tactics.v 23.3 KB
Newer Older
1
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
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.

Ralf Jung's avatar
Ralf Jung committed
38 39 40 41
(** [done] can get slow as it calls "trivial". [fast_done] can solve way less
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 *)
42
Ltac fast_done :=
43 44 45 46 47
  solve
    [ eassumption
    | symmetry; eassumption
    | apply not_symmetry; eassumption
    | reflexivity ].
48 49 50
Tactic Notation "fast_by" tactic(tac) :=
  tac; fast_done.

51
(** A slightly modified version of Ssreflect's finishing tactic [done]. It
52 53 54 55
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]. *)
56
Ltac done :=
57
  solve
58
  [ repeat first
59 60
    [ fast_done
    | solve [trivial]
61 62 63
    (* 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
64
    | solve [symmetry; trivial]
65
    | solve [apply not_symmetry; trivial]
66 67
    | discriminate
    | contradiction
68
    | split
Robbert Krebbers's avatar
Robbert Krebbers committed
69
    | match goal with H : ¬_ |- _ => case H; clear H; fast_done end ]
70
  ].
71 72 73
Tactic Notation "by" tactic(tac) :=
  tac; done.

74 75 76 77 78 79
Ltac done_if b :=
  match b with
  | true => done
  | false => idtac
  end.

80 81 82 83
(** Aliases for trans and etrans that are easier to type *)
Tactic Notation "trans" constr(A) := transitivity A.
Tactic Notation "etrans" := etransitivity.

84 85 86 87 88 89 90 91 92 93 94
(** 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. *)
95 96 97 98 99
Tactic Notation "split_and" :=
  match goal with
  | |- _  _ => split
  | |- Is_true (_ && _) => apply andb_True; split
  end.
100 101
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.
102

103 104 105 106 107 108 109 110 111
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?).

112 113 114
(** 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. *)
115 116 117 118 119 120
Ltac case_match :=
  match goal with
  | H : context [ match ?x with _ => _ end ] |- _ => destruct x eqn:?
  | |- context [ match ?x with _ => _ end ] => destruct x eqn:?
  end.

121 122 123 124
(** 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].
125 126 127 128 129 130

(** 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.

131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155
(** 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
156
hypotheses depend on [H]. *)
157 158 159 160 161 162 163
Tactic Notation "is_non_dependent" constr(H) :=
  match goal with
  | _ : context [ H ] |- _ => fail 1
  | |- context [ H ] => fail 1
  | _ => idtac
  end.

164 165
(** The tactic [var_eq x y] fails if [x] and [y] are unequal, and [var_neq]
does the converse. *)
166 167 168
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
169 170 171 172 173 174 175
(** 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 *)
176 177
Ltac fold_classes :=
  repeat match goal with
178
  | |- context [ ?F ] =>
179 180 181 182 183 184 185 186 187 188 189 190 191 192 193
    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.
194 195
Ltac fold_classes_hyps H :=
  repeat match type of H with
196
  | context [ ?F ] =>
197 198
    progress match type of F with
    | FMap _ =>
199 200
       change F with (@fmap _ F) in H;
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F) in H
201
    | MBind _ =>
202 203
       change F with (@mbind _ F) in H;
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F) in H
204
    | OMap _ =>
205 206
       change F with (@omap _ F) in H;
       repeat change (@omap _ (@omap _ F)) with (@omap _ F) in H
207
    | Alter _ _ _ =>
208 209
       change F with (@alter _ _ _ F) in H;
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F) in H
210 211
    end
  end.
212 213
Tactic Notation "csimpl" "in" hyp(H) :=
  try (progress simpl in H; fold_classes_hyps H).
214
Tactic Notation "csimpl" := try (progress simpl; fold_classes).
215 216
Tactic Notation "csimpl" "in" "*" :=
  repeat_on_hyps (fun H => csimpl in H); csimpl.
217

