @ E {{ Φ }}.
55
Proof.
56
  intros. rewrite-(wp_lift_atomic_det_head_step σ (LitV LitUnit) (<[l:=v]>σ) None)
Robbert Krebbers's avatar
Robbert Krebbers committed
57
    ?right_id //; last (by intros; inv_head_step; eauto); simpl; by eauto.
58
Qed.
59

60
Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Φ :
61
  to_val e1 = Some v1  to_val e2 = Some v2  σ !! l = Some v'  v'  v1 
62
  ( ownP σ   (ownP σ - Φ (LitV $ LitBool false)))
63
   WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
64
Proof.
65
  intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool false) σ None)
Robbert Krebbers's avatar
Robbert Krebbers committed
66
    ?right_id //; last (by intros; inv_head_step; eauto);
67
    simpl; by eauto 10.
Ralf Jung's avatar
Ralf Jung committed
68
Qed.
69

70
Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Φ :
71
  to_val e1 = Some v1  to_val e2 = Some v2  σ !! l = Some v1 
72
  ( ownP σ   (ownP (<[l:=v2]>σ) - Φ (LitV $ LitBool true)))
73
   WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
74
Proof.
75
  intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool true)
Robbert Krebbers's avatar
Robbert Krebbers committed
76
    (<[l:=v2]>σ) None) ?right_id //; last (by intros; inv_head_step; eauto);
77
    simpl; by eauto 10.
Ralf Jung's avatar
Ralf Jung committed
78 79
Qed.

80
(** Base axioms for core primitives of the language: Stateless reductions *)
81
Lemma wp_fork E e Φ :
82
  ( Φ (LitV LitUnit)   WP e {{ _, True }})  WP Fork e @ E {{ Φ }}.
83
Proof.
84
  rewrite -(wp_lift_pure_det_head_step (Fork e) (Lit LitUnit) (Some e)) //=;
Robbert Krebbers's avatar
Robbert Krebbers committed
85
    last by intros; inv_head_step; eauto.
86
  rewrite later_sep -(wp_value _ _ (Lit _)) //.
87
Qed.
88

89
Lemma wp_rec E f x erec e1 e2 v2 Φ :
90
  e1 = Rec f x erec 
91
  to_val e2 = Some v2 
92 93
   WP subst' x e2 (subst' f e1 erec) @ E {{ Φ }}
   WP App e1 e2 @ E {{ Φ }}.
94 95 96 97 98
Proof.
  intros -> ?. rewrite -(wp_lift_pure_det_head_step (App _ _)
    (subst' x e2 (subst' f (Rec f x erec) erec)) None) //= ?right_id;
    intros; inv_head_step; eauto.
Qed.
99

100
Lemma wp_un_op E op l l' Φ :
101
  un_op_eval op l = Some l' 
102
   Φ (LitV l')  WP UnOp op (Lit l) @ E {{ Φ }}.
103
Proof.
104
  intros. rewrite -(wp_lift_pure_det_head_step (UnOp op _) (Lit l') None)
Robbert Krebbers's avatar
Robbert Krebbers committed
105
    ?right_id -?wp_value //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
106
Qed.
107

108
Lemma wp_bin_op E op l1 l2 l' Φ :
109
  bin_op_eval op l1 l2 = Some l' 
110
   Φ (LitV l')  WP BinOp op (Lit l1) (Lit l2) @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
111
Proof.
112
  intros Heval. rewrite -(wp_lift_pure_det_head_step (BinOp op _ _) (Lit l') None)
Robbert Krebbers's avatar
Robbert Krebbers committed
113
    ?right_id -?wp_value //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
114
Qed.
115

116
Lemma wp_if_true E e1 e2 Φ :
117
   WP e1 @ E {{ Φ }}  WP If (Lit (LitBool true)) e1 e2 @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
118
Proof.
119
  rewrite -(wp_lift_pure_det_head_step (If _ _ _) e1 None)
Robbert Krebbers's avatar
Robbert Krebbers committed
120
    ?right_id //; intros; inv_head_step; eauto.
121 122
Qed.

123
Lemma wp_if_false E e1 e2 Φ :
124
   WP e2 @ E {{ Φ }}  WP If (Lit (LitBool false)) e1 e2 @ E {{ Φ }}.
125
Proof.
126
  rewrite -(wp_lift_pure_det_head_step (If _ _ _) e2 None)
Robbert Krebbers's avatar
Robbert Krebbers committed
127
    ?right_id //; intros; inv_head_step; eauto.
128
Qed.
129

130
Lemma wp_fst E e1 v1 e2 v2 Φ :
131
  to_val e1 = Some v1  to_val e2 = Some v2 
132
   Φ v1  WP Fst (Pair e1 e2) @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
133
Proof.
134
  intros. rewrite -(wp_lift_pure_det_head_step (Fst _) e1 None)
Robbert Krebbers's avatar
Robbert Krebbers committed
135
    ?right_id -?wp_value //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
136
Qed.
137

138
Lemma wp_snd E e1 v1 e2 v2 Φ :
139
  to_val e1 = Some v1  to_val e2 = Some v2 
140
   Φ v2  WP Snd (Pair e1 e2) @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
141
Proof.
142
  intros. rewrite -(wp_lift_pure_det_head_step (Snd _) e2 None)
Robbert Krebbers's avatar
Robbert Krebbers committed
143
    ?right_id -?wp_value //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
144
Qed.
145

146
Lemma wp_case_inl E e0 v0 e1 e2 Φ :
147
  to_val e0 = Some v0 
148
   WP App e1 e0 @ E {{ Φ }}  WP Case (InjL e0) e1 e2 @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
149
Proof.
150
  intros. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
Robbert Krebbers's avatar
Robbert Krebbers committed
151
    (App e1 e0) None) ?right_id //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
152
Qed.
153

154
Lemma wp_case_inr E e0 v0 e1 e2 Φ :
155
  to_val e0 = Some v0 
156
   WP App e2 e0 @ E {{ Φ }}  WP Case (InjR e0) e1 e2 @ E {{ Φ }}.
Ralf Jung's avatar
Ralf Jung committed
157
Proof.
158
  intros. rewrite -(wp_lift_pure_det_head_step (Case _ _ _)
Robbert Krebbers's avatar
Robbert Krebbers committed
159
    (App e2 e0) None) ?right_id //; intros; inv_head_step; eauto.
Ralf Jung's avatar
Ralf Jung committed
160
Qed.
161
End lifting.