proofmode.v 9.41 KB
Newer Older
1
From iris.program_logic Require Export weakestpre.
Robbert Krebbers's avatar
Robbert Krebbers committed
2
From iris.proofmode Require Import coq_tactics.
3
From iris.proofmode Require Export tactics.
4
From iris.heap_lang Require Export tactics lifting.
5
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
6
7
Import uPred.

8
9
10
11
12
13
(** wp-specific helper tactics *)
Ltac wp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
  | _ => etrans; [|fast_by apply (wp_bind K)]; simpl
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
14

15
16
17
18
19
20
21
22
23
24
(* Solves side-conditions generated by the wp tactics *)
Ltac wp_done :=
  match goal with
  | |- Closed _ _ => solve_closed
  | |- is_Some (to_val _) => solve_to_val
  | |- to_val _ = Some _ => solve_to_val
  | |- language.to_val _ = Some _ => solve_to_val
  | _ => fast_done
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
25
Ltac wp_value_head := etrans; [|eapply wp_value; wp_done]; simpl.
26
27
28
29
30
31
32
33

Lemma of_val_unlock v e : of_val v = e  of_val (locked v) = e.
Proof. by unlock. Qed.

(* Applied to goals that are equalities of expressions. Will try to unlock the
   LHS once if necessary, to get rid of the lock added by the syntactic sugar. *)
Ltac solve_of_val_unlock := try apply of_val_unlock; reflexivity.

34
35
36
(* Solves side-conditions generated specifically by wp_pure *)
Ltac wp_pure_done :=
  split_and?;
37
  lazymatch goal with
38
39
  | |- of_val _ = _ => solve_of_val_unlock
  | _ => wp_done
40
41
  end.

42
43
44
Lemma tac_wp_pure `{heapG Σ} K Δ Δ' E e1 e2 φ Φ :
  PureExec φ e1 e2 
  φ 
45
  IntoLaterNEnvs 1 Δ Δ' 
46
47
48
  (Δ'  WP fill K e2 @ E {{ Φ }}) 
  (Δ  WP fill K e1 @ E {{ Φ }}).
Proof.
49
  intros ??? HΔ'. rewrite into_laterN_env_sound /=.
Dan Frumin's avatar
Dan Frumin committed
50
  rewrite -lifting.wp_bind HΔ' -wp_pure_step_later //.
51
  by rewrite -ectx_lifting.wp_ectx_bind_inv.
52
Qed.
53

54
Tactic Notation "wp_pure" open_constr(efoc) :=
55
56
57
  iStartProof;
  lazymatch goal with
  | |- _  wp ?E ?e ?Q => reshape_expr e ltac:(fun K e' =>
58
    unify e' efoc;
59
    eapply (tac_wp_pure K);
60
    [unlock; simpl; apply _   (* PureExec *)
61
    |wp_pure_done             (* The pure condition for PureExec *)
62
    |apply _                  (* IntoLaters *)
63
    |simpl_subst; try wp_value_head (* new goal *)])
64
65
   || fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a reduct"
  | _ => fail "wp_pure: not a 'wp'"
66
67
  end.

68
69
70
71
72
73
74
75
76
77
78
79
Tactic Notation "wp_if" := wp_pure (If _ _ _).
Tactic Notation "wp_if_true" := wp_pure (If (Lit (LitBool true)) _ _).
Tactic Notation "wp_if_false" := wp_pure (If (Lit (LitBool false)) _ _).
Tactic Notation "wp_unop" := wp_pure (UnOp _ _).
Tactic Notation "wp_binop" := wp_pure (BinOp _ _ _).
Tactic Notation "wp_op" := wp_unop || wp_binop.
Tactic Notation "wp_rec" := wp_pure (App _ _).
Tactic Notation "wp_lam" := wp_rec.
Tactic Notation "wp_let" := wp_lam.
Tactic Notation "wp_seq" := wp_lam.
Tactic Notation "wp_proj" := wp_pure (Fst _) || wp_pure (Snd _).
Tactic Notation "wp_case" := wp_pure (Case _ _ _).
80
Tactic Notation "wp_match" := wp_case; wp_let.
81
82
83
84
85
86
87
88
89
90
91
92

Tactic Notation "wp_bind" open_constr(efoc) :=
  iStartProof;
  lazymatch goal with
  | |- _  wp ?E ?e ?Q => reshape_expr e ltac:(fun K e' =>
    match e' with
    | efoc => unify e' efoc; wp_bind_core K
    end) || fail "wp_bind: cannot find" efoc "in" e
  | _ => fail "wp_bind: not a 'wp'"
  end.

