tactics.v 56.3 KB
Newer Older
1 2
From iris.proofmode Require Import coq_tactics.
From iris.proofmode Require Import intro_patterns spec_patterns sel_patterns.
3
From iris.base_logic Require Export base_logic.
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 iris.prelude Require Import stringmap hlist.
7
From iris.proofmode Require Import strings.
Robbert Krebbers's avatar
Robbert Krebbers committed
8 9

Declare Reduction env_cbv := cbv [
10
  beq ascii_beq string_beq
11 12
  env_lookup env_lookup_delete env_delete env_app env_replace env_dom
  env_persistent env_spatial env_spatial_is_nil envs_dom
13
  envs_lookup envs_lookup_delete envs_delete envs_snoc envs_app
14 15
    envs_simple_replace envs_replace envs_split
    envs_clear_spatial envs_clear_persistent
16
    envs_split_go envs_split].
Robbert Krebbers's avatar
Robbert Krebbers committed
17 18 19
Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.

20
(** * Misc *)
21 22
(* Tactic Notation tactics cannot return terms *)
Ltac iFresh' H :=
Robbert Krebbers's avatar
Robbert Krebbers committed
23 24
  lazymatch goal with
  |- of_envs ?Δ  _ =>
25 26 27
     (* [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
28 29
     eval vm_compute in (fresh_string_of_set H (of_list Hs))
  | _ => H
Robbert Krebbers's avatar
Robbert Krebbers committed
30
  end.
31
Ltac iFresh := iFresh' "~".
Robbert Krebbers's avatar
Robbert Krebbers committed
32 33 34

Tactic Notation "iTypeOf" constr(H) tactic(tac):=
  let Δ := match goal with |- of_envs ?Δ  _ => Δ end in
35
  lazymatch eval env_cbv in (envs_lookup H Δ) with
Robbert Krebbers's avatar
Robbert Krebbers committed
36 37 38
  | Some (?p,?P) => tac p P
  end.

39
Tactic Notation "iMatchHyp" tactic1(tac) :=
40 41 42 43
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
44
(** * Start a proof *)
45
Ltac iStartProof :=
Robbert Krebbers's avatar
Robbert Krebbers committed
46
  lazymatch goal with
47
  | |- of_envs _  _ => idtac
48
  | |- ?P =>
49
    lazymatch eval hnf in P with
50
    (* need to use the unfolded version of [uPred_valid] due to the hnf *)
51 52 53 54
    | True  _ => apply tac_adequate
    | _  _ => apply uPred.wand_entails, tac_adequate
    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply uPred.iff_equiv, tac_adequate
55
    | _ => fail "iStartProof: not a uPred"
56
    end
Robbert Krebbers's avatar
Robbert Krebbers committed
57 58 59 60 61 62 63 64
  end.

(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
    [env_cbv; reflexivity || fail "iRename:" H1 "not found"
    |env_cbv; reflexivity || fail "iRename:" H2 "not fresh"|].

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92
Local Inductive esel_pat :=
  | ESelPure
  | ESelName : bool  string  esel_pat.

Ltac iElaborateSelPat pat tac :=
  let rec go pat Δ Hs :=
    lazymatch pat with
    | [] => let Hs' := eval cbv in Hs in tac Hs'
    | 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
       go pat Δ' ((ESelName true <$> Hs') ++ Hs)
    | SelSpatial :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_spatial Δ)) in
       let Δ' := eval env_cbv in (envs_clear_spatial Δ) in
       go pat Δ' ((ESelName false <$> Hs') ++ Hs)
    | SelName ?H :: ?pat =>
       lazymatch eval env_cbv in (envs_lookup_delete H Δ) with
       | Some (?p,_,?Δ') => go pat Δ' (ESelName p H :: Hs)
       | None => fail "iElaborateSelPat:" H "not found"
       end
    end in
  lazymatch goal with
  | |- of_envs ?Δ  _ =>
    let pat := sel_pat.parse pat in go pat Δ (@nil esel_pat)
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
93 94
Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
95
    lazymatch Hs with
Robbert Krebbers's avatar
Robbert Krebbers committed
96
    | [] => idtac
97 98
    | ESelPure :: ?Hs => clear; go Hs
    | ESelName _ ?H :: ?Hs =>
Robbert Krebbers's avatar
Robbert Krebbers committed
99 100 101
       eapply tac_clear with _ H _ _; (* (i:=H) *)
         [env_cbv; reflexivity || fail "iClear:" H "not found"|go Hs]
    end in
102 103
  iElaborateSelPat Hs go.

