proofmode.v 7.59 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.
Robbert Krebbers's avatar
Robbert Krebbers committed
4 5 6
From iris.heap_lang Require Export wp_tactics heap.
Import uPred.

7
Ltac wp_strip_later ::= iNext.
Robbert Krebbers's avatar
Robbert Krebbers committed
8 9

Section heap.
10
Context `{heapG Σ}.
11 12 13
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
14

15 16 17
Global Instance into_and_mapsto l q v :
  IntoAnd false (l {q} v) (l {q/2} v) (l {q/2} v).
Proof. by rewrite /IntoAnd heap_mapsto_op_eq Qp_div_2. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
18

19
Lemma tac_wp_alloc Δ Δ' E j e v Φ :
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
20
  to_val e = Some v 
21
  (Δ  heap_ctx)  nclose heapN  E 
22
  IntoLaterEnvs Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
23
  ( l,  Δ'',
24
    envs_app false (Esnoc Enil j (l  v)) Δ' = Some Δ'' 
25
    (Δ''  |={E}=> Φ (LitV (LitLoc l)))) 
Robbert Krebbers's avatar
Robbert Krebbers committed
26 27
  Δ  WP Alloc e @ E {{ Φ }}.
Proof.
Ralf Jung's avatar
Ralf Jung committed
28
  intros ???? HΔ. rewrite -wp_fupd -wp_alloc // -always_and_sep_l.
29
  apply and_intro; first done.
30
  rewrite into_later_env_sound; apply later_mono, forall_intro=> l.
Robbert Krebbers's avatar
Robbert Krebbers committed
31 32 33 34
  destruct (HΔ l) as (Δ''&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.

35 36
Lemma tac_wp_load Δ Δ' E i l q v Φ :
  (Δ  heap_ctx)  nclose heapN  E 
37
  IntoLaterEnvs Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
38
  envs_lookup i Δ' = Some (false, l {q} v)%I 
39
  (Δ'  |={E}=> Φ v) 
40
  Δ  WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
41
Proof.
Ralf Jung's avatar
Ralf Jung committed
42 43
  intros. rewrite -wp_fupd -wp_load // -assoc -always_and_sep_l.
  apply and_intro; first done.
44
  rewrite into_later_env_sound -later_sep envs_lookup_split //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
45 46 47
  by apply later_mono, sep_mono_r, wand_mono.
Qed.

48
Lemma tac_wp_store Δ Δ' Δ'' E i l v e v' Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
49
  to_val e = Some v' 
50
  (Δ  heap_ctx)  nclose heapN  E 
51
  IntoLaterEnvs Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
52 53
  envs_lookup i Δ' = Some (false, l  v)%I 
  envs_simple_replace i false (Esnoc Enil i (l  v')) Δ' = Some Δ'' 
54 55
  (Δ''  |={E}=> Φ (LitV LitUnit)) 
  Δ  WP Store (Lit (LitLoc l)) e @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
56
Proof.
Ralf Jung's avatar
Ralf Jung committed
57 58
  intros. rewrite -wp_fupd -wp_store // -assoc -always_and_sep_l.
  apply and_intro; first done.
59
  rewrite into_later_env_sound -later_sep envs_simple_replace_sound //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
60 61 62
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.

63
Lemma tac_wp_cas_fail Δ Δ' E i l q v e1 v1 e2 v2 Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
64
  to_val e1 = Some v1  to_val e2 = Some v2 
65
  (Δ  heap_ctx)  nclose heapN  E 
66
  IntoLaterEnvs Δ Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
67
  envs_lookup i Δ' = Some (false, l {q} v)%I  v  v1 
68
  (Δ'  |={E}=> Φ (LitV (LitBool false))) 
69
  Δ  WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
70
Proof.
Ralf Jung's avatar
Ralf Jung committed
71 72
  intros. rewrite -wp_fupd -wp_cas_fail // -assoc -always_and_sep_l.
  apply and_intro; first done.
73
  rewrite into_later_env_sound -later_sep envs_lookup_split //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
74 75 76
  by apply later_mono, sep_mono_r, wand_mono.
Qed.

77
Lemma tac_wp_cas_suc Δ Δ' Δ'' E i l v e1 v1 e2 v2 Φ :
Robbert Krebbers's avatar
Robbert Krebbers committed
78
  to_val e1 = Some v1  to_val e2 = Some v2 
79
  (Δ  heap_ctx)  nclose heapN  E 
80
  IntoLaterEnvs Δ Δ' 
81
  envs_lookup i Δ' = Some (false, l  v)%I  v = v1 
Robbert Krebbers's avatar
Robbert Krebbers committed
82
  envs_simple_replace i false (Esnoc Enil i (l  v2)) Δ' = Some Δ'' 
83 84
  (Δ''  |={E}=> Φ (LitV (LitBool true))) 
  Δ  WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
Robbert Krebbers's avatar
Robbert Krebbers committed
85
Proof.
Ralf Jung's avatar
Ralf Jung committed
86 87
  intros; subst. rewrite -wp_fupd -wp_cas_suc // -assoc -always_and_sep_l.
  apply and_intro; first done.
88
  rewrite into_later_env_sound -later_sep envs_simple_replace_sound //; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
89 90 91 92 93
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
End heap.