(** Heap tactics *)
Robbert Krebbers's avatar
Robbert Krebbers committed
93
Section heap.
94
Context `{heapG Σ}.
95
96
97
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val  iProp Σ.
Implicit Types Δ : envs (iResUR Σ).
Robbert Krebbers's avatar
Robbert Krebbers committed
98

99
Lemma tac_wp_alloc Δ Δ' E j K e v Φ :
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
100
  to_val e = Some v 
101
  IntoLaterNEnvs 1 Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
102
  ( l,  Δ'',
103
    envs_app false (Esnoc Enil j (l  v)) Δ' = Some Δ'' 
104
105
    (Δ''  WP fill K (Lit (LitLoc l)) @ E {{ Φ }})) 
  Δ  WP fill K (Alloc e) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
106
Proof.
107
  intros ?? HΔ. rewrite -wp_bind. eapply wand_apply; first exact: wp_alloc.
108
  rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
Robbert Krebbers's avatar
Robbert Krebbers committed
109
110
111
112
  destruct (HΔ l) as (Δ''&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.

113
Lemma tac_wp_load Δ Δ' E i K l q v Φ :
114
  IntoLaterNEnvs 1 Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
115
  envs_lookup i Δ' = Some (false, l {q} v)%I 
116
117
  (Δ'  WP fill K (of_val v) @ E {{ Φ }}) 
  Δ  WP fill K (Load (Lit (LitLoc l))) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
118
Proof.
119
  intros. rewrite -wp_bind. eapply wand_apply; first exact: wp_load.
120
  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
121
122
123
  by apply later_mono, sep_mono_r, wand_mono.
Qed.

124
Lemma tac_wp_store Δ Δ' Δ'' E i K l v e v' Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
125
  to_val e = Some v' 
126
  IntoLaterNEnvs 1 Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
127
128
  envs_lookup i Δ' = Some (false, l  v)%I 
  envs_simple_replace i false (Esnoc Enil i (l  v')) Δ' = Some Δ'' 
129
130
  (Δ''  WP fill K (Lit LitUnit) @ E {{ Φ }}) 
  Δ  WP fill K (Store (Lit (LitLoc l)) e) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
131
Proof.
132
  intros. rewrite -wp_bind. eapply wand_apply; first by eapply wp_store.
133
  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
134
135
136
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.

137
Lemma tac_wp_cas_fail Δ Δ' E i K l q v e1 v1 e2 v2 Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
138
  to_val e1 = Some v1  to_val e2 = Some v2 
139
  IntoLaterNEnvs 1 Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
140
  envs_lookup i Δ' = Some (false, l {q} v)%I  v  v1 
141
142
  (Δ'  WP fill K (Lit (LitBool false)) @ E {{ Φ }}) 
  Δ  WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
143
Proof.
144
  intros. rewrite -wp_bind. eapply wand_apply; first exact: wp_cas_fail.
145
  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
146
147
148
  by apply later_mono, sep_mono_r, wand_mono.
Qed.

149
Lemma tac_wp_cas_suc Δ Δ' Δ'' E i K l v e1 v1 e2 v2 Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
150
  to_val e1 = Some v1  to_val e2 = Some v2 
151
  IntoLaterNEnvs 1 Δ Δ' 
152
  envs_lookup i Δ' = Some (false, l  v)%I  v = v1 
Robbert Krebbers's avatar
Robbert Krebbers committed
153
  envs_simple_replace i false (Esnoc Enil i (l  v2)) Δ' = Some Δ'' 
154
155
  (Δ''  WP fill K (Lit (LitBool true)) @ E {{ Φ }}) 
  Δ  WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
156
Proof.
157
  intros; subst. rewrite -wp_bind. eapply wand_apply; first exact: wp_cas_suc.
158
  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
159
160
161
162
163
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
End heap.

