tactics.v 59.3 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.
8
Set Default Proof Using "Type".
Robbert Krebbers's avatar
Robbert Krebbers committed
9 10

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

21
(** * Misc *)
22 23
(* Tactic Notation tactics cannot return terms *)
Ltac iFresh' H :=
Robbert Krebbers's avatar
Robbert Krebbers committed
24 25
  lazymatch goal with
  |- of_envs ?Δ  _ =>
26 27 28
     (* [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
29 30
     eval vm_compute in (fresh_string_of_set H (of_list Hs))
  | _ => H
Robbert Krebbers's avatar
Robbert Krebbers committed
31
  end.
32
Ltac iFresh := iFresh' "~".
Robbert Krebbers's avatar
Robbert Krebbers committed
33 34 35

Tactic Notation "iTypeOf" constr(H) tactic(tac):=
  let Δ := match goal with |- of_envs ?Δ  _ => Δ end in
36
  lazymatch eval env_cbv in (envs_lookup H Δ) with
Robbert Krebbers's avatar
Robbert Krebbers committed
37 38 39
  | Some (?p,?P) => tac p P
  end.

40
Tactic Notation "iMatchHyp" tactic1(tac) :=
41 42 43 44
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
45
(** * Start a proof *)
46
Ltac iStartProof :=
Robbert Krebbers's avatar
Robbert Krebbers committed
47
  lazymatch goal with
48
  | |- of_envs _  _ => idtac
49
  | |- ?P =>
50
    lazymatch eval hnf in P with
51
    (* need to use the unfolded version of [uPred_valid] due to the hnf *)
52 53 54 55
    | 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
56
    | _ => fail "iStartProof: not a uPred"
57
    end
Robbert Krebbers's avatar
Robbert Krebbers committed
58 59 60 61 62 63 64 65
  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"|].

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 93
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
94 95
Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
96
    lazymatch Hs with
Robbert Krebbers's avatar
Robbert Krebbers committed
97
    | [] => idtac
98 99
    | ESelPure :: ?Hs => clear; go Hs
    | ESelName _ ?H :: ?Hs =>
Robbert Krebbers's avatar
Robbert Krebbers committed
100 101 102
       eapply tac_clear with _ H _ _; (* (i:=H) *)
         [env_cbv; reflexivity || fail "iClear:" H "not found"|go Hs]
    end in
103 104
  iElaborateSelPat Hs go.

105 106
Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
107 108 109

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
110 111
  eapply tac_assumption with H _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iExact:" H "not found"
112
    |let P := match goal with |- FromAssumption _ ?P _ => P end in
113
     apply _ || fail "iExact:" H ":" P "does not match goal"].
Robbert Krebbers's avatar
Robbert Krebbers committed
114 115 116 117 118 119 120 121 122 123 124 125

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.
126

Robbert Krebbers's avatar
Robbert Krebbers committed
127
Tactic Notation "iAssumption" :=
128 129 130 131
  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
132
       [pose proof (_ : FromAssumption p P Q) as Hass;
133 134 135 136 137 138 139 140
        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
141 142 143 144 145

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

(** * Making hypotheses persistent or pure *)
146
Local Tactic Notation "iPersistent" constr(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
147 148
  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
    [env_cbv; reflexivity || fail "iPersistent:" H "not found"
149
    |let Q := match goal with |- IntoPersistentP ?Q _ => Q end in
150
     apply _ || fail "iPersistent:" Q "not persistent"
Robbert Krebbers's avatar
Robbert Krebbers committed
151 152
    |env_cbv; reflexivity|].

153
Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
154 155
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
    [env_cbv; reflexivity || fail "iPure:" H "not found"
156
    |let P := match goal with |- IntoPure ?P _ => P end in
157
     apply _ || fail "iPure:" P "not pure"
Robbert Krebbers's avatar
Robbert Krebbers committed
158 159
    |intros pat].

