tactics.v 83.2 KB
Newer Older
1
From iris.proofmode Require Import coq_tactics.
2
From iris.proofmode Require Import base intro_patterns spec_patterns sel_patterns.
3
From iris.bi Require Export bi big_op.
Robbert Krebbers's avatar
Robbert Krebbers committed
4
From iris.proofmode Require Export classes notation.
5
From iris.proofmode Require Import class_instances.
6
From stdpp Require Import hlist pretty.
7
Set Default Proof Using "Type".
8
Export ident.
Robbert Krebbers's avatar
Robbert Krebbers committed
9
10

Declare Reduction env_cbv := cbv [
11
  option_bind
12
  beq ascii_beq string_beq positive_beq ident_beq
13
14
  env_lookup env_lookup_delete env_delete env_app env_replace env_dom
  env_persistent env_spatial env_spatial_is_nil envs_dom
15
  envs_lookup envs_lookup_delete envs_delete envs_snoc envs_app
16
17
    envs_simple_replace envs_replace envs_split
    envs_clear_spatial envs_clear_persistent
18
    envs_split_go envs_split].
Robbert Krebbers's avatar
Robbert Krebbers committed
19
20
Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.
21
Ltac env_reflexivity := env_cbv; exact eq_refl.
Robbert Krebbers's avatar
Robbert Krebbers committed
22

23
(** * Misc *)
24
(* Tactic Notation tactics cannot return terms *)
25
Ltac iFresh :=
Robbert Krebbers's avatar
Robbert Krebbers committed
26
  lazymatch goal with
27
  |- envs_entails ?Δ _ =>
28
29
30
     (* [vm_compute fails] if any of the hypotheses in [Δ] contain evars, so
     first use [cbv] to compute the domain of [Δ] *)
     let Hs := eval cbv in (envs_dom Δ) in
31
32
33
34
35
36
     eval vm_compute in
       (IAnon (match Hs with
         | [] => 1
         | _ => 1 + foldr Pos.max 1 (omap (maybe IAnon) Hs)
         end))%positive
  | _ => constr:(IAnon 1)
Robbert Krebbers's avatar
Robbert Krebbers committed
37
38
  end.

39
40
41
Ltac iMissingHyps Hs :=
  let Δ :=
    lazymatch goal with
42
    | |- envs_entails ?Δ _ => Δ
43
44
45
46
47
    | |- context[ envs_split _ _ ?Δ ] => Δ
    end in
  let Hhyps := eval env_cbv in (envs_dom Δ) in
  eval vm_compute in (list_difference Hs Hhyps).

48
Ltac iTypeOf H :=
49
  let Δ := match goal with |- envs_entails ?Δ _ => Δ end in
50
  eval env_cbv in (envs_lookup H Δ).
Robbert Krebbers's avatar
Robbert Krebbers committed
51

52
Tactic Notation "iMatchHyp" tactic1(tac) :=
53
54
55
56
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
57
(** * Start a proof *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
58
59
60
61
62
63
64
65
66
67
68
69
Tactic Notation "iStartProof" :=
  lazymatch goal with
  | |- envs_entails _ _ => idtac
  | |- ?φ => eapply (as_valid_2 φ);
               [apply _ || fail "iStartProof: not a Bi entailment"
               |apply tac_adequate]
  end.

(* Same as above, with 2 differences :
   - We can specify a BI in which we want the proof to be done
   - If the goal starts with a let or a ∀, they are automatically
     introduced. *)
70
Tactic Notation "iStartProof" uconstr(PROP) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
71
  lazymatch goal with
72
  | |- @envs_entails ?PROP' _ _ =>
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
73
74
    (* This cannot be shared with the other [iStartProof], because
    type_term has a non-negligeable performance impact. *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
75
    let x := type_term (eq_refl : @eq Type PROP PROP') in idtac
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
76

Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
77
  (* We eta-expand [as_valid_2], in order to make sure that
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
78
79
     [iStartProof PROP] works even if [PROP] is the carrier type. In
     this case, typing this expression will end up unifying PROP with
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
80
81
     [bi_car _], and hence trigger the canonical structures mechanism
     to find the corresponding bi. *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
82
  | |- ?φ => eapply (λ P : PROP, @as_valid_2 φ _ P);
83
               [apply _ || fail "iStartProof: not a Bi entailment"
84
               |apply tac_adequate]
Robbert Krebbers's avatar
Robbert Krebbers committed
85
86
  end.

