option.v 17.4 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 definitions and theorems on the option
data type that are not in the Coq standard library. *)
5
From iris.prelude Require Export tactics.
Robbert Krebbers's avatar
Robbert Krebbers committed
6
7
8
9
10
11
12

Inductive option_reflect {A} (P : A  Prop) (Q : Prop) : option A  Type :=
  | ReflectSome x : P x  option_reflect P Q (Some x)
  | ReflectNone : Q  option_reflect P Q None.

(** * General definitions and theorems *)
(** Basic properties about equality. *)
13
Lemma None_ne_Some {A} (x : A) : None  Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
14
Proof. congruence. Qed.
15
Lemma Some_ne_None {A} (x : A) : Some x  None.
Robbert Krebbers's avatar
Robbert Krebbers committed
16
Proof. congruence. Qed.
17
Lemma eq_None_ne_Some {A} (mx : option A) x : mx = None  mx  Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
18
Proof. congruence. Qed.
19
Instance Some_inj {A} : Inj (=) (=) (@Some A).
Robbert Krebbers's avatar
Robbert Krebbers committed
20
21
Proof. congruence. Qed.

22
23
(** The [from_option] is the eliminator for option. *)
Definition from_option {A B} (f : A  B) (y : B) (mx : option A) : B :=
24
  match mx with None => y | Some x => f x end.
25
26
Instance: Params (@from_option) 3.
Arguments from_option {_ _} _ _ !_ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
27

28
29
30
(* The eliminator again, but with the arguments in different order, which is
sometimes more convenient. *)
Notation default y mx f := (from_option f y mx) (only parsing).
Robbert Krebbers's avatar
Robbert Krebbers committed
31
32
33

(** An alternative, but equivalent, definition of equality on the option
data type. This theorem is useful to prove that two options are the same. *)
34
35
36
Lemma option_eq {A} (mx my: option A): mx = my   x, mx = Some x  my = Some x.
Proof. split; [by intros; by subst |]. destruct mx, my; naive_solver. Qed.
Lemma option_eq_1 {A} (mx my: option A) x : mx = my  mx = Some x  my = Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
37
Proof. congruence. Qed.
38
39
Lemma option_eq_1_alt {A} (mx my : option A) x :
  mx = my  my = Some x  mx = Some x.
Robbert Krebbers's avatar
Robbert Krebbers committed
40
41
Proof. congruence. Qed.

42
43
44
45
Definition is_Some {A} (mx : option A) :=  x, mx = Some x.
Instance: Params (@is_Some) 1.

Lemma mk_is_Some {A} (mx : option A) x : mx = Some x  is_Some mx.
Robbert Krebbers's avatar
Robbert Krebbers committed
46
47
48
49
50
51
Proof. intros; red; subst; eauto. Qed.
Hint Resolve mk_is_Some.
Lemma is_Some_None {A} : ¬is_Some (@None A).
Proof. by destruct 1. Qed.
Hint Resolve is_Some_None.

52
Instance is_Some_pi {A} (mx : option A) : ProofIrrel (is_Some mx).
Robbert Krebbers's avatar
Robbert Krebbers committed
53
Proof.
54
55
  set (P (mx : option A) := match mx with Some _ => True | _ => False end).
  set (f mx := match mx return P mx  is_Some mx with
Robbert Krebbers's avatar
Robbert Krebbers committed
56
    Some _ => λ _, ex_intro _ _ eq_refl | None => False_rect _ end).
57
58
59
60
  set (g mx (H : is_Some mx) :=
    match H return P mx with ex_intro _ p => eq_rect _ _ I _ (eq_sym p) end).
  assert ( mx H, f mx (g mx H) = H) as f_g by (by intros ? [??]; subst).
  intros p1 p2. rewrite <-(f_g _ p1), <-(f_g _ p2). by destruct mx, p1.
Robbert Krebbers's avatar
Robbert Krebbers committed
61
Qed.
62
63
Instance is_Some_dec {A} (mx : option A) : Decision (is_Some mx) :=
  match mx with
Robbert Krebbers's avatar
Robbert Krebbers committed
64
65
66
67
  | Some x => left (ex_intro _ x eq_refl)
  | None => right is_Some_None
  end.

68
69
70
71
72
Definition is_Some_proj {A} {mx : option A} : is_Some mx  A :=
  match mx with Some x => λ _, x | None => False_rect _  is_Some_None end.
Definition Some_dec {A} (mx : option A) : { x | mx = Some x } + { mx = None } :=
  match mx return { x | mx = Some x } + { mx = None } with
  | Some x => inleft (x  eq_refl _)
Robbert Krebbers's avatar
Robbert Krebbers committed
73
74
75
  | None => inright eq_refl
  end.

