tactics.v 77.7 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.
Ralf Jung's avatar
Ralf Jung committed
6
From stdpp 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
Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.
20
Ltac env_reflexivity := env_cbv; exact eq_refl.
Robbert Krebbers's avatar
Robbert Krebbers committed
21

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

35 36 37 38 39 40 41 42 43
Ltac iMissingHyps Hs :=
  let Δ :=
    lazymatch goal with
    | |- of_envs ?Δ  _ => Δ
    | |- 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).

44
Ltac iTypeOf H :=
Robbert Krebbers's avatar
Robbert Krebbers committed
45
  let Δ := match goal with |- of_envs ?Δ  _ => Δ end in
46
  eval env_cbv in (envs_lookup H Δ).
Robbert Krebbers's avatar
Robbert Krebbers committed
47

48
Tactic Notation "iMatchHyp" tactic1(tac) :=
49 50 51 52
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
53 54 55 56 57 58 59 60 61 62 63 64
Class AsValid {M} (φ : Prop) (P : uPred M) := as_entails : φ  P.
Arguments AsValid {_} _%type _%I.

Instance as_valid_valid {M} (P : uPred M) : AsValid (uPred_valid P) P | 0.
Proof. by rewrite /AsValid. Qed.

Instance as_valid_entails {M} (P Q : uPred M) : AsValid (P  Q) (P - Q) | 1.
Proof. split. apply uPred.entails_wand. apply uPred.wand_entails. Qed.

Instance as_valid_equiv {M} (P Q : uPred M) : AsValid (P  Q) (P  Q).
Proof. split. apply uPred.equiv_iff. apply uPred.iff_equiv. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
65
(** * Start a proof *)
66
Ltac iStartProof :=
Robbert Krebbers's avatar
Robbert Krebbers committed
67
  lazymatch goal with
68
  | |- of_envs _  _ => idtac
69
  | |- ?P =>
Robbert Krebbers's avatar
Robbert Krebbers committed
70 71
    apply (proj2 (_ : AsValid P _)), tac_adequate
    || fail "iStartProof: not a uPred"
Robbert Krebbers's avatar
Robbert Krebbers committed
72 73 74 75 76
  end.

(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
77 78
    [env_reflexivity || fail "iRename:" H1 "not found"
    |env_reflexivity || fail "iRename:" H2 "not fresh"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
79

80 81 82 83
Local Inductive esel_pat :=
  | ESelPure
  | ESelName : bool  string  esel_pat.

84
Ltac iElaborateSelPat pat :=
85 86
  let rec go pat Δ Hs :=
    lazymatch pat with
87
    | [] => eval cbv in Hs
88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107
    | 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
108 109
Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
110
    lazymatch Hs with
Robbert Krebbers's avatar
Robbert Krebbers committed
111
    | [] => idtac
112 113
    | ESelPure :: ?Hs => clear; go Hs
    | ESelName _ ?H :: ?Hs =>
Robbert Krebbers's avatar
Robbert Krebbers committed
114
       eapply tac_clear with _ H _ _; (* (i:=H) *)
115
         [env_reflexivity || fail "iClear:" H "not found"|go Hs]
Robbert Krebbers's avatar
Robbert Krebbers committed
116
    end in
117
  let Hs := iElaborateSelPat Hs in go Hs.
118

119 120
Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
121 122 123

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
124
  eapply tac_assumption with H _ _; (* (i:=H) *)
125
    [env_reflexivity || fail "iExact:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
126 127 128
    |apply _ ||
     let P := match goal with |- FromAssumption _ ?P _ => P end in
     fail "iExact:" H ":" P "does not match goal"].
Robbert Krebbers's avatar
Robbert Krebbers committed
129 130 131 132 133 134 135 136

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) =>
137
     first [is_evar i; fail 1 | env_reflexivity]
Robbert Krebbers's avatar
Robbert Krebbers committed
138
  | |- envs_lookup ?i (Envs ?Γp ?Γs) = Some (_, ?P) =>
139
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
140
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
141
     first [is_evar i; fail 1 | env_reflexivity]
142
  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
143
     is_evar i; first [find Γp i P | find Γs i P]; env_reflexivity
Robbert Krebbers's avatar
Robbert Krebbers committed
144
  end.
145

Robbert Krebbers's avatar
Robbert Krebbers committed
146
Tactic Notation "iAssumption" :=
147 148 149 150
  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
151
       [pose proof (_ : FromAssumption p P Q) as Hass;
152
        apply (tac_assumption _ j p P); [env_reflexivity|apply Hass]
153 154 155 156 157 158 159
       |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
160 161 162 163 164