104 105
Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
106 107 108

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
109 110
  eapply tac_assumption with H _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iExact:" H "not found"
111
    |let P := match goal with |- FromAssumption _ ?P _ => P end in
112
     apply _ || fail "iExact:" H ":" P "does not match goal"].
Robbert Krebbers's avatar
Robbert Krebbers committed
113 114 115 116 117 118 119 120 121 122 123 124

Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
    match Γ with
    | Esnoc ?Γ ?j ?Q => first [unify P Q; unify i j| find Γ i P]
    end in
  match goal with
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
     first [is_evar i; fail 1 | env_cbv; reflexivity]
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
     is_evar i; first [find Γp i P | find Γs i P]; env_cbv; reflexivity
  end.
125

Robbert Krebbers's avatar
Robbert Krebbers committed
126
Tactic Notation "iAssumption" :=
127 128 129 130
  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
131
       [pose proof (_ : FromAssumption p P Q) as Hass;
132 133 134 135 136 137 138 139
        apply (tac_assumption _ j p P); [env_cbv; reflexivity|apply Hass]
       |find p Γ Q]
    end in
  match goal with
  | |- of_envs (Envs ?Γp ?Γs)  ?Q =>
     first [find true Γp Q | find false Γs Q
           |fail "iAssumption:" Q "not found"]
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
140 141 142 143 144

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

(** * Making hypotheses persistent or pure *)
145
Local Tactic Notation "iPersistent" constr(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
146 147
  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iPersistent:" H "not found"
148
    |let Q := match goal with |- IntoPersistentP ?Q _ => Q end in
149
     apply _ || fail "iPersistent:" Q "not persistent"
Robbert Krebbers's avatar
Robbert Krebbers committed
150 151
    |env_cbv; reflexivity|].

152
Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
153 154
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
    [env_cbv; reflexivity || fail "iPure:" H "not found"
155
    |let P := match goal with |- IntoPure ?P _ => P end in
156
     apply _ || fail "iPure:" P "not pure"
Robbert Krebbers's avatar
Robbert Krebbers committed
157 158
    |intros pat].

159 160
Tactic Notation "iPureIntro" :=
  eapply tac_pure_intro;
161
    [let P := match goal with |- FromPure ?P _ => P end in
162
     apply _ || fail "iPureIntro:" P "not pure"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
163

164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229
(** Framing *)
Local Ltac iFrameFinish :=
  lazy iota beta;
  try match goal with
  | |- _  True => exact (uPred.pure_intro _ _ I)
  end.

Local Ltac iFramePure t :=
  let φ := type of t in
  eapply (tac_frame_pure _ _ _ _ t);
    [apply _ || fail "iFrame: cannot frame" φ
    |iFrameFinish].

Local Ltac iFrameHyp H :=
  eapply tac_frame with _ H _ _ _;
    [env_cbv; reflexivity || fail "iFrame:" H "not found"
    |let R := match goal with |- Frame ?R _ _ => R end in
     apply _ || fail "iFrame: cannot frame" R
    |iFrameFinish].

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

Local Ltac iFrameAnyPersistent :=
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => repeat iFrameHyp H; go Hs end in
  match goal with
  | |- of_envs ?Δ  _ =>
     let Hs := eval cbv in (env_dom (env_persistent Δ)) in go Hs
  end.

Local Ltac iFrameAnySpatial :=
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
  | |- of_envs ?Δ  _ =>
     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 :=
    match Hs with
    | [] => idtac
230 231 232 233
    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
234
    end
235
  in let Hs := sel_pat.parse Hs in go Hs.
236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257
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.

