tactics.v 21.5 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
2
(* This file is distributed under the terms of the BSD license. *)
3
(** This file collects general purpose tactics that are used throughout
4
the development. *)
5
From Coq Require Import Omega.
6
From Coq Require Export Lia.
7
From stdpp Require Export decidable.
8
Set Default Proof Using "Type*".
9

Robbert Krebbers's avatar
Robbert Krebbers committed
10 11 12 13 14 15 16 17 18 19 20 21 22
Lemma f_equal_dep {A B} (f g :  x : A, B x) x : f = g  f x = g x.
Proof. intros ->; reflexivity. Qed.
Lemma f_equal_help {A B} (f g : A  B) x y : f = g  x = y  f x = g y.
Proof. intros -> ->; reflexivity. Qed.
Ltac f_equal :=
  let rec go :=
    match goal with
    | _ => reflexivity
    | _ => apply f_equal_help; [go|try reflexivity]
    | |- ?f ?x = ?g ?x => apply (f_equal_dep f g); go
    end in
  try go.

23 24 25 26 27
(** We declare hint databases [f_equal], [congruence] and [lia] and containing
solely the tactic corresponding to its name. These hint database are useful in
to be combined in combination with other hint database. *)
Hint Extern 998 (_ = _) => f_equal : f_equal.
Hint Extern 999 => congruence : congruence.
28
Hint Extern 1000 => lia : lia.
Ralf Jung's avatar
Ralf Jung committed
29
Hint Extern 1000 => omega : omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
30 31
Hint Extern 1001 => progress subst : subst. (** backtracking on this one will
be very bad, so use with care! *)
32 33 34

(** The tactic [intuition] expands to [intuition auto with *] by default. This
is rather efficient when having big hint databases, or expensive [Hint Extern]
Robbert Krebbers's avatar
Robbert Krebbers committed
35
declarations as the ones above. *)
36 37
Tactic Notation "intuition" := intuition auto.

38
(* [done] can get slow as it calls "trivial". [fast_done] can solve way less
39 40 41 42
   goals, but it will also always finish quickly.
   We do 'reflexivity' last because for goals of the form ?x = y, if
   we have x = y in the context, we will typically want to use the
   assumption and not reflexivity *)
43
Ltac fast_done :=
44 45 46 47 48
  solve
    [ eassumption
    | symmetry; eassumption
    | apply not_symmetry; eassumption
    | reflexivity ].
49 50 51
Tactic Notation "fast_by" tactic(tac) :=
  tac; fast_done.

52
(** A slightly modified version of Ssreflect's finishing tactic [done]. It
53 54 55 56
also performs [reflexivity] and uses symmetry of negated equalities. Compared
to Ssreflect's [done], it does not compute the goal's [hnf] so as to avoid
unfolding setoid equalities. Note that this tactic performs much better than
Coq's [easy] tactic as it does not perform [inversion]. *)
57
Ltac done :=
58
  solve
59
  [ repeat first
60 61
    [ fast_done
    | solve [trivial]
62 63 64
    (* All the tactics below will introduce themselves anyway, or make no sense
       for goals of product type. So this is a good place for us to do it. *)
    | progress intros
65
    | solve [symmetry; trivial]
66
    | solve [apply not_symmetry; trivial]
67 68
    | discriminate
    | contradiction
69
    | split
Robbert Krebbers's avatar
Robbert Krebbers committed
70
    | match goal with H : ¬_ |- _ => case H; clear H; fast_done end ]
71
  ].
72 73 74
Tactic Notation "by" tactic(tac) :=
  tac; done.

75 76 77 78
(** Aliases for trans and etrans that are easier to type *)
Tactic Notation "trans" constr(A) := transitivity A.
Tactic Notation "etrans" := etransitivity.

79 80 81 82 83 84 85 86 87 88 89
(** Tactics for splitting conjunctions:

- [split_and] : split the goal if is syntactically of the shape [_ ∧ _]
- [split_ands?] : split the goal repeatedly (perhaps zero times) while it is
  of the shape [_ ∧ _].
- [split_ands!] : works similarly, but at least one split should succeed. In
  order to do so, it will head normalize the goal first to possibly expose a
  conjunction.