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

(** * Making hypotheses persistent or pure *)
165
Local Tactic Notation "iPersistent" constr(H) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
166
  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
167
    [env_reflexivity || fail "iPersistent:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
168
    |apply _ ||
169
     let Q := match goal with |- IntoPersistent _ ?Q _ => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
170
     fail "iPersistent:" Q "not persistent"
171
    |env_reflexivity|].
Robbert Krebbers's avatar
Robbert Krebbers committed
172

173
Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
174
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
175
    [env_reflexivity || fail "iPure:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
176 177 178
    |apply _ ||
     let P := match goal with |- IntoPure ?P _ => P end in
     fail "iPure:" P "not pure"
Robbert Krebbers's avatar
Robbert Krebbers committed
179 180
    |intros pat].

181
Tactic Notation "iPureIntro" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
182
  iStartProof;
183
  eapply tac_pure_intro;
Robbert Krebbers's avatar
Robbert Krebbers committed
184 185 186 187
    [apply _ ||
     let P := match goal with |- FromPure ?P _ => P end in
     fail "iPureIntro:" P "not pure"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
188

189 190 191 192 193 194 195 196 197 198 199 200 201 202 203
(** 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 _ _ _;
204
    [env_reflexivity || fail "iFrame:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
205 206 207
    |apply _ ||
     let R := match goal with |- Frame _ ?R _ _ => R end in
     fail "iFrame: cannot frame" R
208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253
    |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 :=
254
    lazymatch Hs with
255
    | [] => idtac
256 257 258 259
    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
260
    end
261
  in let Hs := sel_pat.parse Hs in go Hs.
262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283
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.

284 285 286
(** * Basic introduction tactics *)
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
  try iStartProof;
287 288 289
  lazymatch goal with
  | |- _  _ =>
    eapply tac_forall_intro;
Robbert Krebbers's avatar
Robbert Krebbers committed
290
      [apply _ ||
291 292 293 294 295
       let P := match goal with |- FromForall ?P _ => P end in
       fail "iIntro: cannot turn" P "into a universal quantifier"
      |lazy beta; intros x]
  | |- _ => intros x
  end.
296 297 298 299 300 301

Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
  [ (* (?Q → _) *)
    eapply tac_impl_intro with _ H; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
302
      [env_cbv; apply _ ||
303
       let P := lazymatch goal with |- Persistent ?P => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
304 305
       fail 1 "iIntro: introducing non-persistent" H ":" P
              "into non-empty spatial context"
Robbert Krebbers's avatar
Robbert Krebbers committed
306 307
      |env_reflexivity || fail "iIntro:" H "not fresh"
      |]
308 309
  | (* (_ -∗ _) *)
    eapply tac_wand_intro with _ H; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
310 311
      [env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
312 313 314 315 316 317 318
  | fail 1 "iIntro: nothing to introduce" ].

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
319
      [apply _ || 
320
       let P := match goal with |- IntoPersistent _ ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
321 322 323
       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
324 325
  | (* (?P -∗ _) *)
    eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
326
      [apply _ || 
327
       let P := match goal with |- IntoPersistent _ ?P _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
328 329 330
       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "_" :=
  try iStartProof;
  first
  [ (* (?Q → _) *) apply tac_impl_intro_drop
  | (* (_ -∗ _) *) apply tac_wand_intro_drop
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntroForall" :=
  try iStartProof;
  lazymatch goal with
  | |-  _, ?P => fail
  | |-  _, _ => intro
  | |- _  ( x : _, _) => let x' := fresh x in iIntro (x')
  end.
Local Tactic Notation "iIntro" :=
  try iStartProof;
  lazymatch goal with
  | |- _  ?P => intro
  | |- _  (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  | |- _  (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  end.

Robbert Krebbers's avatar
Robbert Krebbers committed
356
(** * Specialize *)
357 358 359 360 361
Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
362
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
363
Notation "( H $! x1 .. xn 'with' pat )" :=
364
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
365 366
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

367 368 369 370 371 372 373 374 375 376 377
(*
There is some hacky stuff going on here (most probably there is a Coq bug).
Holes -- like unresolved type class instances -- in the argument `xs` are
resolved at arbitrary moments. It seems that tactics like `apply`, `split` and
`eexists` trigger type class search to resolve these holes. To avoid TC being
triggered too eagerly, this tactic uses `refine` at various places instead of
`apply`.