Robbert Krebbers's avatar
Robbert Krebbers committed
258
(** * Specialize *)
259 260 261 262 263
Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
264
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
265
Notation "( H $! x1 .. xn 'with' pat )" :=
266
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
267 268
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

269
Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
270 271 272 273 274
  match xs with
  | hnil => idtac
  | _ =>
    eapply tac_forall_specialize with _ H _ _ _ xs; (* (i:=H) (a:=x) *)
      [env_cbv; reflexivity || fail 1 "iSpecialize:" H "not found"
275 276
      |let P := match goal with |- ForallSpecialize _ ?P _ => P end in
       apply _ || fail 1 "iSpecialize:" P "not a forall of the right arity or type"
Robbert Krebbers's avatar
Robbert Krebbers committed
277
      |cbn [himpl hcurry]; reflexivity|]
278
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
279

280
Local Tactic Notation "iSpecializePat" constr(H) constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
281
  let solve_to_wand H1 :=
282
    let P := match goal with |- IntoWand ?P _ _ => P end in
283
    apply _ || fail "iSpecialize:" P "not an implication/wand" in
Robbert Krebbers's avatar
Robbert Krebbers committed
284 285 286
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
287
    | SForall :: ?pats => try (iSpecializeArgs H1 (hcons _ _)); go H1 pats
288
    | SName ?H2 :: ?pats =>
Robbert Krebbers's avatar
Robbert Krebbers committed
289 290 291
       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
         [env_cbv; reflexivity || fail "iSpecialize:" H2 "not found"
         |env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
292 293
         |let P := match goal with |- IntoWand ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand ?P ?Q _ => Q end in
294
          apply _ || fail "iSpecialize: cannot instantiate" P "with" Q
Robbert Krebbers's avatar
Robbert Krebbers committed
295
         |env_cbv; reflexivity|go H1 pats]
296
    | SGoalPersistent :: ?pats =>
297
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
298 299
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
300 301
         |let Q := match goal with |- PersistentP ?Q => Q end in
          apply _ || fail "iSpecialize:" Q "not persistent"
302 303 304 305
         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
    | SGoalPure :: ?pats =>
306
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
Robbert Krebbers's avatar
Robbert Krebbers committed
307 308
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
309
         |let Q := match goal with |- FromPure ?Q _ => Q end in
310 311 312 313
          apply _ || fail "iSpecialize:" Q "not pure"
         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
314
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs) :: ?pats =>
315
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
316
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
Robbert Krebbers's avatar
Robbert Krebbers committed
317 318
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
319 320 321
         |match m with
          | false => apply elim_modal_dummy
          | true => apply _ || fail "iSpecialize: goal not a modality"
322
          end
323
         |env_cbv; reflexivity || fail "iSpecialize:" Hs "not found"
324
         |iFrame Hs_frame (*goal*)
Robbert Krebbers's avatar
Robbert Krebbers committed
325
         |go H1 pats]
326 327
    end in let pats := spec_pat.parse pat in go H pats.

Robbert Krebbers's avatar
Robbert Krebbers committed
328 329 330
(* The argument [p] denotes whether the conclusion of the specialized term is
persistent. It can either be a Boolean or an introduction pattern, which will be
coerced into [true] when it only contains `#` or `%` patterns at the top-level. *)
331
Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) tactic(tac) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
332
  let p := intro_pat_persistent p in
333 334
  lazymatch t with
  | ITrm ?H ?xs ?pat =>
335 336 337 338 339 340 341 342 343 344 345 346 347
    lazymatch type of H with
    | string =>
      lazymatch p with
      | true =>
         eapply tac_specialize_persistent_helper with _ H _ _ _;
           [env_cbv; reflexivity || fail "iSpecialize:" H "not found"
           |iSpecializeArgs H xs; iSpecializePat H pat; last (iExact H)
           |let Q := match goal with |- PersistentP ?Q => Q end in
            apply _ || fail "iSpecialize:" Q "not persistent"
           |env_cbv; reflexivity|tac H]
      | false => iSpecializeArgs H xs; iSpecializePat H pat; last (tac H)
      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
348
    end
349
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
350

351 352 353 354 355
Tactic Notation "iSpecialize" open_constr(t) :=
  iSpecializeCore t as false (fun _ => idtac).
Tactic Notation "iSpecialize" open_constr(t) "#" :=
  iSpecializeCore t as true (fun _ => idtac).