87

88
89
(** * Simplification *)
Tactic Notation "iEval" tactic(t) :=
90
  iStartProof; try (eapply tac_eval; [t; reflexivity|]).
91
92
93

Tactic Notation "iSimpl" := iEval simpl.

Robbert Krebbers's avatar
Robbert Krebbers committed
94
95
96
(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
97
98
    [env_reflexivity || fail "iRename:" H1 "not found"
    |env_reflexivity || fail "iRename:" H2 "not fresh"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
99

100
101
Local Inductive esel_pat :=
  | ESelPure
102
  | ESelIdent : bool  ident  esel_pat.
103

104
Ltac iElaborateSelPat pat :=
105
106
  let rec go pat Δ Hs :=
    lazymatch pat with
107
    | [] => eval cbv in Hs
108
109
110
111
    | SelPure :: ?pat => go pat Δ (ESelPure :: Hs)
    | SelPersistent :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_persistent Δ)) in
       let Δ' := eval env_cbv in (envs_clear_persistent Δ) in
112
       go pat Δ' ((ESelIdent true <$> Hs') ++ Hs)
113
114
115
    | SelSpatial :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_spatial Δ)) in
       let Δ' := eval env_cbv in (envs_clear_spatial Δ) in
116
117
       go pat Δ' ((ESelIdent false <$> Hs') ++ Hs)
    | SelIdent ?H :: ?pat =>
118
       lazymatch eval env_cbv in (envs_lookup_delete H Δ) with
119
       | Some (?p,_,?Δ') => go pat Δ' (ESelIdent p H :: Hs)
120
121
122
123
       | None => fail "iElaborateSelPat:" H "not found"
       end
    end in
  lazymatch goal with
124
  | |- envs_entails ?Δ _ =>
125
126
127
    let pat := sel_pat.parse pat in go pat Δ (@nil esel_pat)
  end.

128
129
Local Ltac iClearHyp H :=
  eapply tac_clear with _ H _ _; (* (i:=H) *)
130
131
132
133
134
    [env_reflexivity || fail "iClear:" H "not found"
    |env_cbv; apply _ ||
     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iClear:" H ":" P "not affine and the goal not absorbing"
    |].
135

Robbert Krebbers's avatar
Robbert Krebbers committed
136
137
Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
138
    lazymatch Hs with
Robbert Krebbers's avatar
Robbert Krebbers committed
139
    | [] => idtac
140
    | ESelPure :: ?Hs => clear; go Hs
141
    | ESelIdent _ ?H :: ?Hs => iClearHyp H; go Hs
Robbert Krebbers's avatar
Robbert Krebbers committed
142
    end in
143
  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.
144

145
146
Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
147
148
149

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
150
  eapply tac_assumption with _ H _ _; (* (i:=H) *)
151
    [env_reflexivity || fail "iExact:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
152
153
    |apply _ ||
     let P := match goal with |- FromAssumption _ ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
154
155
156
     fail "iExact:" H ":" P "does not match goal"
    |env_cbv; apply _ ||
     fail "iExact:" H "not absorbing and the remaining hypotheses not affine"].
Robbert Krebbers's avatar
Robbert Krebbers committed
157
158
159

Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
160
    lazymatch Γ with
Robbert Krebbers's avatar
Robbert Krebbers committed
161
    | Esnoc ?Γ ?j ?Q => first [unify P Q; unify i j|find Γ i P]
Robbert Krebbers's avatar
Robbert Krebbers committed
162
163
164
    end in
  match goal with
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
165
     first [is_evar i; fail 1 | env_reflexivity]
Robbert Krebbers's avatar
Robbert Krebbers committed
166
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
167
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
168
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
169
     first [is_evar i; fail 1 | env_reflexivity]
170
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
171
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
Robbert Krebbers's avatar
Robbert Krebbers committed
172
  end.
173

Robbert Krebbers's avatar
Robbert Krebbers committed
174
Tactic Notation "iAssumption" :=
175
176
  let Hass := fresh in
  let rec find p Γ Q :=
177
    lazymatch Γ with
178
    | Esnoc ?Γ ?j ?P => first
179
       [pose proof (_ : FromAssumption p P Q) as Hass;
Robbert Krebbers's avatar
Robbert Krebbers committed
180
181
182
183
184
185
186
187
188
        eapply (tac_assumption _ _ j p P);
          [env_reflexivity
          |apply Hass
          |env_cbv; apply _ ||
           fail 1 "iAssumption:" j "not absorbing and the remaining hypotheses not affine"]
       |assert (P = False%I) as Hass by reflexivity;
        apply (tac_false_destruct _ j p P);
          [env_reflexivity
          |exact Hass]
189
190
       |find p Γ Q]
    end in
191
  lazymatch goal with
192
  | |- envs_entails (Envs ?Γp ?Γs) ?Q =>
193
194
195
     first [find true Γp Q | find false Γs Q
           |fail "iAssumption:" Q "not found"]
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
196
197
198
199
200

(** * False *)
Tactic Notation "iExFalso" := apply tac_ex_falso.

(** * Making hypotheses persistent or pure *)
201
Local Tactic Notation "iPersistent" constr(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
202
  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
203
    [env_reflexivity || fail "iPersistent:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
204
    |apply _ ||
Robbert Krebbers's avatar
Robbert Krebbers committed
205
206
207
     let P := match goal with |- IntoPersistent _ ?P _ => P end in
     fail "iPersistent:" P "not persistent"
    |env_cbv; apply _ ||
208
209
     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iPersistent:" P "not affine and the goal not absorbing"
210
    |env_reflexivity|].
Robbert Krebbers's avatar
Robbert Krebbers committed
211

212
Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
213
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
214
    [env_reflexivity || fail "iPure:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
215
216
217
    |apply _ ||
     let P := match goal with |- IntoPure ?P _ => P end in
     fail "iPure:" P "not pure"
Robbert Krebbers's avatar
Robbert Krebbers committed
218
    |env_cbv; apply _ ||
219
220
     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iPure:" P "not affine and the goal not absorbing"
Robbert Krebbers's avatar
Robbert Krebbers committed
221
222
    |intros pat].

Robbert Krebbers's avatar
Robbert Krebbers committed
223
224
225
Tactic Notation "iEmpIntro" :=
  iStartProof;
  eapply tac_emp_intro;
226
227
    [env_cbv; apply _ ||
     fail "iEmpIntro: spatial context contains non-affine hypotheses"].
Robbert Krebbers's avatar
Robbert Krebbers committed
228

229
Tactic Notation "iPureIntro" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
230
  iStartProof;
231
  eapply tac_pure_intro;
Robbert Krebbers's avatar
Robbert Krebbers committed
232
233
234
235
    [apply _ ||
     let P := match goal with |- FromPure ?P _ => P end in
     fail "iPureIntro:" P "not pure"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
236

237
238
239
240
(** Framing *)
Local Ltac iFrameFinish :=
  lazy iota beta;
  try match goal with
241
242
  | |- envs_entails _ True => exact (bi.pure_intro _ _ I)
  | |- envs_entails _ emp => iEmpIntro
243
244
245
  end.