Note that [split_and] differs from [split] by only splitting conjunctions. The
[split] tactic splits any inductive with one constructor. *)
90 91 92 93 94
Tactic Notation "split_and" :=
  match goal with
  | |- _  _ => split
  | |- Is_true (_ && _) => apply andb_True; split
  end.
95 96
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.
97

98 99 100 101 102 103 104 105 106
Tactic Notation "destruct_and" "?" :=
  repeat match goal with
  | H : False |- _ => destruct H
  | H : _  _ |- _ => destruct H
  | H : Is_true (bool_decide _) |- _ => apply (bool_decide_unpack _) in H
  | H : Is_true (_ && _) |- _ => apply andb_True in H; destruct H
  end.
Tactic Notation "destruct_and" "!" := progress (destruct_and?).

107 108 109
(** The tactic [case_match] destructs an arbitrary match in the conclusion or
assumptions, and generates a corresponding equality. This tactic is best used
together with the [repeat] tactical. *)
110 111 112 113 114 115
Ltac case_match :=
  match goal with
  | H : context [ match ?x with _ => _ end ] |- _ => destruct x eqn:?
  | |- context [ match ?x with _ => _ end ] => destruct x eqn:?
  end.

116 117 118 119
(** The tactic [unless T by tac_fail] succeeds if [T] is not provable by
the tactic [tac_fail]. *)
Tactic Notation "unless" constr(T) "by" tactic3(tac_fail) :=
  first [assert T by tac_fail; fail 1 | idtac].
120 121 122 123 124 125

(** The tactic [repeat_on_hyps tac] repeatedly applies [tac] in unspecified
order on all hypotheses until it cannot be applied to any hypothesis anymore. *)
Tactic Notation "repeat_on_hyps" tactic3(tac) :=
  repeat match goal with H : _ |- _ => progress tac H end.

126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
(** The tactic [clear dependent H1 ... Hn] clears the hypotheses [Hi] and
their dependencies. *)
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) :=
  clear dependent H1; clear dependent H2.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) :=
  clear dependent H1 H2; clear dependent H3.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) :=
  clear dependent H1 H2 H3; clear dependent H4.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4)
  hyp(H5) := clear dependent H1 H2 H3 H4; clear dependent H5.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) := clear dependent H1 H2 H3 H4 H5; clear dependent H6.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) := clear dependent H1 H2 H3 H4 H5 H6; clear dependent H7.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7; clear dependent H8.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) hyp(H9) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7 H8; clear dependent H9.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) hyp(H9) hyp(H10) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7 H8 H9; clear dependent H10.

(** The tactic [is_non_dependent H] determines whether the goal's conclusion or
151
hypotheses depend on [H]. *)
152 153 154 155 156 157 158
Tactic Notation "is_non_dependent" constr(H) :=
  match goal with
  | _ : context [ H ] |- _ => fail 1
  | |- context [ H ] => fail 1
  | _ => idtac
  end.

159 160
(** The tactic [var_eq x y] fails if [x] and [y] are unequal, and [var_neq]
does the converse. *)
161 162 163
Ltac var_eq x1 x2 := match x1 with x2 => idtac | _ => fail 1 end.
Ltac var_neq x1 x2 := match x1 with x2 => fail 1 | _ => idtac end.

Robbert Krebbers's avatar
Robbert Krebbers committed
164 165 166 167 168 169 170
(** Operational type class projections in recursive calls are not folded back
appropriately by [simpl]. The tactic [csimpl] uses the [fold_classes] tactics
to refold recursive calls of [fmap], [mbind], [omap] and [alter]. A
self-contained example explaining the problem can be found in the following
Coq-club message:

https://sympa.inria.fr/sympa/arc/coq-club/2012-10/msg00147.html *)
171 172
Ltac fold_classes :=
  repeat match goal with
173
  | |- context [ ?F ] =>
174 175 176 177 178 179 180 181 182 183 184 185 186 187 188
    progress match type of F with
    | FMap _ =>
       change F with (@fmap _ F);
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F)
    | MBind _ =>
       change F with (@mbind _ F);
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F)
    | OMap _ =>
       change F with (@omap _ F);
       repeat change (@omap _ (@omap _ F)) with (@omap _ F)
    | Alter _ _ _ =>
       change F with (@alter _ _ _ F);
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F)
    end
  end.
189 190
Ltac fold_classes_hyps H :=
  repeat match type of H with
191
  | context [ ?F ] =>
192 193
    progress match type of F with
    | FMap _ =>
194 195
       change F with (@fmap _ F) in H;
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F) in H
196
    | MBind _ =>
197 198
       change F with (@mbind _ F) in H;
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F) in H
199
    | OMap _ =>
200 201
       change F with (@omap _ F) in H;
       repeat change (@omap _ (@omap _ F)) with (@omap _ F) in H
202
    | Alter _ _ _ =>
203 204
       change F with (@alter _ _ _ F) in H;
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F) in H
205 206
    end
  end.
207 208
Tactic Notation "csimpl" "in" hyp(H) :=
  try (progress simpl in H; fold_classes_hyps H).
209
Tactic Notation "csimpl" := try (progress simpl; fold_classes).
210 211
Tactic Notation "csimpl" "in" "*" :=
  repeat_on_hyps (fun H => csimpl in H); csimpl.
212