Tactic Notation "wp_apply" open_constr(lem) :=
94
  lazymatch goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
95
  | |- _  wp ?E ?e ?Q => reshape_expr e ltac:(fun K e' =>
96
    wp_bind_core K; iApply lem; try iNext)
97
  | _ => fail "wp_apply: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
98 99 100
  end.

Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
101
  lazymatch goal with
102 103 104
  | |- _  wp ?E ?e ?Q =>
    first
      [reshape_expr e ltac:(fun K e' =>
105
         match eval hnf in e' with Alloc _ => wp_bind_core K end)
106
      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
107
    eapply tac_wp_alloc with _ H _;
108 109 110
      [let e' := match goal with |- to_val ?e' = _ => e' end in
       wp_done || fail "wp_alloc:" e' "not a value"
      |iAssumption || fail "wp_alloc: cannot find heap_ctx"
111
      |solve_ndisj || fail "wp_alloc: cannot open heap invariant"
112 113 114 115 116
      |apply _
      |first [intros l | fail 1 "wp_alloc:" l "not fresh"];
        eexists; split;
          [env_cbv; reflexivity || fail "wp_alloc:" H "not fresh"
          |wp_finish]]
117
  | _ => fail "wp_alloc: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
118
  end.
119

120 121
Tactic Notation "wp_alloc" ident(l) :=
  let H := iFresh in wp_alloc l as H.
Robbert Krebbers's avatar
Robbert Krebbers committed
122 123

Tactic Notation "wp_load" :=
124
  lazymatch goal with
125 126 127
  | |- _  wp ?E ?e ?Q =>
    first
      [reshape_expr e ltac:(fun K e' =>
128
         match eval hnf in e' with Load _ => wp_bind_core K end)
129 130 131
      |fail 1 "wp_load: cannot find 'Load' in" e];
    eapply tac_wp_load;
      [iAssumption || fail "wp_load: cannot find heap_ctx"
132
      |solve_ndisj || fail "wp_load: cannot open heap invariant"
133 134
      |apply _
      |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
135
       iAssumptionCore || fail "wp_load: cannot find" l "↦ ?"
136
      |wp_finish]
137
  | _ => fail "wp_load: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
138 139 140
  end.

Tactic Notation "wp_store" :=
141
  lazymatch goal with
142 143 144
  | |- _  wp ?E ?e ?Q =>
    first
      [reshape_expr e ltac:(fun K e' =>
145
         match eval hnf in e' with Store _ _ => wp_bind_core K end)
146 147 148 149 150
      |fail 1 "wp_store: cannot find 'Store' in" e];
    eapply tac_wp_store;
      [let e' := match goal with |- to_val ?e' = _ => e' end in
       wp_done || fail "wp_store:" e' "not a value"
      |iAssumption || fail "wp_store: cannot find heap_ctx"
151
      |solve_ndisj || fail "wp_store: cannot open heap invariant"
152 153 154 155
      |apply _
      |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
       iAssumptionCore || fail "wp_store: cannot find" l "↦ ?"
      |env_cbv; reflexivity
156
      |wp_finish]
157
  | _ => fail "wp_store: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
158 159 160
  end.

Tactic Notation "wp_cas_fail" :=
161
  lazymatch goal with
162 163 164
  | |- _  wp ?E ?e ?Q =>
    first
      [reshape_expr e ltac:(fun K e' =>
165
         match eval hnf in e' with CAS _ _ _ => wp_bind_core K end)
166 167 168 169 170 171 172
      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
    eapply tac_wp_cas_fail;
      [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"
      |iAssumption || fail "wp_cas_fail: cannot find heap_ctx"
173
      |solve_ndisj || fail "wp_cas_fail: cannot open heap invariant"
174 175 176 177 178
      |apply _
      |let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
       iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?"
      |try congruence
      |wp_finish]
179
  | _ => fail "wp_cas_fail: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
180 181 182
  end.

Tactic Notation "wp_cas_suc" :=
183
  lazymatch goal with
184 185 186
  | |- _  wp ?E ?e ?Q =>
    first
      [reshape_expr e ltac:(fun K e' =>
187
         match eval hnf in e' with CAS _ _ _ => wp_bind_core K end)
188 189 190 191 192 193 194
      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
    eapply tac_wp_cas_suc;
      [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"
      |iAssumption || fail "wp_cas_suc: cannot find heap_ctx"
195
      |solve_ndisj || fail "wp_cas_suc: cannot open heap invariant"
196 197 198 199 200 201
      |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
      |wp_finish]
202
  | _ => fail "wp_cas_suc: not a 'wp'"
Robbert Krebbers's avatar
Robbert Krebbers committed
203
  end.