tactics.v 14.9 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects general purpose tactics that are used throughout
the development. *)
5
6
7
From Coq Require Import Omega.
From Coq Require Export Psatz.
From prelude Require Export base.
Robbert Krebbers's avatar
Robbert Krebbers committed
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

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.

(** 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.
Hint Extern 1000 => lia : lia.
Ralf Jung's avatar
Ralf Jung committed
28
Hint Extern 1000 => omega : omega.
Robbert Krebbers's avatar
Robbert Krebbers committed
29
30
Hint Extern 1001 => progress subst : subst. (** backtracking on this one will
be very bad, so use with care! *)
Robbert Krebbers's avatar
Robbert Krebbers committed
31
32
33

(** 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
34
declarations as the ones above. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
35
36
37
38
39
40
41
42
43
44
45
46
Tactic Notation "intuition" := intuition auto.

(** A slightly modified version of Ssreflect's finishing tactic [done]. It
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]. *)
Ltac done :=
  trivial; intros; solve
  [ repeat first
    [ solve [trivial]
    | solve [symmetry; trivial]
Robbert Krebbers's avatar
Robbert Krebbers committed
47
    | eassumption
Robbert Krebbers's avatar
Robbert Krebbers committed
48
49
50
51
52
53
54
55
56
    | reflexivity
    | discriminate
    | contradiction
    | solve [apply not_symmetry; trivial]
    | split ]
  | match goal with H : ¬_ |- _ => solve [destruct H; trivial] end ].
Tactic Notation "by" tactic(tac) :=
  tac; done.

57
58
59
60
61
62
63
64
65
66
67
68
69
70
(** 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. *)
Tactic Notation "split_and" := match goal with |- _  _ => split end.
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.
Robbert Krebbers's avatar
Robbert Krebbers committed
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128

(** 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. *)
Ltac case_match :=
  match goal with
  | H : context [ match ?x with _ => _ end ] |- _ => destruct x eqn:?
  | |- context [ match ?x with _ => _ end ] => destruct x eqn:?
  end.

(** 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].

(** 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.

(** 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
hypotheses depend on [H]. *)
Tactic Notation "is_non_dependent" constr(H) :=
  match goal with
  | _ : context [ H ] |- _ => fail 1
  | |- context [ H ] => fail 1
  | _ => idtac
  end.

(** The tactic [var_eq x y] fails if [x] and [y] are unequal, and [var_neq]
does the converse. *)
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
129
130
131
132
133
134
135
(** 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 *)
Robbert Krebbers's avatar
Robbert Krebbers committed
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
Ltac fold_classes :=
  repeat match goal with
  | |- appcontext [ ?F ] =>
    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.
Ltac fold_classes_hyps H :=
  repeat match type of H with
  | appcontext [ ?F ] =>
    progress match type of F with
    | FMap _ =>
       change F with (@fmap _ F) in H;
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F) in H
    | MBind _ =>
       change F with (@mbind _ F) in H;
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F) in H
    | OMap _ =>
       change F with (@omap _ F) in H;
       repeat change (@omap _ (@omap _ F)) with (@omap _ F) in H
    | Alter _ _ _ =>
       change F with (@alter _ _ _ F) in H;
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F) in H
    end
  end.
Tactic Notation "csimpl" "in" hyp(H) :=
  try (progress simpl in H; fold_classes_hyps H).
Tactic Notation "csimpl" := try (progress simpl; fold_classes).
Tactic Notation "csimpl" "in" "*" :=
  repeat_on_hyps (fun H => csimpl in H); csimpl.