Robbert Krebbers's avatar
Robbert Krebbers committed
213
(** The tactic [simplify_eq] repeatedly substitutes, discriminates,
214 215
and injects equalities, and tries to contradict impossible inequalities. *)
Tactic Notation "simplify_eq" := repeat
216
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
217 218
  | H : _  _ |- _ => by case H; try clear H
  | H : _ = _  False |- _ => by case H; try clear H
219 220
  | H : ?x = _ |- _ => subst x
  | H : _ = ?x |- _ => subst x
221
  | H : _ = _ |- _ => discriminate H
222
  | H : _  _ |- _ => apply leibniz_equiv in H
223 224
  | H : ?f _ = ?f _ |- _ => apply (inj f) in H
  | H : ?f _ _ = ?f _ _ |- _ => apply (inj2 f) in H; destruct H
Robbert Krebbers's avatar
Robbert Krebbers committed
225
    (* before [injection] to circumvent bug #2939 in some situations *)
226
  | H : ?f _ = ?f _ |- _ => progress injection H as H
Robbert Krebbers's avatar
Robbert Krebbers committed
227
    (* first hyp will be named [H], subsequent hyps will be given fresh names *)
228 229 230 231 232
  | H : ?f _ _ = ?f _ _ |- _ => progress injection H as H
  | H : ?f _ _ _ = ?f _ _ _ |- _ => progress injection H as H
  | H : ?f _ _ _ _ = ?f _ _ _ _ |- _ => progress injection H as H
  | H : ?f _ _ _ _ _ = ?f _ _ _ _ _ |- _ => progress injection H as H
  | H : ?f _ _ _ _ _ _ = ?f _ _ _ _ _ _ |- _ => progress injection H as H
233
  | H : ?x = ?x |- _ => clear H
234 235 236 237
    (* unclear how to generalize the below *)
  | H1 : ?o = Some ?x, H2 : ?o = Some ?y |- _ =>
    assert (y = x) by congruence; clear H2
  | H1 : ?o = Some ?x, H2 : ?o = None |- _ => congruence
238 239
  | H : @existT ?A _ _ _ = existT _ _ |- _ =>
     apply (Eqdep_dec.inj_pair2_eq_dec _ (decide_rel (@eq A))) in H
240
  end.
241 242 243
Tactic Notation "simplify_eq" "/=" :=
  repeat (progress csimpl in * || simplify_eq).
Tactic Notation "f_equal" "/=" := csimpl in *; f_equal.
244

Robbert Krebbers's avatar
Robbert Krebbers committed
245
Ltac setoid_subst_aux R x :=
Robbert Krebbers's avatar
Robbert Krebbers committed
246
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
247
  | H : R x ?y |- _ =>
Robbert Krebbers's avatar
Robbert Krebbers committed
248 249 250 251 252 253 254 255
     is_var x;
     try match y with x _ => fail 2 end;
     repeat match goal with
     | |- context [ x ] => setoid_rewrite H
     | H' : context [ x ] |- _ =>
        try match H' with H => fail 2 end;
        setoid_rewrite H in H'
     end;
256
     clear x H
Robbert Krebbers's avatar
Robbert Krebbers committed
257 258 259
  end.
Ltac setoid_subst :=
  repeat match goal with
260
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
261 262
  | H : @equiv ?A ?e ?x _ |- _ => setoid_subst_aux (@equiv A e) x
  | H : @equiv ?A ?e _ ?x |- _ => symmetry in H; setoid_subst_aux (@equiv A e) x
Robbert Krebbers's avatar
Robbert Krebbers committed
263 264
  end.