160 161
Tactic Notation "iPureIntro" :=
  eapply tac_pure_intro;
162
    [let P := match goal with |- FromPure ?P _ => P end in
163
     apply _ || fail "iPureIntro:" P "not pure"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
164

165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230
(** 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
231 232 233 234
    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
235
    end
236
  in let Hs := sel_pat.parse Hs in go Hs.
237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258
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
259
(** * Specialize *)
260 261 262 263 264
Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
265
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
266
Notation "( H $! x1 .. xn 'with' pat )" :=
267
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
268 269
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

270
Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
271 272 273 274 275
  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"
276 277
      |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
278
      |cbn [himpl hcurry]; reflexivity|]
279
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
280

281
Local Tactic Notation "iSpecializePat" constr(H) constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
282
  let solve_to_wand H1 :=
283
    let P := match goal with |- IntoWand ?P _ _ => P end in
284
    apply _ || fail "iSpecialize:" P "not an implication/wand" in
Robbert Krebbers's avatar
Robbert Krebbers committed
285 286 287
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
288
    | SForall :: ?pats => try (iSpecializeArgs H1 (hcons _ _)); go H1 pats
289
    | SName ?H2 :: ?pats =>
Robbert Krebbers's avatar
Robbert Krebbers committed
290 291 292
       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"
293 294
         |let P := match goal with |- IntoWand ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand ?P ?Q _ => Q end in
295
          apply _ || fail "iSpecialize: cannot instantiate" P "with" Q
Robbert Krebbers's avatar
Robbert Krebbers committed
296
         |env_cbv; reflexivity|go H1 pats]
297
    | SGoalPersistent :: ?pats =>
298
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
299 300
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
301 302
         |let Q := match goal with |- PersistentP ?Q => Q end in
          apply _ || fail "iSpecialize:" Q "not persistent"
303 304 305 306
         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
    | SGoalPure :: ?pats =>
307
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
Robbert Krebbers's avatar
Robbert Krebbers committed
308 309
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
310
         |let Q := match goal with |- FromPure ?Q _ => Q end in
311 312 313 314
          apply _ || fail "iSpecialize:" Q "not pure"
         |env_cbv; reflexivity
         |(*goal*)
         |go H1 pats]
315
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs) :: ?pats =>
316
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
317
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
Robbert Krebbers's avatar
Robbert Krebbers committed
318 319
         [env_cbv; reflexivity || fail "iSpecialize:" H1 "not found"
         |solve_to_wand H1
320 321 322
         |match m with
          | false => apply elim_modal_dummy
          | true => apply _ || fail "iSpecialize: goal not a modality"
323
          end
324
         |env_cbv; reflexivity || fail "iSpecialize:" Hs "not found"
325
         |iFrame Hs_frame (*goal*)
Robbert Krebbers's avatar
Robbert Krebbers committed
326
         |go H1 pats]
327 328
    end in let pats := spec_pat.parse pat in go H pats.

Robbert Krebbers's avatar
Robbert Krebbers committed
329
(* The argument [p] denotes whether the conclusion of the specialized term is
330 331 332 333 334
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
`#` or `%` patterns at the top-level. *)
Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
335
  let p := intro_pat_persistent p in
336 337
  lazymatch t with
  | ITrm ?H ?xs ?pat =>
338 339 340 341 342 343 344 345 346
    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"
347 348
           |env_cbv; reflexivity|(* goal *)]
      | false => iSpecializeArgs H xs; iSpecializePat H pat
349 350
      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
351
    end
352
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
353

354
Tactic Notation "iSpecialize" open_constr(t) :=
355 356 357
  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.
358

Robbert Krebbers's avatar
Robbert Krebbers committed
359
(** * Pose proof *)
360 361
(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
362