76
77
78
79
Lemma eq_None_not_Some {A} (mx : option A) : mx = None  ¬is_Some mx.
Proof. destruct mx; unfold is_Some; naive_solver. Qed.
Lemma not_eq_None_Some {A} (mx : option A) : mx  None  is_Some mx.
Proof. rewrite eq_None_not_Some; apply dec_stable; tauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
80
81

(** Lifting a relation point-wise to option *)
82
83
84
Inductive option_Forall2 {A B} (R: A  B  Prop) : option A  option B  Prop :=
  | Some_Forall2 x y : R x y  option_Forall2 R (Some x) (Some y)
  | None_Forall2 : option_Forall2 R None None.
Robbert Krebbers's avatar
Robbert Krebbers committed
85
86
87
88
89
90
91
92
93
Definition option_relation {A B} (R: A  B  Prop) (P: A  Prop) (Q: B  Prop)
    (mx : option A) (my : option B) : Prop :=
  match mx, my with
  | Some x, Some y => R x y
  | Some x, None => P x
  | None, Some y => Q y
  | None, None => True
  end.

94
95
96
97
98
99
100
101
102
103
104
105
106
Section Forall2.
  Context {A} (R : relation A).

  Global Instance option_Forall2_refl : Reflexive R  Reflexive (option_Forall2 R).
  Proof. intros ? [?|]; by constructor. Qed.
  Global Instance option_Forall2_sym : Symmetric R  Symmetric (option_Forall2 R).
  Proof. destruct 2; by constructor. Qed.
  Global Instance option_Forall2_trans : Transitive R  Transitive (option_Forall2 R).
  Proof. destruct 2; inversion_clear 1; constructor; etrans; eauto. Qed.
  Global Instance option_Forall2_equiv : Equivalence R  Equivalence (option_Forall2 R).
  Proof. destruct 1; split; apply _. Qed.
End Forall2.

Robbert Krebbers's avatar
Robbert Krebbers committed
107
(** Setoids *)
108
109
Instance option_equiv `{Equiv A} : Equiv (option A) := option_Forall2 ().

Robbert Krebbers's avatar
Robbert Krebbers committed
110
Section setoids.
111
  Context `{Equiv A} `{!Equivalence (() : relation A)}.
112
  Implicit Types mx my : option A.
113
114

  Lemma equiv_option_Forall2 mx my : mx  my  option_Forall2 () mx my.
115
  Proof. done. Qed.
116

117
  Global Instance option_equivalence : Equivalence (() : relation (option A)).
118
  Proof. apply _. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
119
120
121
  Global Instance Some_proper : Proper (() ==> ()) (@Some A).
  Proof. by constructor. Qed.
  Global Instance option_leibniz `{!LeibnizEquiv A} : LeibnizEquiv (option A).
122
  Proof. intros x y; destruct 1; fold_leibniz; congruence. Qed.
123

124
  Lemma equiv_None mx : mx  None  mx = None.
125
  Proof. split; [by inversion_clear 1|by intros ->]. Qed.
126
  Lemma equiv_Some_inv_l mx my x :
127
128
    mx  my  mx = Some x   y, my = Some y  x  y.
  Proof. destruct 1; naive_solver. Qed.
129
130
  Lemma equiv_Some_inv_r mx my y :
    mx  my  my = Some y   x, mx = Some x  x  y.
131
  Proof. destruct 1; naive_solver. Qed.
132
133
134
135
  Lemma equiv_Some_inv_l' my x : Some x  my   x', Some x' = my  x  x'.
  Proof. intros ?%(equiv_Some_inv_l _ _ x); naive_solver. Qed.
  Lemma equiv_Some_inv_r' mx y : mx  Some y   y', mx = Some y'  y  y'.
  Proof. intros ?%(equiv_Some_inv_r _ _ y); naive_solver. Qed.
136

Robbert Krebbers's avatar
Robbert Krebbers committed
137
138
  Global Instance is_Some_proper : Proper (() ==> iff) (@is_Some A).
  Proof. inversion_clear 1; split; eauto. Qed.
139
140
141
  Global Instance from_option_proper {B} (R : relation B) (f : A  B) :
    Proper (() ==> R) f  Proper (R ==> () ==> R) (from_option f).
  Proof. destruct 3; simpl; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
142
143
End setoids.

144
145
Typeclasses Opaque option_equiv.

Robbert Krebbers's avatar
Robbert Krebbers committed
146
(** Equality on [option] is decidable. *)
147
148
149
150
151
152
Instance option_eq_None_dec {A} (mx : option A) : Decision (mx = None) :=
  match mx with Some _ => right (Some_ne_None _) | None => left eq_refl end.
Instance option_None_eq_dec {A} (mx : option A) : Decision (None = mx) :=
  match mx with Some _ => right (None_ne_Some _) | None => left eq_refl end.
Instance option_eq_dec {A} {dec :  x y : A, Decision (x = y)}
  (mx my : option A) : Decision (mx = my).
Robbert Krebbers's avatar
Robbert Krebbers committed
153
154
Proof.
 refine
155
156
  match mx, my with
  | Some x, Some y => cast_if (decide (x = y))
Robbert Krebbers's avatar
Robbert Krebbers committed
157
158
159
160
161
162
  | None, None => left _ | _, _ => right _
  end; clear dec; abstract congruence.
Defined.