Robbert Krebbers's avatar
Robbert Krebbers committed
178
(** The tactic [simplify_eq] repeatedly substitutes, discriminates,
179
180
and injects equalities, and tries to contradict impossible inequalities. *)
Tactic Notation "simplify_eq" := repeat
Robbert Krebbers's avatar
Robbert Krebbers committed
181
182
183
184
185
186
  match goal with
  | H : _  _ |- _ => by destruct H
  | H : _ = _  False |- _ => by destruct H
  | H : ?x = _ |- _ => subst x
  | H : _ = ?x |- _ => subst x
  | H : _ = _ |- _ => discriminate H
187
188
  | 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
189
190
191
192
193
194
195
196
    (* before [injection] to circumvent bug #2939 in some situations *)
  | H : ?f _ = ?f _ |- _ => injection H as H
    (* first hyp will be named [H], subsequent hyps will be given fresh names *)
  | H : ?f _ _ = ?f _ _ |- _ => injection H as H
  | H : ?f _ _ _ = ?f _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ = ?f _ _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ _ = ?f _ _ _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ _ _ = ?f _ _ _ _ _ _ |- _ => injection H as H
Robbert Krebbers's avatar
Robbert Krebbers committed
197
198
199
200
201
202
  | H : ?x = ?x |- _ => clear H
    (* 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
  end.
203
204
205
Tactic Notation "simplify_eq" "/=" :=
  repeat (progress csimpl in * || simplify_eq).
Tactic Notation "f_equal" "/=" := csimpl in *; f_equal.
Robbert Krebbers's avatar
Robbert Krebbers committed
206

Robbert Krebbers's avatar
Robbert Krebbers committed
207
Ltac setoid_subst_aux R x :=
Robbert Krebbers's avatar
Robbert Krebbers committed
208
  match goal with
Robbert Krebbers's avatar
Robbert Krebbers committed
209
  | H : R x ?y |- _ =>
Robbert Krebbers's avatar
Robbert Krebbers committed
210
211
212
213
214
215
216
217
     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;
218
     clear x H
Robbert Krebbers's avatar
Robbert Krebbers committed
219
220
221
  end.
Ltac setoid_subst :=
  repeat match goal with
222
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
223
224
  | 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
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
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
  end.

(** 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) :=
  let rec go l :=
  match l with ?x :: ?l => tac x || go l end in go l.

(** Given H : [A_1  ...  A_n  B] (where each [A_i] is non-dependent), the
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. *)
Tactic Notation "efeed" constr(H) "using" tactic3(tac) "by" tactic3 (bytac) :=
  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;
      [bytac | go (H HT1); clear HT1 ]
  | ?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.
Tactic Notation "efeed" constr(H) "using" tactic3(tac) :=
  efeed H using tac by idtac.

(** 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') :=
  efeed H using (fun p => pose proof p as H').
Tactic Notation "efeed" "pose" "proof" constr(H) :=
  efeed H using (fun p => pose proof p).

Tactic Notation "feed" "specialize" hyp(H) :=
  feed (fun p => specialize p) H.
Tactic Notation "efeed" "specialize" hyp(H) :=
  efeed H using (fun p => specialize p).

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.

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

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.

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. *)
Lemma forall_and_distr (A : Type) (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.

Tactic Notation "naive_solver" tactic(tac) :=
  unfold iff, not in *;
  repeat match goal with
  | H : context [ _, _  _ ] |- _ =>
    repeat setoid_rewrite forall_and_distr in H; revert H
  end;
  let rec go n :=
  repeat match goal with
  (**i intros *)
  | |-  _, _ => intro
  (**i simplification of assumptions *)
  | H : False |- _ => destruct H
  | H : _  _ |- _ => destruct H
  | H :  _, _  |- _ => destruct H
Robbert Krebbers's avatar
Robbert Krebbers committed
333
  | H : ?P  ?Q, H2 : ?P |- _ => specialize (H H2)
Robbert Krebbers's avatar
Robbert Krebbers committed
334
  (**i simplify and solve equalities *)
335
  | |- _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
  (**i solve the goal *)
  | |- _ =>
    solve
    [ eassumption
    | symmetry; eassumption
    | apply not_symmetry; eassumption
    | reflexivity ]
  (**i operations that generate more subgoals *)
  | |- _  _ => split
  | H : _  _ |- _ => destruct H
  (**i solve the goal using the user supplied tactic *)
  | |- _ => solve [tac]
  end;
  (**i use recursion to enable backtracking on the following clauses. *)
  match goal with
  (**i instantiation of the conclusion *)
  | |-  x, _ => eexists; go n
  | |- _  _ => 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. *)
      match goal with 
      | H : _  _ |- _ =>
        is_non_dependent H;
        eapply H; clear H; go n'
      | H : _  _ |- _ =>
        is_non_dependent H;
        try (eapply H; fail 2);
        efeed pose proof H; clear H; go n'
      end
    end
  end
  in iter (fun n' => go n') (eval compute in (seq 0 6)).
Tactic Notation "naive_solver" := naive_solver eauto.