265 266
(** f_equiv works on goals of the form [f _ = f _], for any relation and any
number of arguments. It looks for an appropriate [Proper] instance, and applies
267
it. The tactic is somewhat limited, since it cannot be used to backtrack on
Ralf Jung's avatar
Ralf Jung committed
268
the Proper instances that has been found. To that end, we try to avoid the
269
trivial instance in which the resulting goals have an [eq]. More generally,
Ralf Jung's avatar
Ralf Jung committed
270
we try to "maintain" the relation of the current goal. For example,
271
when having [Proper (equiv ==> dist) f] and [Proper (dist ==> dist) f], it will
Ralf Jung's avatar
Ralf Jung committed
272
favor the second because the relation (dist) stays the same. *)
273
Ltac f_equiv :=
274 275
  match goal with
  | _ => reflexivity
276
  | |- pointwise_relation _ _ _ _ => intros ?
277 278
  (* We support matches on both sides, *if* they concern the same variable, or
     variables in some relation. *)
279
  | |- ?R (match ?x with _ => _ end) (match ?x with _ => _ end) =>
280
    destruct x
281 282
  | H : ?R ?x ?y |- ?R2 (match ?x with _ => _ end) (match ?y with _ => _ end) =>
     destruct H
283
  (* First assume that the arguments need the same relation as the result *)
284 285 286 287 288 289 290 291 292 293
  | |- ?R (?f ?x) (?f _) => apply (_ : Proper (R ==> R) f)
  (* For the case in which R is polymorphic, or an operational type class,
  like equiv. *)
  | |- (?R _) (?f ?x) (?f _) => apply (_ : Proper (R _ ==> _) f)
  | |- (?R _ _) (?f ?x) (?f _) => apply (_ : Proper (R _ _ ==> _) f)
  | |- (?R _ _ _) (?f ?x) (?f _) => apply (_ : Proper (R _ _ _ ==> _) f)
  | |- (?R _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ ==> R _ ==> _) f)
  | |- (?R _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ ==> R _ _ ==> _) f)
  | |- (?R _ _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> _) f)
  | |- (?R _ _ _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ _ _ ==> R _ _ _ _ ==> _) f)
294 295 296 297 298 299
  (* Next, try to infer the relation. Unfortunately, there is an instance
     of Proper for (eq ==> _), which will always be matched. *)
  (* TODO: Can we exclude that instance? *)
  (* TODO: If some of the arguments are the same, we could also
     query for "pointwise_relation"'s. But that leads to a combinatorial
     explosion about which arguments are and which are not the same. *)
300 301
  | |- ?R (?f ?x) (?f _) => apply (_ : Proper (_ ==> R) f)
  | |- ?R (?f ?x ?y) (?f _ _) => apply (_ : Proper (_ ==> _ ==> R) f)
302 303
   (* In case the function symbol differs, but the arguments are the same,
      maybe we have a pointwise_relation in our context. *)
304
  | H : pointwise_relation _ ?R ?f ?g |- ?R (?f ?x) (?g ?x) => apply H
305
  end;
306
  try reflexivity.
307

308 309 310
(* The tactic [preprocess_solve_proper] unfolds the first head symbol, so that
we proceed by repeatedly using [f_equiv]. *)
Ltac preprocess_solve_proper :=
311 312 313
  (* Introduce everything *)
  intros;
  repeat lazymatch goal with
314 315
  | |- Proper _ _ => intros ???
  | |- (_ ==> _)%signature _ _ => intros ???
316
  | |- pointwise_relation _ _ _ _ => intros ?
317 318
  | |- ?R ?f _ => try let f' := constr:(λ x, f x) in intros ?
  end; simpl;
319 320
  (* Unfold the head symbol, which is the one we are proving a new property about *)
  lazymatch goal with
321 322 323 324
  | |- ?R (?f _ _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _) (?f _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _) (?f _ _ _ _ _) => unfold f
325 326 327 328 329
  | |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
  | |- ?R (?f _ _ _) (?f _ _ _) => unfold f
  | |- ?R (?f _ _) (?f _ _) => unfold f
  | |- ?R (?f _) (?f _) => unfold f
  end;
330 331 332 333 334 335 336 337
  simplify_eq.