TODO: Investigate what really is going on. Is there a related to Cog bug #5752?
When should holes in an `open_constr` be resolved?
*)
378
Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
379
  let rec go xs :=
380
    lazymatch xs with
381
    | hnil => apply id (* Finally, trigger TC *)
382
    | hcons ?x ?xs =>
383
       eapply tac_forall_specialize with _ H _ _ _; (* (i:=H) (a:=x) *)
384 385
         [env_reflexivity || fail "iSpecialize:" H "not found"
         |typeclasses eauto ||
Robbert Krebbers's avatar
Robbert Krebbers committed
386
          let P := match goal with |- IntoForall ?P _ => P end in
387 388 389 390
          fail "iSpecialize: cannot instantiate" P "with" x
         |lazymatch goal with (* Force [A] in [ex_intro] to deal with coercions. *)
          | |-  _ : ?A, _ => refine (@ex_intro A _ x (conj _ _))
          end; [env_reflexivity|go xs]]
391 392
    end in
  go xs.
Robbert Krebbers's avatar
Robbert Krebbers committed
393

394
Local Tactic Notation "iSpecializePat" open_constr(H) constr(pat) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
395
  let solve_to_wand H1 :=
Robbert Krebbers's avatar
Robbert Krebbers committed
396
    apply _ ||
397
    let P := match goal with |- IntoWand _ ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
398
    fail "iSpecialize:" P "not an implication/wand" in
Robbert Krebbers's avatar
Robbert Krebbers committed
399 400 401
  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
402 403 404
    | SForall :: ?pats =>
       idtac "the * specialization pattern is deprecated because it is applied implicitly";
       go H1 pats
405
    | SName ?H2 :: ?pats =>
Robbert Krebbers's avatar
Robbert Krebbers committed
406
       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
407 408
         [env_reflexivity || fail "iSpecialize:" H2 "not found"
         |env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
409
         |apply _ ||
410 411
          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
412
          fail "iSpecialize: cannot instantiate" P "with" Q
413
         |env_reflexivity|go H1 pats]
414 415
    | SPureGoal ?d :: ?pats =>
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
416
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
417
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
418 419 420
         |apply _ ||
          let Q := match goal with |- FromPure ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
421
         |env_reflexivity
422
         |done_if d (*goal*)
423
         |go H1 pats]
424 425
    | SGoal (SpecGoal GPersistent false ?Hs_frame [] ?d) :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
426
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
427
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
428
         |apply _ ||
429
          let Q := match goal with |- Persistent ?Q => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
430
          fail "iSpecialize:" Q "not persistent"
431
         |env_reflexivity
432
         |iFrame Hs_frame; done_if d (*goal*)
433
         |go H1 pats]
434 435 436
    | SGoal (SpecGoal GPersistent _ _ _ _) :: ?pats =>
       fail "iSpecialize: cannot select hypotheses for persistent premise"
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs ?d) :: ?pats =>
437
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
438
       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
439
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
440
         |solve_to_wand H1
441 442 443
         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
444
          end
445 446
         |env_reflexivity ||
          let Hs' := iMissingHyps Hs' in
447
          fail "iSpecialize: hypotheses" Hs' "not found"
448
         |iFrame Hs_frame; done_if d (*goal*)
Robbert Krebbers's avatar
Robbert Krebbers committed
449
         |go H1 pats]
450 451
    | SAutoFrame GPersistent :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
452
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
453
         |solve_to_wand H1
Robbert Krebbers's avatar
Robbert Krebbers committed
454
         |apply _ ||
455
          let Q := match goal with |- Persistent ?Q => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
456
          fail "iSpecialize:" Q "not persistent"
457
         |env_reflexivity
458 459 460 461
         |solve [iFrame "∗ #"]
         |go H1 pats]
    | SAutoFrame ?m :: ?pats =>
       eapply tac_specialize_frame with _ H1 _ _ _ _ _ _;
462
         [env_reflexivity || fail "iSpecialize:" H1 "not found"
463 464 465 466 467
         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
          end
Robbert Krebbers's avatar
Robbert Krebbers committed
468 469
         |iFrame "∗ #"; apply tac_unlock ||
          fail "iSpecialize: premise cannot be solved by framing"
470
         |reflexivity]; iIntro H1; go H1 pats
471 472
    end in let pats := spec_pat.parse pat in go H pats.