Local Ltac iFramePure t :=
246
  iStartProof;
247
248
249
250
251
252
  let φ := type of t in
  eapply (tac_frame_pure _ _ _ _ t);
    [apply _ || fail "iFrame: cannot frame" φ
    |iFrameFinish].

Local Ltac iFrameHyp H :=
253
  iStartProof;
254
  eapply tac_frame with _ H _ _ _;
255
    [env_reflexivity || fail "iFrame:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
256
257
258
    |apply _ ||
     let R := match goal with |- Frame _ ?R _ _ => R end in
     fail "iFrame: cannot frame" R
259
260
261
262
263
264
    |iFrameFinish].

Local Ltac iFrameAnyPure :=
  repeat match goal with H : _ |- _ => iFramePure H end.

Local Ltac iFrameAnyPersistent :=
265
  iStartProof;
266
267
268
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => repeat iFrameHyp H; go Hs end in
  match goal with
269
  | |- envs_entails ?Δ _ =>
270
271
272
273
     let Hs := eval cbv in (env_dom (env_persistent Δ)) in go Hs
  end.

Local Ltac iFrameAnySpatial :=
274
  iStartProof;
275
276
277
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
278
  | |- envs_entails ?Δ _ =>
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
     let Hs := eval cbv in (env_dom (env_spatial Δ)) in go Hs
  end.

Tactic Notation "iFrame" := iFrameAnySpatial.

Tactic Notation "iFrame" "(" constr(t1) ")" :=
  iFramePure t1.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" :=
  iFramePure t1; iFrame ( t2 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" :=
  iFramePure t1; iFrame ( t2 t3 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ).

Tactic Notation "iFrame" constr(Hs) :=
  let rec go Hs :=
307
    lazymatch Hs with
308
    | [] => idtac
309
310
311
    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
312
    | SelIdent ?H :: ?Hs => iFrameHyp H; go Hs
313
    end
314
  in let Hs := sel_pat.parse Hs in go Hs.
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
Tactic Notation "iFrame" "(" constr(t1) ")" constr(Hs) :=
  iFramePure t1; iFrame Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")"
    constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ) Hs.

337
338
(** * Basic introduction tactics *)
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
339
340
341
342
343
344
345
346
347
348
349
350
351
352
  (* In the case the goal starts with an [let x := _ in _], we do not
     want to unfold x and start the proof mode. Instead, we want to
     use intros. So [iStartProof] has to be called only if [intros]
     fails *)
  intros x ||
    (iStartProof;
     lazymatch goal with
     | |- envs_entails _ _ =>
       eapply tac_forall_intro;
       [apply _ ||
              let P := match goal with |- FromForall ?P _ => P end in
              fail "iIntro: cannot turn" P "into a universal quantifier"
       |lazy beta; intros x]
     end).
353
354
355
356
357

Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
  [ (* (?Q → _) *)
358
    eapply tac_impl_intro with _ H _; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
359
      [env_cbv; apply _ ||
360
       let P := lazymatch goal with |- Persistent ?P => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
361
362
       fail 1 "iIntro: introducing non-persistent" H ":" P
              "into non-empty spatial context"
Robbert Krebbers's avatar
Robbert Krebbers committed
363
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
364
      |apply _
Robbert Krebbers's avatar
Robbert Krebbers committed
365
      |]
366
367
  | (* (_ -∗ _) *)
    eapply tac_wand_intro with _ H; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
368
369
      [env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
370
  | fail "iIntro: nothing to introduce" ].
371
372
373
374
375
376

Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
  [ (* (?P → _) *)
    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
377
      [apply _ ||
378
       let P := match goal with |- IntoPersistent _ ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
379
       fail 1 "iIntro:" P "not persistent"
Robbert Krebbers's avatar
Robbert Krebbers committed
380
381
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
382
383
  | (* (?P -∗ _) *)
    eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
384
      [apply _ ||
385
       let P := match goal with |- IntoPersistent _ ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
386
387
       fail 1 "iIntro:" P "not persistent"
      |apply _ ||
388
389
       let P := match goal with |- TCOr (Affine ?P) _ => P end in
       fail 1 "iIntro:" P "not affine and the goal not absorbing"
Robbert Krebbers's avatar
Robbert Krebbers committed
390
391
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
392
  | fail "iIntro: nothing to introduce" ].
393
394
395