(** The tactic [solve_proper] solves goals of the form "Proper (R1 ==> R2)", for
any number of relations. The actual work is done by repeatedly applying
[f_equiv]. *)
Ltac solve_proper :=
  preprocess_solve_proper;
  solve [repeat (f_equiv; try eassumption)].
338

339 340 341 342 343 344 345
(** The tactic [intros_revert tac] introduces all foralls/arrows, performs tac,
and then reverts them. *)
Ltac intros_revert tac :=
  lazymatch goal with
  | |-  _, _ => let H := fresh in intro H; intros_revert tac; revert H
  | |- _ => tac
  end.
346

347 348 349 350
(** Given a tactic [tac2] generating a list of terms, [iter tac1 tac2]
runs [tac x] for each element [x] until [tac x] succeeds. If it does not
suceed for any element of the generated list, the whole tactic wil fail. *)
Tactic Notation "iter" tactic(tac) tactic(l) :=
351
  let rec go l :=
352
  match l with ?x :: ?l => tac x || go l end in go l.
353

Robbert Krebbers's avatar
Robbert Krebbers committed
354
(** Given [H : A_1 → ... → A_n → B] (where each [A_i] is non-dependent), the
355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373
tactic [feed tac H tac_by] creates a subgoal for each [A_i] and calls [tac p]
with the generated proof [p] of [B]. *)
Tactic Notation "feed" tactic(tac) constr(H) :=
  let rec go H :=
  let T := type of H in
  lazymatch eval hnf in T with
  | ?T1  ?T2 =>
    (* Use a separate counter for fresh names to make it more likely that
    the generated name is "fresh" with respect to those generated before
    calling the [feed] tactic. In particular, this hack makes sure that
    tactics like [let H' := fresh in feed (fun p => pose proof p as H') H] do
    not break. *)
    let HT1 := fresh "feed" in assert T1 as HT1;
      [| go (H HT1); clear HT1 ]
  | ?T1 => tac H
  end in go H.

(** The tactic [efeed tac H] is similar to [feed], but it also instantiates
dependent premises of [H] with evars. *)
374
Tactic Notation "efeed" constr(H) "using" tactic3(tac) "by" tactic3 (bytac) :=
375 376 377 378 379
  let rec go H :=
  let T := type of H in
  lazymatch eval hnf in T with
  | ?T1  ?T2 =>
    let HT1 := fresh "feed" in assert T1 as HT1;
380
      [bytac | go (H HT1); clear HT1 ]
381 382 383 384 385 386
  | ?T1  _ =>
    let e := fresh "feed" in evar (e:T1);
    let e' := eval unfold e in e in
    clear e; go (H e')
  | ?T1 => tac H
  end in go H.
387 388
Tactic Notation "efeed" constr(H) "using" tactic3(tac) :=
  efeed H using tac by idtac.
389 390 391 392 393 394 395 396 397

(** The following variants of [pose proof], [specialize], [inversion], and
[destruct], use the [feed] tactic before invoking the actual tactic. *)
Tactic Notation "feed" "pose" "proof" constr(H) "as" ident(H') :=
  feed (fun p => pose proof p as H') H.
Tactic Notation "feed" "pose" "proof" constr(H) :=
  feed (fun p => pose proof p) H.

Tactic Notation "efeed" "pose" "proof" constr(H) "as" ident(H') :=
398
  efeed H using (fun p => pose proof p as H').
399
Tactic Notation "efeed" "pose" "proof" constr(H) :=
400
  efeed H using (fun p => pose proof p).
401 402 403 404

Tactic Notation "feed" "specialize" hyp(H) :=
  feed (fun p => specialize p) H.
Tactic Notation "efeed" "specialize" hyp(H) :=
405
  efeed H using (fun p => specialize p).
406 407 408 409 410 411 412 413 414 415 416

Tactic Notation "feed" "inversion" constr(H) :=
  feed (fun p => let H':=fresh in pose proof p as H'; inversion H') H.
Tactic Notation "feed" "inversion" constr(H) "as" simple_intropattern(IP) :=
  feed (fun p => let H':=fresh in pose proof p as H'; inversion H' as IP) H.

Tactic Notation "feed" "destruct" constr(H) :=
  feed (fun p => let H':=fresh in pose proof p as H'; destruct H') H.
Tactic Notation "feed" "destruct" constr(H) "as" simple_intropattern(IP) :=
  feed (fun p => let H':=fresh in pose proof p as H'; destruct H' as IP) H.

417 418 419 420 421 422 423
(** The block definitions are taken from [Coq.Program.Equality] and can be used
by tactics to separate their goal from hypotheses they generalize over. *)
Definition block {A : Type} (a : A) := a.

Ltac block_goal := match goal with [ |- ?T ] => change (block T) end.
Ltac unblock_goal := unfold block in *.

