aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Program/Tactics.v
blob: c1d958b9d35d46deb99d2d93a5f1eb7e64a15356 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
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
129
130
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
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(** This module implements various tactics used to simplify the goals produced by Program,
   which are also generally useful. *)

(** Debugging tactics to show the goal during evaluation. *)

Ltac show_goal := match goal with [ |- ?T ] => idtac T end.

Ltac show_hyp id :=
  match goal with
    | [ H := ?b : ?T |- _ ] =>
      match H with
        | id => idtac id ":=" b ":" T
      end
    | [ H : ?T |- _ ] =>
      match H with
        | id => idtac id  ":"  T
      end
  end.

Ltac show_hyps :=
  try match reverse goal with
        | [ H : ?T |- _ ] => show_hyp H ; fail
      end.

(** The [do] tactic but using a Coq-side nat. *)

Ltac do_nat n tac :=
  match n with
    | 0 => idtac
    | S ?n' => tac ; do_nat n' tac
  end.

(** Do something on the last hypothesis, or fail *)

Ltac on_last_hyp tac :=
  lazymatch goal with [ H : _ |- _ ] => tac H end.

(** Destructs one pair, without care regarding naming. *)

Ltac destruct_one_pair :=
 match goal with
   | [H : (_ /\ _) |- _] => destruct H
   | [H : prod _ _ |- _] => destruct H
 end.

(** Repeateadly destruct pairs. *)

Ltac destruct_pairs := repeat (destruct_one_pair).

(** Destruct one existential package, keeping the name of the hypothesis for the first component. *)

Ltac destruct_one_ex :=
  let tac H := let ph := fresh "H" in (destruct H as [H ph]) in
  let tac2 H := let ph := fresh "H" in let ph' := fresh "H" in 
    (destruct H as [H ph ph']) 
  in
  let tacT H := let ph := fresh "X" in (destruct H as [H ph]) in
  let tacT2 H := let ph := fresh "X" in let ph' := fresh "X" in 
    (destruct H as [H ph ph']) 
  in
    match goal with
      | [H : (ex _) |- _] => tac H
      | [H : (sig ?P) |- _ ] => tac H
      | [H : (sigT ?P) |- _ ] => tacT H
      | [H : (ex2 _ _) |- _] => tac2 H
      | [H : (sig2 ?P _) |- _ ] => tac2 H
      | [H : (sigT2 ?P _) |- _ ] => tacT2 H
    end.

(** Repeateadly destruct existentials. *)

Ltac destruct_exists := repeat (destruct_one_ex).

(** Repeateadly destruct conjunctions and existentials. *)

Ltac destruct_conjs := repeat (destruct_one_pair || destruct_one_ex).

(** Destruct an existential hypothesis [t] keeping its name for the first component
   and using [Ht] for the second *)

Tactic Notation "destruct" "exist" ident(t) ident(Ht) := destruct t as [t Ht].

(** Destruct a disjunction keeping its name in both subgoals. *)

Tactic Notation "destruct" "or" ident(H) := destruct H as [H|H].

(** Discriminate that also work on a [x <> x] hypothesis. *)

Ltac discriminates :=
  match goal with
    | [ H : ?x <> ?x |- _ ] => elim H ; reflexivity
    | _ => discriminate
  end.

(** Revert the last hypothesis. *)

Ltac revert_last :=
  match goal with
    [ H : _ |- _ ] => revert H
  end.

(** Repeatedly reverse the last hypothesis, putting everything in the goal. *)

Ltac reverse := repeat revert_last.

(** Reverse everything up to hypothesis id (not included). *)

Ltac revert_until id :=
  on_last_hyp ltac:(fun id' =>
    match id' with
      | id => idtac
      | _ => revert id' ; revert_until id
    end).

(** Clear duplicated hypotheses *)

Ltac clear_dup :=
  match goal with
    | [ H : ?X |- _ ] =>
      match goal with
        | [ H' : ?Y |- _ ] =>
          match H with
            | H' => fail 2
            | _ => unify X Y ; (clear H' || clear H)
          end
      end
  end.

Ltac clear_dups := repeat clear_dup.

(** Try to clear everything except some hyp *)

Ltac clear_except hyp := 
  repeat match goal with [ H : _ |- _ ] =>
           match H with
             | hyp => fail 1
             | _ => clear H
           end
         end.

(** A non-failing subst that substitutes as much as possible. *)

Ltac subst_no_fail :=
  repeat (match goal with
            [ H : ?X = ?Y |- _ ] => subst X || subst Y
          end).

Tactic Notation "subst" "*" := subst_no_fail.

Ltac on_application f tac T :=
  match T with
    | context [f ?x ?y ?z ?w ?v ?u ?a ?b ?c] => tac (f x y z w v u a b c)
    | context [f ?x ?y ?z ?w ?v ?u ?a ?b] => tac (f x y z w v u a b)
    | context [f ?x ?y ?z ?w ?v ?u ?a] => tac (f x y z w v u a)
    | context [f ?x ?y ?z ?w ?v ?u] => tac (f x y z w v u)
    | context [f ?x ?y ?z ?w ?v] => tac (f x y z w v)
    | context [f ?x ?y ?z ?w] => tac (f x y z w)
    | context [f ?x ?y ?z] => tac (f x y z)
    | context [f ?x ?y] => tac (f x y)
    | context [f ?x] => tac (f x)
  end.