363
- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
364
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
365 366 367 368
- [∀ (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]. *)
369
Tactic Notation "iIntoValid" open_constr(t) :=
370
  let rec go t :=
371 372
    let tT := type of t in
    lazymatch eval hnf in tT with
373 374
    | True  _ => apply t
    | _  _ => apply (uPred.entails_wand _ _ t)
375 376
    (* need to use the unfolded version of [⊣⊢] due to the hnf *)
    | uPred_equiv' _ _ => apply (uPred.equiv_iff _ _ t)
377
    | ?P  ?Q => let H := fresh in assert P as H; [|go uconstr:(t H); clear H]
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')
383 384
    end in
  go t.
385

386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
(* 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
be performed before (if false) or after (if true) [tac H] is called. *)
Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
  try iStartProof;
  let Htmp := iFresh in
  let t :=
    lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let spec_tac _ :=
    lazymatch lem with
    | ITrm ?t ?xs ?pat => iSpecializeCore (ITrm Htmp xs pat) as p
    | _ => idtac
    end in
  let go goal_tac :=
401 402 403 404 405
    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"
406
         |goal_tac ()]
407 408
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
409
         [iIntoValid t
410
         |env_cbv; reflexivity || fail "iPoseProof:" Htmp "not fresh"
411
         |goal_tac ()]
412
    end;
413 414 415 416
    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
417 418
  end.

419
Tactic Notation "iPoseProof" open_constr(lem) "as" constr(H) :=
420
  iPoseProofCore lem as false false (fun Htmp => iRename Htmp into H).
Robbert Krebbers's avatar
Robbert Krebbers committed
421 422

(** * Apply *)
423
Tactic Notation "iApply" open_constr(lem) :=
424 425 426 427 428 429
  let lem := (* add a `*` to specialize all top-level foralls *)
    lazymatch lem with
    | ITrm ?t ?xs ?pat => constr:(ITrm t xs ("*" +:+ pat))
    | _ => constr:(ITrm lem hnil "*")
    end in
  iPoseProofCore lem as false true (fun H => first
430 431
    [iExact H
    |eapply tac_apply with _ H _ _ _;
432 433 434
      [env_cbv; reflexivity
      |apply _
      |lazy beta (* reduce betas created by instantiation *)]]).
Robbert Krebbers's avatar
Robbert Krebbers committed
435 436

(** * Revert *)
437
Local Tactic Notation "iForallRevert" ident(x) :=
438 439 440 441 442 443
  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
444 445
  let A := type of x in
  lazymatch type of A with
446 447 448
  | 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
449 450

Tactic Notation "iRevert" constr(Hs) :=
451 452
  let rec go Hs :=
    lazymatch Hs with
453
    | [] => idtac
454 455 456 457 458 459 460
    | 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]
461
    end in
462
  iElaborateSelPat Hs go.
Robbert Krebbers's avatar
Robbert Krebbers committed
463

464
Tactic Notation "iRevert" "(" ident(x1) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
465
  iForallRevert x1.
466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491
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
492
    constr(Hs) :=
493 494 495 496 497 498 499 500 501 502 503 504 505
  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
506 507 508

(** * Disjunction *)
Tactic Notation "iLeft" :=
509
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
510
  eapply tac_or_l;
511
    [let P := match goal with |- FromOr ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
512 513
     apply _ || fail "iLeft:" P "not a disjunction"|].
Tactic Notation "iRight" :=
514
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
515
  eapply tac_or_r;
516
    [let P := match goal with |- FromOr ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
517 518
     apply _ || fail "iRight:" P "not a disjunction"|].

519
Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
520 521
  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
    [env_cbv; reflexivity || fail "iOrDestruct:" H "not found"
522
    |let P := match goal with |- IntoOr ?P _ _ => P end in
523
     apply _ || fail "iOrDestruct: cannot destruct" P
Robbert Krebbers's avatar
Robbert Krebbers committed
524 525 526 527 528
    |env_cbv; reflexivity || fail "iOrDestruct:" H1 "not fresh"
    |env_cbv; reflexivity || fail "iOrDestruct:" H2 "not fresh"| |].