Local Tactic Notation "iIntro" "_" :=
  first
396
  [ (* (?Q → _) *) iStartProof; apply tac_impl_intro_drop
Robbert Krebbers's avatar
Robbert Krebbers committed
397
  | (* (_ -∗ _) *)
398
    iStartProof; apply tac_wand_intro_drop;
Robbert Krebbers's avatar
Robbert Krebbers committed
399
400
401
402
      [apply _ ||
       let P := match goal with |- TCOr (Affine ?P) _ => P end in
       fail 1 "iIntro:" P "not affine and the goal not absorbing"
      |]
403
404
405
406
407
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntroForall" :=
  lazymatch goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
408
  | |-  _, ?P => fail (* actually an →, this is handled by iIntro below *)
409
  | |-  _, _ => intro
410
411
412
413
414
415
  | |- let _ := _ in _ => intro
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ ( x : _, _) => let x' := fresh x in iIntro (x')
    end
416
417
418
419
  end.
Local Tactic Notation "iIntro" :=
  lazymatch goal with
  | |- _  ?P => intro
420
421
422
423
424
425
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    | |- envs_entails _ (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    end
426
427
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
428
(** * Specialize *)
429
430
431
432
433
Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
434
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
435
Notation "( H $! x1 .. xn 'with' pat )" :=
436
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
437
438
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

439
(*
Robbert Krebbers's avatar
Robbert Krebbers committed
440
441
442
443
444
There is some hacky stuff going on here: because of Coq bug #6583, unresolved
type classes in the arguments `xs` are resolved at arbitrary moments. Tactics
like `apply`, `split` and `eexists` wrongly trigger type class search to resolve
these holes. To avoid TC being triggered too eagerly, this tactic uses `refine`
at most places instead of `apply`.
445
*)
446
Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
447
  let rec go xs :=
448
    lazymatch xs with
449
    | hnil => apply id (* Finally, trigger TC *)
450
    | hcons ?x ?xs =>
451
       eapply tac_forall_specialize with _ H _ _ _; (* (i:=H) (a:=x) *)
452
453
         [env_reflexivity || fail "iSpecialize:" H "not found"
         |typeclasses eauto ||
Robbert Krebbers's avatar
Robbert Krebbers committed
454
          let P := match goal with |- IntoForall ?P _ => P end in
455
          fail "iSpecialize: cannot instantiate" P "with" x
Robbert Krebbers's avatar
Robbert Krebbers committed
456
457
458
         |match goal with (* Force [A] in [ex_intro] to deal with coercions. *)
          | |-  _ : ?A, _ => refine (@ex_intro A _ x (conj _ _)); [|]
          (* If the existentially quantified predicate is non-dependent and [x]
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
459
          is a hole, [refine] will generate an additional goal. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
460
          | |-  _ : ?A, _ => refine (@ex_intro A _ x (conj _ _));[shelve| |]
461
          end; [env_reflexivity|go xs]]
462
463
    end in
  go xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
464