(** A variant of [apply] using [refine], doing as much conversion as necessary. *)

Ltac rapply p :=
  refine (p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _ _) ||
  refine (p _ _ _ _ _ _) ||
  refine (p _ _ _ _ _) ||
  refine (p _ _ _ _) ||
  refine (p _ _ _) ||
  refine (p _ _) ||
  refine (p _) ||
  refine p.

(** Tactical [on_call f tac] applies [tac] on any application of [f] in the hypothesis or goal. *)

Ltac on_call f tac :=
  match goal with
    | |- ?T  => on_application f tac T
    | H : ?T |- _  => on_application f tac T
  end.

(* Destructs calls to f in hypothesis or conclusion, useful if f creates a subset object. *)

Ltac destruct_call f :=
  let tac t := (destruct t) in on_call f tac.

Ltac destruct_calls f := repeat destruct_call f.

Ltac destruct_call_in f H :=
  let tac t := (destruct t) in
  let T := type of H in
    on_application f tac T.

Ltac destruct_call_as f l :=
  let tac t := (destruct t as l) in on_call f tac.

Ltac destruct_call_as_in f l H :=
  let tac t := (destruct t as l) in
  let T := type of H in
    on_application f tac T.

Tactic Notation "destruct_call" constr(f) := destruct_call f.

(** Permit to name the results of destructing the call to [f]. *)

Tactic Notation "destruct_call" constr(f) "as" simple_intropattern(l) :=
  destruct_call_as f l.

(** Specify the hypothesis in which the call occurs as well. *)

Tactic Notation "destruct_call" constr(f) "in" hyp(id) :=
  destruct_call_in f id.

Tactic Notation "destruct_call" constr(f) "as" simple_intropattern(l) "in" hyp(id) :=
  destruct_call_as_in f l id.

(** A marker for prototypes to destruct. *)

Definition fix_proto {A : Type} (a : A) := a.

Ltac destruct_rec_calls :=
  match goal with
    | [ H : fix_proto _ |- _ ] => destruct_calls H ; clear H
  end.

Ltac destruct_all_rec_calls :=
  repeat destruct_rec_calls ; unfold fix_proto in *.

(** Try to inject any potential constructor equality hypothesis. *)

Ltac autoinjection tac :=
  match goal with
    | [ H : ?f ?a = ?f' ?a' |- _ ] => tac H
  end.

Ltac inject H := progress (inversion H ; subst*; clear_dups) ; clear H.

Ltac autoinjections := repeat (clear_dups ; autoinjection ltac:inject).

(** Destruct an hypothesis by first copying it to avoid dependencies. *)

Ltac destruct_nondep H := let H0 := fresh "H" in assert(H0 := H); destruct H0.

(** If bang appears in the goal, it means that we have a proof of False and the goal is solved. *)

Ltac bang :=
  match goal with
    | |- ?x =>
      match x with
        | appcontext [False_rect _ ?p] => elim p
      end
  end.

(** A tactic to show contradiction by first asserting an automatically provable hypothesis. *)
Tactic Notation "contradiction" "by" constr(t) :=
  let H := fresh in assert t as H by auto with * ; contradiction.

(** A tactic that adds [H:=p:typeof(p)] to the context if no hypothesis of the same type appears in the goal.
   Useful to do saturation using tactics. *)

Ltac add_hypothesis H' p :=
  match type of p with
    ?X =>
    match goal with
      | [ H : X |- _ ] => fail 1
      | _ => set (H':=p) ; try (change p with H') ; clearbody H'
    end
  end.

(** A tactic to replace an hypothesis by another term. *)

Ltac replace_hyp H c :=
  let H' := fresh "H" in
    assert(H' := c) ; clear H ; rename H' into H.

(** A tactic to refine an hypothesis by supplying some of its arguments. *)

Ltac refine_hyp c :=
  let tac H := replace_hyp H c in
    match c with
      | ?H _ => tac H
      | ?H _ _ => tac H
      | ?H _ _ _ => tac H
      | ?H _ _ _ _ => tac H
      | ?H _ _ _ _ _ => tac H
      | ?H _ _ _ _ _ _ => tac H
      | ?H _ _ _ _ _ _ _ => tac H
      | ?H _ _ _ _ _ _ _ _ => tac H
    end.

(** The default simplification tactic used by Program is defined by [program_simpl], sometimes [auto]
   is not enough, better rebind using [Obligation Tactic := tac] in this case,
   possibly using [program_simplify] to use standard goal-cleaning tactics. *)

Ltac program_simplify :=
simpl; intros ; destruct_all_rec_calls ; repeat (destruct_conjs; simpl proj1_sig in * );
  subst*; autoinjections ; try discriminates ;
    try (solve [ red ; intros ; destruct_conjs ; autoinjections ; discriminates ]).

(** Restrict automation to propositional obligations. *)

Ltac program_solve_wf :=
  match goal with
    | |- well_founded _ => auto with *
    | |- ?T => match type of T with Prop => auto end
  end.

Create HintDb program discriminated.

Ltac program_simpl := program_simplify ; try typeclasses eauto with program ; try program_solve_wf.

Obligation Tactic := program_simpl.

Definition obligation (A : Type) {a : A} := a.