(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
529
  iStartProof;
530 531 532
  lazymatch goal with
  | |- _  _ =>
    eapply tac_and_split;
533
      [let P := match goal with |- FromAnd ?P _ _ => P end in
534 535
       apply _ || fail "iSplit:" P "not a conjunction"| |]
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
536 537

Tactic Notation "iSplitL" constr(Hs) :=
538
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
539 540
  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
541
    [let P := match goal with |- FromSep ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
542
     apply _ || fail "iSplitL:" P "not a separating conjunction"
543
    |env_cbv; reflexivity || fail "iSplitL: hypotheses" Hs
544
                                  "not found in the context"| |].
Robbert Krebbers's avatar
Robbert Krebbers committed
545
Tactic Notation "iSplitR" constr(Hs) :=
546
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
547 548
  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
549
    [let P := match goal with |- FromSep ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
550
     apply _ || fail "iSplitR:" P "not a separating conjunction"
551
    |env_cbv; reflexivity || fail "iSplitR: hypotheses" Hs
552
                                  "not found in the context"| |].
Robbert Krebbers's avatar
Robbert Krebbers committed
553 554 555 556

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

557 558 559 560 561 562 563 564 565 566 567 568 569
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
570 571 572 573 574

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"
575 576
    |let P1 := match goal with |- FromSep _ ?P1 _ => P1 end in
     let P2 := match goal with |- FromSep _ _ ?P2 => P2 end in
577
     apply _ || fail "iCombine: cannot combine" P1 "and" P2
Robbert Krebbers's avatar
Robbert Krebbers committed
578 579 580
    |env_cbv; reflexivity || fail "iCombine:" H "not fresh"|].

(** * Existential *)
Robbert Krebbers's avatar
Robbert Krebbers committed
581
Tactic Notation "iExists" uconstr(x1) :=
582
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
583
  eapply tac_exist;
584
    [let P := match goal with |- FromExist ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
585
     apply _ || fail "iExists:" P "not an existential"
Robbert Krebbers's avatar
Robbert Krebbers committed
586
    |cbv beta; eexists x1].
Robbert Krebbers's avatar
Robbert Krebbers committed
587

Robbert Krebbers's avatar
Robbert Krebbers committed
588
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
589
  iExists x1; iExists x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
590
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
591
  iExists x1; iExists x2, x3.
Robbert Krebbers's avatar
Robbert Krebbers committed
592 593
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
594
  iExists x1; iExists x2, x3, x4.
Robbert Krebbers's avatar
Robbert Krebbers committed
595 596
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
597
  iExists x1; iExists x2, x3, x4, x5.
Robbert Krebbers's avatar
Robbert Krebbers committed
598 599
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
600
  iExists x1; iExists x2, x3, x4, x5, x6.
Robbert Krebbers's avatar
Robbert Krebbers committed
601 602
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
603
  iExists x1; iExists x2, x3, x4, x5, x6, x7.
Robbert Krebbers's avatar
Robbert Krebbers committed
604 605 606
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
607 608
  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.

609 610
Local Tactic Notation "iExistDestruct" constr(H)
    "as" simple_intropattern(x) constr(Hx) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
611 612
  eapply tac_exist_destruct with H _ Hx _ _; (* (i:=H) (j:=Hx) *)
    [env_cbv; reflexivity || fail "iExistDestruct:" H "not found"
613
    |let P := match goal with |- IntoExist ?P _ => P end in
614
     apply _ || fail "iExistDestruct: cannot destruct" P|];
615 616 617 618
  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
619

Robbert Krebbers's avatar
Robbert Krebbers committed
620 621
(** * Always *)
Tactic Notation "iAlways":=
622
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
623 624 625 626
  apply tac_always_intro;
    [reflexivity || fail "iAlways: spatial context non-empty"|].