465
Local Tactic Notation "iSpecializePat" open_constr(H) constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
466
  let solve_to_wand H1 :=
Robbert Krebbers's avatar
Robbert Krebbers committed
467
    apply _ ||
Robbert Krebbers's avatar
Robbert Krebbers committed
468
    let P := match goal with |- IntoWand _ _ ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
469
    fail "iSpecialize:" P "not an implication/wand" in
Robbert Krebbers's avatar
Robbert Krebbers committed
470
471
472
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
473
    | SForall :: ?pats =>
Ralf Jung's avatar
Ralf Jung committed
474
       idtac "[IPM] The * specialization pattern is deprecated because it is applied implicitly.";
475
       go H1 pats
476
    | SIdent ?H2 :: ?pats =>
Robbert Krebbers's avatar
Robbert Krebbers committed
477
       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
478
479
         [env_reflexivity || fail "iSpecialize:" H2 "not found"
         |env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
480
         |apply _ ||
Robbert Krebbers's avatar
Robbert Krebbers committed
481
482
          let P := match goal with |- IntoWand _ _ ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand _ _ ?P ?Q _ => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
483
          fail "iSpecialize: cannot instantiate" P "with" Q
484
         |env_reflexivity|go H1 pats]
485
486
    | SPureGoal ?d :: ?pats =>
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
487
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
488
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
489
490
491
         |apply _ ||
          let Q := match goal with |- FromPure ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
492
         |env_reflexivity
493
         |done_if d (*goal*)
494
         |go H1 pats]
495
    | SGoal (SpecGoal GPersistent false ?Hs_frame [] ?d) :: ?pats =>
496
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _ _;
497
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
498
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
499
         |apply _ ||
500
          let Q := match goal with |- Persistent ?Q => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
501
          fail "iSpecialize:" Q "not persistent"
502
         |apply _
503
         |env_reflexivity
504
         |iFrame Hs_frame; done_if d (*goal*)
505
         |go H1 pats]
506
507
508
    | SGoal (SpecGoal GPersistent _ _ _ _) :: ?pats =>
       fail "iSpecialize: cannot select hypotheses for persistent premise"
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs ?d) :: ?pats =>
509
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
510
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
511
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
512
         |solve_to_wand H1
513
         |lazymatch m with
514
          | GSpatial => apply add_modal_id
515
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
516
          end
517
518
         |env_reflexivity ||
          let Hs' := iMissingHyps Hs' in
519
          fail "iSpecialize: hypotheses" Hs' "not found"
520
         |iFrame Hs_frame; done_if d (*goal*)
Robbert Krebbers's avatar
Robbert Krebbers committed
521
         |go H1 pats]
522
523
    | SAutoFrame GPersistent :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
524
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
525
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
526
         |apply _ ||
527
          let Q := match goal with |- Persistent ?Q => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
528
          fail "iSpecialize:" Q "not persistent"
529
         |env_reflexivity
530
531
532
533
         |solve [iFrame "∗ #"]
         |go H1 pats]
    | SAutoFrame ?m :: ?pats =>
       eapply tac_specialize_frame with _ H1 _ _ _ _ _ _;
534
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
535
536
         |solve_to_wand H1
         |lazymatch m with
537
          | GSpatial => apply add_modal_id
538
539
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
          end
Robbert Krebbers's avatar
Robbert Krebbers committed
540
541
         |iFrame "∗ #"; apply tac_unlock ||
          fail "iSpecialize: premise cannot be solved by framing"
542
         |reflexivity]; iIntro H1; go H1 pats
543
544
    end in let pats := spec_pat.parse pat in go H pats.

Robbert Krebbers's avatar
Robbert Krebbers committed
545
(* The argument [p] denotes whether the conclusion of the specialized term is
546
547
548
persistent. If so, one can use all spatial hypotheses for both proving the
premises and the remaning goal. The argument [p] can either be a Boolean or an
introduction pattern, which will be coerced into [true] when it solely contains
549
550
551
552
553
`#` or `%` patterns at the top-level.