Robbert Krebbers's avatar
Robbert Krebbers committed
356
(** * Pose proof *)
357 358
(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
359

360
- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
361
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
362 363 364 365
- [∀ (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]. *)
366
Tactic Notation "iIntoValid" open_constr(t) :=
367
  let rec go t :=
368 369
    let tT := type of t in
    lazymatch eval hnf in tT with
370 371
    | True  _ => apply t
    | _  _ => apply (uPred.entails_wand _ _ t)
372 373
    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply (uPred.equiv_iff _ _ t)
374
    | ?P  ?Q => let H := fresh in assert P as H; [|go uconstr:(t H); clear H]
375 376 377 378 379 380 381 382
    |  _ : ?T, _ =>
       (* Put [T] inside an [id] to avoid TC inference from being invoked. *)
       (* This is a workarround for Coq bug #4969. *)
       let e := fresh in evar (e:id T);
       let e' := eval unfold e in e in clear e; go (t e')
    end
  in go t.

383
Tactic Notation "iPoseProofCore" open_constr(lem) "as" constr(p) tactic(tac) :=
384 385
  let pose_trm t tac :=
    let Htmp := iFresh in
386
    iStartProof;
387 388 389 390 391 392 393 394
    lazymatch type of t with
    | string =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
         [env_cbv; reflexivity || fail "iPoseProof:" t "not found"
         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |tac Htmp]
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
395
         [iIntoValid t
396 397 398 399 400 401 402
         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
         |tac Htmp]
    end;
    try (apply _) (* solve TC constraints. It is essential that this happens
    after the continuation [tac] has been called. *)
  in lazymatch lem with
  | ITrm ?t ?xs ?pat =>
403
     pose_trm t ltac:(fun Htmp => iSpecializeCore (ITrm Htmp xs pat) as p tac)
404
  | _ => pose_trm lem tac
Robbert Krebbers's avatar
Robbert Krebbers committed
405 406
  end.

407
Tactic Notation "iPoseProof" open_constr(lem) "as" constr(H) :=
408
  iPoseProofCore lem as false (fun Htmp => iRename Htmp into H).
Robbert Krebbers's avatar
Robbert Krebbers committed
409 410

(** * Apply *)
411
Tactic Notation "iApply" open_constr(lem) :=
412 413 414 415
  let finish H := first
    [iExact H
    |eapply tac_apply with _ H _ _ _;
       [env_cbv; reflexivity || fail 1 "iApply:" H "not found"
416
       |let P := match goal with |- IntoWand ?P _ _ => P end in
417
        apply _ || fail 1 "iApply: cannot apply" P
418
       |lazy beta (* reduce betas created by instantiation *)]] in
419 420
  lazymatch lem with
  | ITrm ?t ?xs ?pat =>
421
     iPoseProofCore t as false (fun Htmp =>
422 423
       iSpecializeArgs Htmp xs;
       try (iSpecializeArgs Htmp (hcons _ _));
424
       iSpecializePat Htmp pat; last finish Htmp)
425
  | _ =>
426
     iPoseProofCore lem as false (fun Htmp =>
427
       try (iSpecializeArgs Htmp (hcons _ _));
428
       finish Htmp)
429
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
430 431

(** * Revert *)
432
Local Tactic Notation "iForallRevert" ident(x) :=
433 434 435 436 437 438
  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
439 440
  let A := type of x in
  lazymatch type of A with
441 442 443
  | 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
444 445

Tactic Notation "iRevert" constr(Hs) :=
446 447
  let rec go Hs :=
    lazymatch Hs with
448
    | [] => idtac
449 450 451 452 453 454 455
    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
         [env_cbv; reflexivity || fail "iRevert:" H "not found"
         |env_cbv; go Hs]
456
    end in
457
  iElaborateSelPat Hs go.
Robbert Krebbers's avatar
Robbert Krebbers committed
458