Robbert Krebbers's avatar
Robbert Krebbers committed
473
(* The argument [p] denotes whether the conclusion of the specialized term is
474 475 476
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
477 478 479 480 481
`#` 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). *)
482
Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
483
  let p := intro_pat_persistent p in
484 485
  let t :=
    match type of t with string => constr:(ITrm t hnil "") | _ => t end in
486 487
  lazymatch t with
  | ITrm ?H ?xs ?pat =>
488
    let pat := spec_pat.parse pat in
489 490
    lazymatch type of H with
    | string =>
491 492 493 494 495 496 497 498
      (* 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. *)
      lazymatch eval compute in
        (bool_decide (pat  []) && p && negb (existsb spec_pat_modal pat)) with
499 500
      | true =>
         eapply tac_specialize_persistent_helper with _ H _ _ _;
501
           [env_reflexivity || fail "iSpecialize:" H "not found"
502
           |iSpecializeArgs H xs; iSpecializePat H pat; last (iExact H)
Robbert Krebbers's avatar
Robbert Krebbers committed
503
           |apply _ ||
504
            let Q := match goal with |- Persistent ?Q => Q end in
Robbert Krebbers's avatar
Robbert Krebbers committed
505
            fail "iSpecialize:" Q "not persistent"
506
           |env_reflexivity|(* goal *)]
507
      | false => iSpecializeArgs H xs; iSpecializePat H pat
508 509
      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
510
    end
511
  | _ => fail "iSpecialize:" t "should be a proof mode term"
512
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
513

514
Tactic Notation "iSpecialize" open_constr(t) :=
515 516 517
  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.
518

Robbert Krebbers's avatar
Robbert Krebbers committed
519
(** * Pose proof *)
520 521
(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
522

523
- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
524
- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
525 526 527 528
- [∀ (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]. *)
529
Tactic Notation "iIntoValid" open_constr(t) :=
530
  let rec go t :=
531
    let tT := type of t in
532 533 534 535 536 537 538 539 540 541 542
    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. *)
      (* 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')
    | _ =>
      let tT' := eval cbv zeta in tT in apply (proj1 (_ : AsValid tT' _) t)
      || fail "iPoseProof: not a uPred"
    end in
543
  go t.
544

545 546 547 548 549 550 551 552 553 554 555 556 557 558 559
(* 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 :=
560 561 562
    lazymatch type of t with
    | string =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
563 564
         [env_reflexivity || fail "iPoseProof:" t "not found"
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
565
         |goal_tac ()]
566 567
    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
568
         [iIntoValid t
569
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
570
         |goal_tac ()]
571
    end;
572 573 574 575
    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
576 577 578
  end.

(** * Apply *)
579
Tactic Notation "iApplyHyp" constr(H) :=
580 581
  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
582
      [env_reflexivity
583 584
      |apply _
      |lazy beta (* reduce betas created by instantiation *)]
585
    |iSpecializePat H "[]"; last go H] in
586 587 588 589 590 591 592 593
  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
594 595

(** * Revert *)
596
Local Tactic Notation "iForallRevert" ident(x) :=
597 598 599 600 601 602
  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
603
  iStartProof;
604 605
  let A := type of x in
  lazymatch type of A with
606 607 608
  | 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
609 610

Tactic Notation "iRevert" constr(Hs) :=
611 612
  let rec go Hs :=
    lazymatch Hs with
613
    | [] => idtac
614 615 616 617 618
    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
619
         [env_reflexivity || fail "iRevert:" H "not found"
620
         |env_cbv; go Hs]
621
    end in
Robbert Krebbers's avatar
Robbert Krebbers committed
622
  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.
Robbert Krebbers's avatar
Robbert Krebbers committed
623

624
Tactic Notation "iRevert" "(" ident(x1) ")" :=
Robbert Krebbers's avatar
Robbert Krebbers committed
625
  iForallRevert x1.
626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651
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
652
    constr(Hs) :=
653 654 655 656 657 658 659 660 661 662 663 664 665
  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
666 667 668

(** * Disjunction *)
Tactic Notation "iLeft" :=
669
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
670
  eapply tac_or_l;
Robbert Krebbers's avatar
Robbert Krebbers committed
671 672 673 674
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
675
Tactic Notation "iRight" :=
676
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
677
  eapply tac_or_r;
Robbert Krebbers's avatar
Robbert Krebbers committed
678 679 680 681
    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iRight:" P "not a disjunction"
    |].
Robbert Krebbers's avatar
Robbert Krebbers committed
682

683
Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
684
  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
685
    [env_reflexivity || fail "iOrDestruct:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
686 687 688
    |apply _ ||
     let P := match goal with |- IntoOr ?P _ _ => P end in
     fail "iOrDestruct: cannot destruct" P
689
    |env_reflexivity || fail "iOrDestruct:" H1 "not fresh"
Robbert Krebbers's avatar
Robbert Krebbers committed
690 691
    |env_reflexivity || fail "iOrDestruct:" H2 "not fresh"
    | |].
Robbert Krebbers's avatar
Robbert Krebbers committed
692 693 694