Robbert Krebbers's avatar
Robbert Krebbers committed
218
(** The tactic [simplify_eq] repeatedly substitutes, discriminates,
219 220
and injects equalities, and tries to contradict impossible inequalities. *)
Tactic Notation "simplify_eq" := repeat
221
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
222 223
  | H : _  _ |- _ => by case H; try clear H
  | H : _ = _  False |- _ => by case H; try clear H
224 225
  | H : ?x = _ |- _ => subst x
  | H : _ = ?x |- _ => subst x
226
  | H : _ = _ |- _ => discriminate H
227
  | H : _  _ |- _ => apply leibniz_equiv in H
228 229
  | 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
230
    (* before [injection] to circumvent bug #2939 in some situations *)
231
  | H : ?f _ = ?f _ |- _ => progress injection H as H
Robbert Krebbers's avatar
Robbert Krebbers committed
232
    (* first hyp will be named [H], subsequent hyps will be given fresh names *)
233 234 235 236 237
  | 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
238
  | H : ?x = ?x |- _ => clear H
239 240 241 242
    (* 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
243
  | H : @existT ?A _ _ _ = existT _ _ |- _ =>
244
     apply (Eqdep_dec.inj_pair2_eq_dec _ (decide_rel (=@{A}))) in H
245
  end.
246 247 248
Tactic Notation "simplify_eq" "/=" :=
  repeat (progress csimpl in * || simplify_eq).
Tactic Notation "f_equal" "/=" := csimpl in *; f_equal.
249

Robbert Krebbers's avatar
Robbert Krebbers committed
250
Ltac setoid_subst_aux R x :=
Robbert Krebbers's avatar
Robbert Krebbers committed
251
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
252
  | H : R x ?y |- _ =>
Robbert Krebbers's avatar
Robbert Krebbers committed
253 254 255 256 257 258 259 260
     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;
261
     clear x H
Robbert Krebbers's avatar
Robbert Krebbers committed
262 263 264
  end.
Ltac setoid_subst :=
  repeat match goal with
265
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
266 267
  | 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
268 269
  end.

270 271
(** 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
272
it. The tactic is somewhat limited, since it cannot be used to backtrack on
Ralf Jung's avatar
Ralf Jung committed
273
the Proper instances that has been found. To that end, we try to avoid the
274
trivial instance in which the resulting goals have an [eq]. More generally,
Ralf Jung's avatar
Ralf Jung committed
275
we try to "maintain" the relation of the current goal. For example,
276
when having [Proper (equiv ==> dist) f] and [Proper (dist ==> dist) f], it will
Ralf Jung's avatar
Ralf Jung committed
277
favor the second because the relation (dist) stays the same. *)
278
Ltac f_equiv :=
279
  match goal with
280
  | |- pointwise_relation _ _ _ _ => intros ?
281 282
  (* We support matches on both sides, *if* they concern the same variable, or
     variables in some relation. *)
283
  | |- ?R (match ?x with _ => _ end) (match ?x with _ => _ end) =>
284
    destruct x
285 286
  | H : ?R ?x ?y |- ?R2 (match ?x with _ => _ end) (match ?y with _ => _ end) =>
     destruct H
287
  (* First assume that the arguments need the same relation as the result *)
288 289 290 291
  | |- ?R (?f _) _ => simple apply (_ : Proper (R ==> R) f)
  | |- ?R (?f _ _) _ => simple apply (_ : Proper (R ==> R ==> R) f)
  | |- ?R (?f _ _ _) _ => simple apply (_ : Proper (R ==> R ==> R ==> R) f)
  | |- ?R (?f _ _ _ _) _ => simple apply (_ : Proper (R ==> R ==> R ==> R ==> R) f)
292 293
  (* For the case in which R is polymorphic, or an operational type class,
  like equiv. *)
294 295 296 297 298 299 300 301 302 303 304 305
  | |- (?R _) (?f _) _ => simple apply (_ : Proper (R _ ==> _) f)
  | |- (?R _ _) (?f _) _ => simple apply (_ : Proper (R _ _ ==> _) f)
  | |- (?R _ _ _) (?f _) _ => simple apply (_ : Proper (R _ _ _ ==> _) f)
  | |- (?R _) (?f _ _) _ => simple apply (_ : Proper (R _ ==> R _ ==> _) f)
  | |- (?R _ _) (?f _ _) _ => simple apply (_ : Proper (R _ _ ==> R _ _ ==> _) f)
  | |- (?R _ _ _) (?f _ _) _ => simple apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> _) f)
  | |- (?R _) (?f _ _ _) _ => simple apply (_ : Proper (R _ ==> R _ ==> R _ ==> _) f)
  | |- (?R _ _) (?f _ _ _) _ => simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _ ==> _) f)
  | |- (?R _ _ _) (?f _ _ _) _ => simple apply (_ : Proper (R _ _ _ ==> R _ _ _ R _ _ _ ==> _) f)
  | |- (?R _) (?f _ _ _ _) _ => simple apply (_ : Proper (R _ ==> R _ ==> R _ ==> R _ ==> _) f)
  | |- (?R _ _) (?f _ _ _ _) _ => simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _ ==> R _ _ ==> _) f)
  | |- (?R _ _ _) (?f _ _ _ _) _ => simple apply (_ : Proper (R _ _ _ ==> R _ _ _ R _ _ _ ==> R _ _ _ ==> _) f)
Ralf Jung's avatar
Ralf Jung committed
306 307
  (* Next, try to infer the relation. Unfortunately, very often, it will turn
     the goal into a Leibniz equality so we get stuck. *)
308
  (* TODO: Can we exclude that instance? *)
309 310 311 312
  | |- ?R (?f _) _ => simple apply (_ : Proper (_ ==> R) f)
  | |- ?R (?f _ _) _ => simple apply (_ : Proper (_ ==> _ ==> R) f)
  | |- ?R (?f _ _ _) _ => simple apply (_ : Proper (_ ==> _ ==> _ ==> R) f)
  | |- ?R (?f _ _ _ _) _ => simple apply (_ : Proper (_ ==> _ ==> _ ==> _ ==> R) f)
Ralf Jung's avatar
Ralf Jung committed
313 314 315 316 317
  (* In case the function symbol differs, but the arguments are the same,
     maybe we have a pointwise_relation in our context. *)
  (* TODO: If only 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. *)
318
  | H : pointwise_relation _ ?R ?f ?g |- ?R (?f ?x) (?g ?x) => simple apply H
319
  | H : pointwise_relation _ (pointwise_relation _ ?R) ?f ?g |- ?R (?f ?x ?y) (?g ?x ?y) => simple apply H
320
  end;
321
  try simple apply reflexivity.
322
Tactic Notation "f_equiv" "/=" := csimpl in *; f_equiv.
323