Tactic Notation "wp_apply" open_constr(lem) :=
164
165
166
167
168
169
170
171
172
173
  iPoseProofCore lem as false true (fun H =>
    lazymatch goal with
    | |- _  wp ?E ?e ?Q =>
      reshape_expr e ltac:(fun K e' =>
        wp_bind_core K; iApplyHyp H; try iNext; simpl) ||
      lazymatch iTypeOf H with
      | Some (_,?P) => fail "wp_apply: cannot apply" P
      end
    | _ => fail "wp_apply: not a 'wp'"
    end).
Robbert Krebbers's avatar
Robbert Krebbers committed
174
175

Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
176
  iStartProof;
177
  lazymatch goal with
178
179
  | |- _  wp ?E ?e ?Q =>
    first
180
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_alloc _ _ _ H K))
181
      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
182
183
184
185
186
187
188
    [let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_alloc:" e' "not a value"
    |apply _
    |first [intros l | fail 1 "wp_alloc:" l "not fresh"];
      eexists; split;
        [env_cbv; reflexivity || fail "wp_alloc:" H "not fresh"
        |simpl; try wp_value_head]]
189
  | _ => fail "wp_alloc: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
190
  end.
191

192
193
Tactic Notation "wp_alloc" ident(l) :=
  let H := iFresh in wp_alloc l as H.
Robbert Krebbers's avatar
Robbert Krebbers committed
194
195

Tactic Notation "wp_load" :=
196
  iStartProof;
197
  lazymatch goal with
198
199
  | |- _  wp ?E ?e ?Q =>
    first
200
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_load _ _ _ _ K))
201
      |fail 1 "wp_load: cannot find 'Load' in" e];
202
203
204
205
    [apply _
    |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
     iAssumptionCore || fail "wp_load: cannot find" l "↦ ?"
    |simpl; try wp_value_head]
206
  | _ => fail "wp_load: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
207
208
209
  end.

Tactic Notation "wp_store" :=
210
  iStartProof;
211
  lazymatch goal with
212
213
  | |- _  wp ?E ?e ?Q =>
    first
214
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_store _ _ _ _ _ K))
215
      |fail 1 "wp_store: cannot find 'Store' in" e];
216
217
218
219
220
221
222
    [let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_store:" e' "not a value"
    |apply _
    |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
     iAssumptionCore || fail "wp_store: cannot find" l "↦ ?"
    |env_cbv; reflexivity
    |simpl; try first [wp_pure (Seq (Lit LitUnit) _)|wp_value_head]]
223
  | _ => fail "wp_store: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
224
225
226
  end.

Tactic Notation "wp_cas_fail" :=
227
  iStartProof;
228
  lazymatch goal with
229
230
  | |- _  wp ?E ?e ?Q =>
    first
231
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_cas_fail _ _ _ _ K))
232
      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
233
234
235
236
237
238
239
240
241
    [let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_cas_fail:" e' "not a value"
    |let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_cas_fail:" e' "not a value"
    |apply _
    |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
     iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?"
    |try congruence
    |simpl; try wp_value_head]
242
  | _ => fail "wp_cas_fail: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
243
244
245
  end.

Tactic Notation "wp_cas_suc" :=
246
  iStartProof;
247
  lazymatch goal with
248
249
  | |- _  wp ?E ?e ?Q =>
    first
250
      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_cas_suc _ _ _ _ _ K))
251
      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
252
253
254
255
256
257
258
259
260
261
    [let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_cas_suc:" e' "not a value"
    |let e' := match goal with |- to_val ?e' = _ => e' end in
     wp_done || fail "wp_cas_suc:" e' "not a value"
    |apply _
    |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
     iAssumptionCore || fail "wp_cas_suc: cannot find" l "↦ ?"
    |try congruence
    |env_cbv; reflexivity
    |simpl; try wp_value_head]
262
  | _ => fail "wp_cas_suc: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
263
  end.