(** * Later *)
627
Tactic Notation "iNext" open_constr(n) :=
628
  iStartProof;
629 630 631 632 633 634 635 636 637 638
  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
639

640
(** * Update modality *)
641
Tactic Notation "iModIntro" :=
642
  iStartProof;
643 644 645 646 647 648 649 650 651 652
  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
653 654
    |env_cbv; reflexivity|].

655
(** * Basic destruct tactic *)
656
Local Tactic Notation "iDestructHyp" constr(H) "as" constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
657 658 659
  let rec go Hz pat :=
    lazymatch pat with
    | IAnom => idtac
660
    | IDrop => iClear Hz
Robbert Krebbers's avatar
Robbert Krebbers committed
661
    | IFrame => iFrame Hz
Robbert Krebbers's avatar
Robbert Krebbers committed
662
    | IName ?y => iRename Hz into y
663
    | IList [[]] => iExFalso; iExact Hz
664 665
    | 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
666
    | IList [[?pat1; ?pat2]] =>
667
       let Hy := iFresh in iAndDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
Robbert Krebbers's avatar
Robbert Krebbers committed
668
    | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]
Robbert Krebbers's avatar
Robbert Krebbers committed
669 670
    | IPureElim => iPure Hz as ?
    | IAlwaysElim ?pat => iPersistent Hz; go Hz pat
671
    | IModalElim ?pat => iModCore Hz; go Hz pat
Robbert Krebbers's avatar
Robbert Krebbers committed
672 673 674 675
    | _ => fail "iDestruct:" pat "invalid"
    end
  in let pat := intro_pat.parse_one pat in go H pat.

676
Local Tactic Notation "iDestructHyp" constr(H) "as" "(" simple_intropattern(x1) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
677 678
    constr(pat) :=
  iExistDestruct H as x1 H; iDestructHyp H as @ pat.
679 680 681 682 683 684 685 686
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
687
    constr(pat) :=
688 689
  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
690
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
691 692 693
    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
694
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
695 696 697
    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
698
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
699
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7) ")"
Robbert Krebbers's avatar
Robbert Krebbers committed
700
    constr(pat) :=
701 702
  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
703 704
    simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4)
    simple_intropattern(x5) simple_intropattern(x6) simple_intropattern(x7)
705 706
    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
707 708

(** * Introduction tactic *)
709
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
710
  try iStartProof;
711 712 713 714 715
  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"|]
716
    |(* (?P -∗ _) *) eapply tac_wand_intro_pure;
717
      [let P := match goal with |- IntoPure ?P _ => P end in
718
       apply _ || fail "iIntro:" P "not pure"|]
Robbert Krebbers's avatar
Robbert Krebbers committed
719 720
    |(* ⌜∀ _, _⌝ *) apply tac_pure_forall_intro
    |(* ⌜_ → _⌝ *) apply tac_pure_impl_intro];
721
  intros x.
722

Robbert Krebbers's avatar
Robbert Krebbers committed
723 724 725
Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
726
  [ (* (?Q → _) *)
727
    eapply tac_impl_intro with _ H; (* (i:=H) *)
728 729
      [reflexivity || fail 1 "iIntro: introducing" H
                             "into non-empty spatial context"
730
      |env_cbv; reflexivity || fail "iIntro:" H "not fresh"|]
731
  | (* (_ -∗ _) *)
732
    eapply tac_wand_intro with _ H; (* (i:=H) *)
733 734
      [env_cbv; reflexivity || fail 1 "iIntro:" H "not fresh"|]
  | fail 1 "iIntro: nothing to introduce" ].
735

Robbert Krebbers's avatar
Robbert Krebbers committed
736 737 738
Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
739
  [ (* (?P → _) *)
740
    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
741
      [let P := match goal with |- IntoPersistentP ?P _ => P end in
742 743
       apply _ || fail 1 "iIntro: " P " not persistent"
      |env_cbv; reflexivity