Ralf Jung's avatar
Ralf Jung committed
324
(** The tactic [solve_proper_unfold] unfolds the first head symbol, so that
325
we proceed by repeatedly using [f_equiv]. *)
326 327
Ltac solve_proper_unfold :=
  (* Try unfolding the head symbol, which is the one we are proving a new property about *)
328
  lazymatch goal with
329 330 331 332
  | |- ?R (?f _ _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _) (?f _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _) (?f _ _ _ _ _) => unfold f
333 334 335 336
  | |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
  | |- ?R (?f _ _ _) (?f _ _ _) => unfold f
  | |- ?R (?f _ _) (?f _ _) => unfold f
  | |- ?R (?f _) (?f _) => unfold f
337
  end.
Ralf Jung's avatar
Ralf Jung committed
338 339 340
(** [solve_proper_prepare] does some preparation work before the main
[solve_proper] loop.  Having this as a separate tactic is useful for debugging
[solve_proper] failure. *)
341
Ltac solve_proper_prepare :=
342 343 344 345 346 347
  (* Introduce everything *)
  intros;
  repeat lazymatch goal with
  | |- Proper _ _ => intros ???
  | |- (_ ==> _)%signature _ _ => intros ???
  | |- pointwise_relation _ _ _ _ => intros ?
Ralf Jung's avatar
Ralf Jung committed
348
  | |- ?R ?f _ => let f' := constr:(λ x, f x) in intros ?
349
  end; simplify_eq;
350
  (* We try with and without unfolding. We have to backtrack on
351
     that because unfolding may succeed, but then the proof may fail. *)
352 353 354 355 356 357 358
  (solve_proper_unfold + idtac); simpl.
(** The tactic [solve_proper_core tac] solves goals of the form "Proper (R1 ==> R2)", for
any number of relations. The actual work is done by repeatedly applying
[tac]. *)
Ltac solve_proper_core tac :=
  solve_proper_prepare;
  (* Now do the job. *)
359
  solve [repeat first [eassumption | tac ()] ].
360 361

(** Finally, [solve_proper] tries to apply [f_equiv] in a loop. *)
362
Ltac solve_proper := solve_proper_core ltac:(fun _ => f_equiv).
363

364 365 366 367 368 369 370
(** 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.
371

372 373 374 375
(** 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) :=
376
  let rec go l :=
377
  match l with ?x :: ?l => tac x || go l end in go l.
378