459
Tactic Notation "iRevert" "(" ident(x1) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
460
  iForallRevert x1.
461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
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
487
    constr(Hs) :=
488 489 490 491 492 493 494 495 496 497 498 499 500
  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
501 502 503

(** * Disjunction *)
Tactic Notation "iLeft" :=
504
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
505
  eapply tac_or_l;
506
    [let P := match goal with |- FromOr ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
507 508
     apply _ || fail "iLeft:" P "not a disjunction"|].
Tactic Notation "iRight" :=
509
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
510
  eapply tac_or_r;
511
    [let P := match goal with |- FromOr ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
512 513
     apply _ || fail "iRight:" P "not a disjunction"|].

514
Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
515 516
  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_cbv; reflexivity || fail "iOrDestruct:" H "not found"
517
    |let P := match goal with |- IntoOr ?P _ _ => P end in
518
     apply _ || fail "iOrDestruct: cannot destruct" P
Robbert Krebbers's avatar
Robbert Krebbers committed
519 520 521 522 523
    |env_cbv; reflexivity || fail "iOrDestruct:" H1 "not fresh"
    |env_cbv; reflexivity || fail "iOrDestruct:" H2 "not fresh"| |].

(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
524
  iStartProof;
525 526 527
  lazymatch goal with
  | |- _  _ =>
    eapply tac_and_split;
528
      [let P := match goal with |- FromAnd ?P _ _ => P end in
529 530
       apply _ || fail "iSplit:" P "not a conjunction"| |]
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
531 532

Tactic Notation "iSplitL" constr(Hs) :=
533
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
534 535
  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
536
    [let P := match goal with |- FromSep ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
537
     apply _ || fail "iSplitL:" P "not a separating conjunction"
538
    |env_cbv; reflexivity || fail "iSplitL: hypotheses" Hs
539
                                  "not found in the context"| |].
Robbert Krebbers's avatar
Robbert Krebbers committed
540
Tactic Notation "iSplitR" constr(Hs) :=
541
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
542 543
  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
544
    [let P := match goal with |- FromSep ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
545
     apply _ || fail "iSplitR:" P "not a separating conjunction"
546
    |env_cbv; reflexivity || fail "iSplitR: hypotheses" Hs
547
                                  "not found in the context"| |].
Robbert Krebbers's avatar
Robbert Krebbers committed
548 549 550 551

Tactic Notation "iSplitL" := iSplitR "".
Tactic Notation "iSplitR" := iSplitL "".

552 553 554 555 556 557 558 559 560 561 562 563 564
Local Tactic Notation "iAndDestruct" constr(H) "as" constr(H1) constr(H2) :=
  eapply tac_and_destruct with _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_cbv; reflexivity || fail "iAndDestruct:" H "not found"
    |let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     apply _ || fail "iAndDestruct: cannot destruct" P
    |env_cbv; reflexivity || fail "iAndDestruct:" H1 "or" H2 " not fresh"|].

Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(lr) constr(H') :=
  eapply tac_and_destruct_choice with _ H _ lr H' _ _ _;
    [env_cbv; reflexivity || fail "iAndDestruct:" H "not found"
    |let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     apply _ || fail "iAndDestruct: cannot destruct" P
    |env_cbv; reflexivity || fail "iAndDestruct:" H' " not fresh"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
565 566 567 568 569

Tactic Notation "iCombine" constr(H1) constr(H2) "as" constr(H) :=
  eapply tac_combine with _ _ _ H1 _ _ H2 _ _ H _;
    [env_cbv; reflexivity || fail "iCombine:" H1 "not found"
    |env_cbv; reflexivity || fail "iCombine:" H2 "not found"
570 571
    |let P1 := match goal with |- FromSep _ ?P1 _ => P1 end in
     let P2 := match goal with |- FromSep _ _ ?P2 => P2 end in
572
     apply _ || fail "iCombine: cannot combine" P1 "and" P2
Robbert Krebbers's avatar
Robbert Krebbers committed
573 574 575
    |env_cbv; reflexivity || fail "iCombine:" H "not fresh"|].