424 425 426 427 428

(** The following tactic can be used to add support for patterns to tactic notation:
It will search for the first subterm of the goal matching [pat], and then call [tac]
with that subterm. *)
Ltac find_pat pat tac :=
429 430 431 432 433
  match goal with
  |- context [?x] =>
      unify pat x with typeclass_instances;
      tryif tac x then idtac else fail 2
  end.
434

435
(** Coq's [firstorder] tactic fails or loops on rather small goals already. In 
436 437 438 439
particular, on those generated by the tactic [unfold_elem_ofs] which is used
to solve propositions on collections. The [naive_solver] tactic implements an
ad-hoc and incomplete [firstorder]-like solver using Ltac's backtracking
mechanism. The tactic suffers from the following limitations:
440
- It might leave unresolved evars as Ltac provides no way to detect that.
441 442
- To avoid the tactic becoming too slow, we allow a universally quantified
  hypothesis to be instantiated only once during each search path.
443 444 445
- It does not perform backtracking on instantiation of universally quantified
  assumptions.

446 447 448 449
We use a counter to make the search breath first. Breath first search ensures
that a minimal number of hypotheses is instantiated, and thus reduced the
posibility that an evar remains unresolved.

450 451 452
Despite these limitations, it works much better than Coq's [firstorder] tactic
for the purposes of this development. This tactic either fails or proves the
goal. *)
453 454 455 456
Lemma forall_and_distr (A : Type) (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.

457 458 459 460 461 462
(** The tactic [no_new_unsolved_evars tac] executes [tac] and fails if it
creates any new evars. This trick is by Jonathan Leivent, see:
https://coq.inria.fr/bugs/show_bug.cgi?id=3872 *)

Ltac no_new_unsolved_evars tac := exact ltac:(tac).

463 464
Tactic Notation "naive_solver" tactic(tac) :=
  unfold iff, not in *;
465 466
  repeat match goal with
  | H : context [ _, _  _ ] |- _ =>
467
    repeat setoid_rewrite forall_and_distr in H; revert H
468
  end;
469
  let rec go n :=
470 471 472 473 474
  repeat match goal with
  (**i intros *)
  | |-  _, _ => intro
  (**i simplification of assumptions *)
  | H : False |- _ => destruct H
475
  | H : _  _ |- _ =>
476
     (* Work around bug https://coq.inria.fr/bugs/show_bug.cgi?id=2901 *)
477 478 479 480 481
     let H1 := fresh in let H2 := fresh in
     destruct H as [H1 H2]; try clear H
  | H :  _, _  |- _ =>
     let x := fresh in let Hx := fresh in
     destruct H as [x Hx]; try clear H
Robbert Krebbers's avatar
Robbert Krebbers committed
482
  | H : ?P  ?Q, H2 : ?P |- _ => specialize (H H2)
483 484
  | H : Is_true (bool_decide _) |- _ => apply (bool_decide_unpack _) in H
  | H : Is_true (_ && _) |- _ => apply andb_True in H; destruct H
485
  (**i simplify and solve equalities *)
486
  | |- _ => progress simplify_eq/=
487
  (**i solve the goal *)
488
  | |- _ => fast_done
489 490
  (**i operations that generate more subgoals *)
  | |- _  _ => split
491 492
  | |- Is_true (bool_decide _) => apply (bool_decide_pack _)
  | |- Is_true (_ && _) => apply andb_True; split
493 494
  | H : _  _ |- _ =>
     let H1 := fresh in destruct H as [H1|H1]; try clear H
495 496 497
  (**i solve the goal using the user supplied tactic *)
  | |- _ => solve [tac]
  end;
498 499 500
  (**i use recursion to enable backtracking on the following clauses. *)
  match goal with
  (**i instantiation of the conclusion *)
501
  | |-  x, _ => no_new_unsolved_evars ltac:(eexists; go n)
502 503 504 505 506 507
  | |- _  _ => first [left; go n | right; go n]
  | _ =>
    (**i instantiations of assumptions. *)
    lazymatch n with
    | S ?n' =>
      (**i we give priority to assumptions that fit on the conclusion. *)
508
      match goal with
509 510
      | H : _  _ |- _ =>
        is_non_dependent H;
511 512
        no_new_unsolved_evars
          ltac:(first [eapply H | efeed pose proof H]; clear H; go n')
513 514 515
      end
    end
  end
516
  in iter (fun n' => go n') (eval compute in (seq 1 6)).
517
Tactic Notation "naive_solver" := naive_solver eauto.