Robbert Krebbers's avatar
Robbert Krebbers committed
379
(** Given [H : A_1 → ... → A_n → B] (where each [A_i] is non-dependent), the
380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398
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. *)
399
Tactic Notation "efeed" constr(H) "using" tactic3(tac) "by" tactic3 (bytac) :=
400 401 402 403 404
  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;
405
      [bytac | go (H HT1); clear HT1 ]
406 407 408 409 410 411
  | ?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.
412 413
Tactic Notation "efeed" constr(H) "using" tactic3(tac) :=
  efeed H using tac by idtac.
414 415 416 417 418 419 420 421 422

(** 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') :=
423
  efeed H using (fun p => pose proof p as H').
424
Tactic Notation "efeed" "pose" "proof" constr(H) :=
425
  efeed H using (fun p => pose proof p).
426 427 428 429

Tactic Notation "feed" "specialize" hyp(H) :=
  feed (fun p => specialize p) H.
Tactic Notation "efeed" "specialize" hyp(H) :=
430
  efeed H using (fun p => specialize p).
431 432 433 434 435 436 437 438 439 440 441

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.

442 443 444 445 446 447 448
(** 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 *.

449 450 451 452 453

(** 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 :=
454 455 456 457 458
  match goal with
  |- context [?x] =>
      unify pat x with typeclass_instances;
      tryif tac x then idtac else fail 2
  end.
459

460
(** Coq's [firstorder] tactic fails or loops on rather small goals already. In 
461 462 463 464
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:
465
- It might leave unresolved evars as Ltac provides no way to detect that.
466 467
- To avoid the tactic becoming too slow, we allow a universally quantified
  hypothesis to be instantiated only once during each search path.
468 469 470
- It does not perform backtracking on instantiation of universally quantified
  assumptions.

471 472 473 474
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.

475 476 477
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. *)
478 479 480 481
Lemma forall_and_distr (A : Type) (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.

482 483 484 485 486 487
(** 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).

488 489
Tactic Notation "naive_solver" tactic(tac) :=
  unfold iff, not in *;
490 491
  repeat match goal with
  | H : context [ _, _  _ ] |- _ =>
492
    repeat setoid_rewrite forall_and_distr in H; revert H
493
  end;
494
  let rec go n :=
495 496 497 498 499
  repeat match goal with
  (**i intros *)
  | |-  _, _ => intro
  (**i simplification of assumptions *)
  | H : False |- _ => destruct H
500
  | H : _  _ |- _ =>
501
     (* Work around bug https://coq.inria.fr/bugs/show_bug.cgi?id=2901 *)
502 503 504 505 506
     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
507
  | H : ?P  ?Q, H2 : ?P |- _ => specialize (H H2)
508 509
  | H : Is_true (bool_decide _) |- _ => apply (bool_decide_unpack _) in H
  | H : Is_true (_ && _) |- _ => apply andb_True in H; destruct H
510
  (**i simplify and solve equalities *)
511
  | |- _ => progress simplify_eq/=
512
  (**i solve the goal *)
513
  | |- _ => fast_done
514 515
  (**i operations that generate more subgoals *)
  | |- _  _ => split
516 517
  | |- Is_true (bool_decide _) => apply (bool_decide_pack _)
  | |- Is_true (_ && _) => apply andb_True; split
518 519
  | H : _  _ |- _ =>
     let H1 := fresh in destruct H as [H1|H1]; try clear H
520 521
  | H : Is_true (_ || _) |- _ =>
     apply orb_True in H; let H1 := fresh in destruct H as [H1|H1]; try clear H
522 523 524
  (**i solve the goal using the user supplied tactic *)
  | |- _ => solve [tac]
  end;
525 526 527
  (**i use recursion to enable backtracking on the following clauses. *)
  match goal with
  (**i instantiation of the conclusion *)
528
  | |-  x, _ => no_new_unsolved_evars ltac:(eexists; go n)
529
  | |- _  _ => first [left; go n | right; go n]
530
  | |- Is_true (_ || _) => apply orb_True; first [left; go n | right; go n]
531 532 533 534 535
  | _ =>
    (**i instantiations of assumptions. *)
    lazymatch n with
    | S ?n' =>
      (**i we give priority to assumptions that fit on the conclusion. *)
536
      match goal with
537 538
      | H : _  _ |- _ =>
        is_non_dependent H;
539 540
        no_new_unsolved_evars
          ltac:(first [eapply H | efeed pose proof H]; clear H; go n')
541 542 543
      end
    end
  end
544
  in iter (fun n' => go n') (eval compute in (seq 1 6)).
545
Tactic Notation "naive_solver" := naive_solver eauto.