In case the specialization pattern in [t] states that the modality of the goal
should be kept for one of the premises (i.e. [>[H1 .. Hn]] is used) then [p]
defaults to [false] (i.e. spatial hypotheses are not preserved). *)
554
Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
555
  let p := intro_pat_persistent p in
556
  let t :=
557
558
559
560
561
    match type of t with
    | string => constr:(ITrm (INamed t) hnil "")
    | ident => constr:(ITrm t hnil "")
    | _ => t
    end in
562
563
  lazymatch t with
  | ITrm ?H ?xs ?pat =>
564
    let pat := spec_pat.parse pat in
565
    let H := lazymatch type of H with string => constr:(INamed H) | _ => H end in
Robbert Krebbers's avatar
Robbert Krebbers committed
566
    iSpecializeArgs H xs;
567
    lazymatch type of H with
568
    | ident =>
569
570
571
572
573
574
      (* The lemma [tac_specialize_persistent_helper] allows one to use all
      spatial hypotheses for both proving the premises of the lemma we
      specialize as well as those of the remaining goal. We can only use it when
      the result of the specialization is persistent, and no modality is
      eliminated. As an optimization, we do not use this when only universal
      quantifiers are instantiated. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
575
      let pat := spec_pat.parse pat in
576
      lazymatch eval compute in
577
        (p && bool_decide (pat  []) && negb (existsb spec_pat_modal pat)) with
578
      | true =>
579
         (* FIXME: do something reasonable when the BI is not affine *)
Robbert Krebbers's avatar
Robbert Krebbers committed
580
         eapply tac_specialize_persistent_helper with _ H _ _ _ _;
581
           [env_reflexivity || fail "iSpecialize:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
582
           |iSpecializePat H pat; last (iExact H)
Robbert Krebbers's avatar
Robbert Krebbers committed
583
           |apply _ ||
Robbert Krebbers's avatar
Robbert Krebbers committed
584
            let Q := match goal with |- IntoPersistent _ ?Q _ => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
585
            fail "iSpecialize:" Q "not persistent"
Robbert Krebbers's avatar
Robbert Krebbers committed
586
           |env_cbv; apply _ ||
587
            let Q := match goal with |- TCAnd _ (Affine ?Q) => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
588
589
590
591
            fail "iSpecialize:" Q "not affine"
           |env_reflexivity
           |(* goal *)]
      | false => iSpecializePat H pat
592
593
      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
594
    end
595
  | _ => fail "iSpecialize:" t "should be a proof mode term"
596
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
597

598
Tactic Notation "iSpecialize" open_constr(t) :=
599
600
601
  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.
602

Robbert Krebbers's avatar
Robbert Krebbers committed
603
(** * Pose proof *)
604
605
(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
606

607
- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
608
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
609
610
611
612
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊣⊢ P2], in which case [Q] becomes [P1 ↔ P2]

The tactic instantiates each dependent argument [x_i] with an evar and generates
a goal [P] for non-dependent arguments [x_i : P]. *)
613
Tactic Notation "iIntoValid" open_constr(t) :=
614
  let rec go t :=
615
    let tT := type of t in
616
617
618
619
    lazymatch eval hnf in tT with
    | ?P  ?Q => let H := fresh in assert P as H; [|go uconstr:(t H); clear H]
    |  _ : ?T, _ =>
      (* Put [T] inside an [id] to avoid TC inference from being invoked. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
620
      (* This is a workarround for Coq bug #6583. *)
621
622
623
      let e := fresh in evar (e:id T);
      let e' := eval unfold e in e in clear e; go (t e')
    | _ =>
624
      let tT' := eval cbv zeta in tT in apply (as_valid_1 tT');
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
625
        [apply _ || fail "iPoseProof: not a BI assertion"|exact t]
626
    end in
627
  go t.
628

629
630
(* The tactic [tac] is called with a temporary fresh name [H]. The argument
[lazy_tc] denotes whether type class inference on the premises of [lem] should
631
632
633
634
635
636
be performed before (if false) or after (if true) [tac H] is called.