(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
695
  iStartProof;
696 697 698
  lazymatch goal with
  | |- _  _ =>
    eapply tac_and_split;
Robbert Krebbers's avatar
Robbert Krebbers committed
699
      [apply _ ||
700
       let P := match goal with |- FromAnd _ ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
701
       fail "iSplit:" P "not a conjunction"| |]
702
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
703 704

Tactic Notation "iSplitL" constr(Hs) :=
705
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
706 707
  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
708
    [apply _ ||
709
     let P := match goal with |- FromAnd _ ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
710
     fail "iSplitL:" P "not a separating conjunction"
711 712
    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
713
     fail "iSplitL: hypotheses" Hs "not found"
714 715
    | |].

Robbert Krebbers's avatar
Robbert Krebbers committed
716
Tactic Notation "iSplitR" constr(Hs) :=
717
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
718 719
  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
Robbert Krebbers's avatar
Robbert Krebbers committed
720
    [apply _ ||
721
     let P := match goal with |- FromAnd _ ?P _ _ => P end in
Robbert Krebbers's avatar
Robbert Krebbers committed
722
     fail "iSplitR:" P "not a separating conjunction"
723 724
    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
725
     fail "iSplitR: hypotheses" Hs "not found"
726
    | |].
Robbert Krebbers's avatar
Robbert Krebbers committed
727 728 729 730

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

731 732
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) *)
733
    [env_reflexivity || fail "iAndDestruct:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
734 735 736
    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestruct: cannot destruct" P
737
    |env_reflexivity || fail "iAndDestruct:" H1 "or" H2 " not fresh"|].
738 739 740

Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(lr) constr(H') :=
  eapply tac_and_destruct_choice with _ H _ lr H' _ _ _;
741
    [env_reflexivity || fail "iAndDestructChoice:" H "not found"
Robbert Krebbers's avatar
Robbert Krebbers committed
742 743 744
    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestructChoice: cannot destruct" P
745
    |env_reflexivity || fail "iAndDestructChoice:" H' " not fresh"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
746

747 748 749 750
(** * Combinining hypotheses *)
Tactic Notation "iCombine" constr(Hs) "as" constr(H) :=
  let Hs := words Hs in
  eapply tac_combine with _ _ Hs _ _ H _;
751 752
    [env_reflexivity ||
     let Hs := iMissingHyps Hs in
753
     fail "iCombine: hypotheses" Hs "not found"
754
    |apply _
755
    |env_reflexivity || fail "iCombine:" H "not fresh"|].
Robbert Krebbers's avatar
Robbert Krebbers committed
756

757 758 759
Tactic Notation "iCombine" constr(H1) constr(H2) "as" constr(H) :=
  iCombine [H1;H2] as H.

Robbert Krebbers's avatar
Robbert Krebbers committed
760
(** * Existential *)
Robbert Krebbers's avatar
Robbert Krebbers committed
761
Tactic Notation "iExists" uconstr(x1) :=
762
  iStartProof;
Robbert Krebbers's avatar
Robbert Krebbers committed
763
  eapply tac_exist;
Robbert Krebbers's avatar
Robbert Krebbers committed
764 765 766
    [apply _ ||
     let P := match goal with |- FromExist ?P _ => P end in
     fail "iExists:" P "not an existential"
Robbert Krebbers's avatar
Robbert Krebbers committed
767
    |cbv beta; eexists x1].
Robbert Krebbers's avatar
Robbert Krebbers committed
768

Robbert Krebbers's avatar
Robbert Krebbers committed
769
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
770
  iExists x1; iExists x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
771
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
772
  iExists x1; iExists x2, x3.
Robbert Krebbers's avatar
Robbert Krebbers committed
773 774
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
775
  iExists x1; iExists x2, x3, x4.
Robbert Krebbers's avatar
Robbert Krebbers committed
776 777
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
778
  iExists x1; iExists x2, x3, x4, x5.
Robbert Krebbers's avatar
Robbert Krebbers committed
779 780
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
781
  iExists x1; iExists x2, x3, x4, x5, x6.
Robbert Krebbers's avatar
Robbert Krebbers committed
782 783
Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
784
  iExists x1; iExists x2, x3, x4, x5, x6, x7.
Robbert Krebbers's avatar
Robbert Krebbers committed
785 786 787
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
788 789
  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.