(** * Existential *)
Robbert Krebbers's avatar
Robbert Krebbers committed
576
Tactic Notation "iExists" uconstr(x1) :=
577
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
578
  eapply tac_exist;
579
    [let P := match goal with |- FromExist ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
580
     apply _ || fail "iExists:" P "not an existential"
Robbert Krebbers's avatar
Robbert Krebbers committed
581
    |cbv beta; eexists x1].
Robbert Krebbers's avatar
Robbert Krebbers committed
582

Robbert Krebbers's avatar
Robbert Krebbers committed
583
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
584
  iExists x1; iExists x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
585
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
586
  iExists x1; iExists x2, x3.
Robbert Krebbers's avatar
Robbert Krebbers committed
587 588
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
589
  iExists x1; iExists x2, x3, x4.
Robbert Krebbers's avatar
Robbert Krebbers committed
590 591
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
592
  iExists x1; iExists x2, x3, x4, x5.
Robbert Krebbers's avatar
Robbert Krebbers committed
593 594
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
595
  iExists x1; iExists x2, x3, x4, x5, x6.
Robbert Krebbers's avatar
Robbert Krebbers committed
596 597
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
598
  iExists x1; iExists x2, x3, x4, x5, x6, x7.
Robbert Krebbers's avatar
Robbert Krebbers committed
599 600 601
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) ","
    uconstr(x8) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
602 603
  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.

604 605
Local Tactic Notation "iExistDestruct" constr(H)
    "as" simple_intropattern(x) constr(Hx) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
606 607
  eapply tac_exist_destruct with H _ Hx _ _; (* (i:=H) (j:=Hx) *)
    [env_cbv; reflexivity || fail "iExistDestruct:" H "not found"
608
    |let P := match goal with |- IntoExist ?P _ => P end in
609
     apply _ || fail "iExistDestruct: cannot destruct" P|];
610 611 612 613
  let y := fresh in
  intros y; eexists; split;
    [env_cbv; reflexivity || fail "iExistDestruct:" Hx "not fresh"
    |revert y; intros x].
Robbert Krebbers's avatar
Robbert Krebbers committed
614

Robbert Krebbers's avatar
Robbert Krebbers committed
615 616
(** * Always *)
Tactic Notation "iAlways":=
617
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
618 619 620 621
  apply tac_always_intro;
    [reflexivity || fail "iAlways: spatial context non-empty"|].

(** * Later *)
622
Tactic Notation "iNext" open_constr(n) :=
623
  iStartProof;
624 625 626 627 628 629 630 631 632 633
  let P := match goal with |- _  ?P => P end in
  try lazymatch n with 0 => fail 1 "iNext: cannot strip 0 laters" end;
  eapply (tac_next _ _ n);
    [apply _ || fail "iNext:" P "does not contain" n "laters"
    |lazymatch goal with
     | |- IntoLaterNEnvs 0 _ _ => fail "iNext:" P "does not contain laters"
     | _ => apply _
     end|].

Tactic Notation "iNext":= iNext _.
Robbert Krebbers's avatar
Robbert Krebbers committed
634

635
(** * Update modality *)
636
Tactic Notation "iModIntro" :=
637
  iStartProof;
638 639 640 641 642 643 644 645 646 647
  eapply tac_modal_intro;
    [let P := match goal with |- IntoModal _ ?P => P end in
     apply _ || fail "iModIntro:" P "not a modality"|].

Tactic Notation "iModCore" constr(H) :=
  eapply tac_modal_elim with _ H _ _ _ _;
    [env_cbv; reflexivity || fail "iMod:" H "not found"
    |let P := match goal with |- ElimModal ?P _ _ _ => P end in
     let Q := match goal with |- ElimModal _ _ ?Q _ => Q end in
     apply _ || fail "iMod: cannot eliminate modality " P "in" Q
Robbert Krebbers's avatar
Robbert Krebbers committed
648 649
    |env_cbv; reflexivity|].