The tactic [iApply] uses laxy type class inference, so that evars can first be
instantiated by matching with the goal, whereas [iDestruct] does not, because
eliminations may not be performed when type classes have not been resolved.
*)
637
638
Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
639
  iStartProof;
640
  let Htmp := iFresh in
641
642
  let t := lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let t := lazymatch type of t with string => constr:(INamed t) | _ => t end in
643
644
645
646
647
648
  let spec_tac _ :=
    lazymatch lem with
    | ITrm ?t ?xs ?pat => iSpecializeCore (ITrm Htmp xs pat) as p
    | _ => idtac
    end in
  let go goal_tac :=
649
    lazymatch type of t with
650
    | ident =>
651
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
652
653
         [env_reflexivity || fail "iPoseProof:" t "not found"
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
654
         |goal_tac ()]
655
656
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
657
         [iIntoValid t
658
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
659
         |goal_tac ()]
660
    end;
661
662
663
664
    try (apply _) in
  lazymatch eval compute in lazy_tc with
  | true => go ltac:(fun _ => spec_tac (); last (tac Htmp))
  | false => go spec_tac; last (tac Htmp)
Robbert Krebbers's avatar
Robbert Krebbers committed
665
666
667
  end.

(** * Apply *)
668
Tactic Notation "iApplyHyp" constr(H) :=
669
670
  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
671
      [env_reflexivity
672
673
      |apply _
      |lazy beta (* reduce betas created by instantiation *)]
674
    |iSpecializePat H "[]"; last go H] in
675
676
677
678
679
680
681
682
  iExact H ||
  go H ||
  lazymatch iTypeOf H with
  | Some (_,?Q) => fail "iApply: cannot apply" Q
  end.

Tactic Notation "iApply" open_constr(lem) :=
  iPoseProofCore lem as false true (fun H => iApplyHyp H).
Robbert Krebbers's avatar
Robbert Krebbers committed
683
684

(** * Revert *)
685
Local Tactic Notation "iForallRevert" ident(x) :=
686
687
688
689
690
691
  let err x :=
    intros x;
    iMatchHyp (fun H P =>
      lazymatch P with
      | context [x] => fail 2 "iRevert:" x "is used in hypothesis" H
      end) in
Robbert Krebbers's avatar
Robbert Krebbers committed
692
  iStartProof;
693
694
  let A := type of x in
  lazymatch type of A with
695
696
697
  | Prop => revert x; first [apply tac_pure_revert|err x]
  | _ => revert x; first [apply tac_forall_revert|err x]
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
698
699

Tactic Notation "iRevert" constr(Hs) :=
700
701
  let rec go Hs :=
    lazymatch Hs with
702
    | [] => idtac
703
704
705
    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
706
    | ESelIdent _ ?H :: ?Hs =>
707
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
708
         [env_reflexivity || fail "iRevert:" H "not found"
709
         |env_cbv; go Hs]
710
    end in
Robbert Krebbers's avatar
Robbert Krebbers committed
711
  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.
Robbert Krebbers's avatar
Robbert Krebbers committed
712

713
Tactic Notation "iRevert" "(" ident(x1) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
714
  iForallRevert x1.
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" :=
  iForallRevert x2; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" :=
  iForallRevert x3; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")" :=
  iForallRevert x4; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" :=
  iForallRevert x5; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" :=
  iForallRevert x6; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" :=
  iForallRevert x7; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" :=
  iForallRevert x8; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).

Tactic Notation "iRevert" "(" ident(x1) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
741
    constr(Hs) :=
742
743
744
745
746
747
748
749
750
751
752
753
754
  iRevert Hs; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 x8 ).
Robbert Krebbers's avatar
Robbert Krebbers committed
755
756
757

(** * Disjunction *)
Tactic Notation "iLeft" :=
758
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
759
  eapply tac_or_l;
Robbert Krebbers's avatar
Robbert Krebbers committed
760
761
762
763
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
764
Tactic Notation "iRight" :=
765
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
766
  eapply tac_or_r;
Robbert Krebbers's avatar
Robbert Krebbers committed
767
768
769
770
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iRight:" P "not a disjunction"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
771

772
Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
773
  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
774
    [env_reflexivity || fail "iOrDestruct:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
775
776
777
    |apply _ ||
     let P