650
(** * Basic destruct tactic *)
651
Local Tactic Notation "iDestructHyp" constr(H) "as" constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
652 653 654
  let rec go Hz pat :=
    lazymatch pat with
    | IAnom => idtac
655
    | IDrop => iClear Hz
Robbert Krebbers's avatar
Robbert Krebbers committed
656
    | IFrame => iFrame Hz
Robbert Krebbers's avatar
Robbert Krebbers committed
657
    | IName ?y => iRename Hz into y
658
    | IList [[]] => iExFalso; iExact Hz
659 660
    | IList [[?pat1; IDrop]] => iAndDestructChoice Hz as true Hz; go Hz pat1
    | IList [[IDrop; ?pat2]] => iAndDestructChoice Hz as false Hz; go Hz pat2
Robbert Krebbers's avatar
Robbert Krebbers committed
661
    | IList [[?pat1; ?pat2]] =>
662
       let Hy := iFresh in iAndDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
Robbert Krebbers's avatar
Robbert Krebbers committed
663
    | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]
Robbert Krebbers's avatar
Robbert Krebbers committed
664 665
    | IPureElim => iPure Hz as ?
    | IAlwaysElim ?pat => iPersistent Hz; go Hz pat
666
    | IModalElim ?pat => iModCore Hz; go Hz pat
Robbert Krebbers's avatar
Robbert Krebbers committed
667 668 669 670
    | _ => fail "iDestruct:" pat "invalid"
    end
  in let pat := intro_pat.parse_one pat in go H pat.

671
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
672 673
    constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as @ pat.
674 675 676 677 678 679 680 681
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
682
    constr(pat) :=
683 684
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
Robbert Krebbers's avatar
Robbert Krebbers committed
685
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
686 687 688
    simple_intropattern(x5) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
Robbert Krebbers's avatar
Robbert Krebbers committed
689
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
690 691 692
    simple_intropattern(x5) simple_intropattern(x6) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
Robbert Krebbers's avatar
Robbert Krebbers committed
693
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
694
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
695
    constr(pat) :=
696 697
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 x7 ) pat.
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1)
Robbert Krebbers's avatar
Robbert Krebbers committed
698 699
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7)
700 701
    simple_intropattern(x8) ")" constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as ( x2 x3 x4 x5 x6 x7 x8 ) pat.
Robbert Krebbers's avatar
Robbert Krebbers committed
702 703

(** * Introduction tactic *)
704
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
705
  try iStartProof;
706 707 708 709 710
  try first
    [(* (∀ _, _) *) apply tac_forall_intro
    |(* (?P → _) *) eapply tac_impl_intro_pure;
      [let P := match goal with |- IntoPure ?P _ => P end in
       apply _ || fail "iIntro:" P "not pure"|]
711
    |(* (?P -∗ _) *) eapply tac_wand_intro_pure;
712
      [let P := match goal with |- IntoPure ?P _ => P end in
713
       apply _ || fail "iIntro:" P "not pure"|]
Robbert Krebbers's avatar
Robbert Krebbers committed
714 715
    |(* ⌜∀ _, _⌝ *) apply tac_pure_forall_intro
    |(* ⌜_ → _⌝ *) apply tac_pure_impl_intro];
716
  intros x.
717

Robbert Krebbers's avatar
Robbert Krebbers committed
718 719 720
Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
721
  [ (* (?Q → _) *)
722
    eapply tac_impl_intro with _ H; (* (i:=H) *)
723 724
      [reflexivity || fail 1 "iIntro: introducing" H
                             "into non-empty spatial context"
725
      |env_cbv; reflexivity || fail "iIntro:" H "not fresh"|]
726
  | (* (_ -∗ _) *)
727
    eapply tac_wand_intro with _ H; (* (i:=H) *)
728 729
      [env_cbv; reflexivity || fail 1 "iIntro:" H "not fresh"|]
  | fail 1 "iIntro: nothing to introduce" ].
730

Robbert Krebbers's avatar
Robbert Krebbers committed
731 732 733
Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
734
  [ (* (?P → _) *)
735
    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
736
      [let P := match goal with |- IntoPersistentP ?P _ => P end in