(** * Monadic operations *)
Instance option_ret: MRet option := @Some.
163
164
165
166
Instance option_bind: MBind option := λ A B f mx,
  match mx with Some x => f x | None => None end.
Instance option_join: MJoin option := λ A mmx,
  match mmx with Some mx => mx | None => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
167
Instance option_fmap: FMap option := @option_map.
168
169
Instance option_guard: MGuard option := λ P dec A f,
  match dec with left H => f H | _ => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
170

171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
Lemma fmap_is_Some {A B} (f : A  B) mx : is_Some (f <$> mx)  is_Some mx.
Proof. unfold is_Some; destruct mx; naive_solver. Qed.
Lemma fmap_Some {A B} (f : A  B) mx y :
  f <$> mx = Some y   x, mx = Some x  y = f x.
Proof. destruct mx; naive_solver. Qed.
Lemma fmap_None {A B} (f : A  B) mx : f <$> mx = None  mx = None.
Proof. by destruct mx. Qed.
Lemma option_fmap_id {A} (mx : option A) : id <$> mx = mx.
Proof. by destruct mx. Qed.
Lemma option_fmap_compose {A B} (f : A  B) {C} (g : B  C) mx :
  g  f <$> mx = g <$> f <$> mx.
Proof. by destruct mx. Qed.
Lemma option_fmap_ext {A B} (f g : A  B) mx :
  ( x, f x = g x)  f <$> mx = g <$> mx.
Proof. intros; destruct mx; f_equal/=; auto. Qed.
Lemma option_fmap_setoid_ext `{Equiv A, Equiv B} (f g : A  B) mx :
  ( x, f x  g x)  f <$> mx  g <$> mx.
Proof. destruct mx; constructor; auto. Qed.
Lemma option_fmap_bind {A B C} (f : A  B) (g : B  option C) mx :
  (f <$> mx) = g = mx = g  f.
Proof. by destruct mx. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
192
Lemma option_bind_assoc {A B C} (f : A  option B)
193
194
195
196
197
198
199
  (g : B  option C) (mx : option A) : (mx = f) = g = mx = (mbind g  f).
Proof. by destruct mx; simpl. Qed.
Lemma option_bind_ext {A B} (f g : A  option B) mx my :
  ( x, f x = g x)  mx = my  mx = f = my = g.
Proof. destruct mx, my; naive_solver. Qed.
Lemma option_bind_ext_fun {A B} (f g : A  option B) mx :
  ( x, f x = g x)  mx = f = mx = g.
Robbert Krebbers's avatar
Robbert Krebbers committed
200
Proof. intros. by apply option_bind_ext. Qed.
201
202
203
204
205
206
207
208
Lemma bind_Some {A B} (f : A  option B) (mx : option A) y :
  mx = f = Some y   x, mx = Some x  f x = Some y.
Proof. destruct mx; naive_solver. Qed.
Lemma bind_None {A B} (f : A  option B) (mx : option A) :
  mx = f = None  mx = None   x, mx = Some x  f x = None.
Proof. destruct mx; naive_solver. Qed.
Lemma bind_with_Some {A} (mx : option A) : mx = Some = mx.
Proof. by destruct mx. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
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

(** ** Inverses of constructors *)
(** We can do this in a fancy way using dependent types, but rewrite does
not particularly like type level reductions. *)
Class Maybe {A B : Type} (c : A  B) :=
  maybe : B  option A.
Arguments maybe {_ _} _ {_} !_ /.
Class Maybe2 {A1 A2 B : Type} (c : A1  A2  B) :=
  maybe2 : B  option (A1 * A2).
Arguments maybe2 {_ _ _} _ {_} !_ /.
Class Maybe3 {A1 A2 A3 B : Type} (c : A1  A2  A3  B) :=
  maybe3 : B  option (A1 * A2 * A3).
Arguments maybe3 {_ _ _ _} _ {_} !_ /.
Class Maybe4 {A1 A2 A3 A4 B : Type} (c : A1  A2  A3  A4  B) :=
  maybe4 : B  option (A1 * A2 * A3 * A4).
Arguments maybe4 {_ _ _ _ _} _ {_} !_ /.

Instance maybe_comp `{Maybe B C c1, Maybe A B c2} : Maybe (c1  c2) := λ x,
  maybe c1 x = maybe c2.
Arguments maybe_comp _ _ _ _ _ _ _ !_ /.

Instance maybe_inl {A B} : Maybe (@inl A B) := λ xy,
  match xy with inl x => Some x | _ => None end.
Instance maybe_inr {A B} : Maybe (@inr A B) := λ xy,
  match xy with inr y => Some y | _ => None end.
Instance maybe_Some {A} : Maybe (@Some A) := id.
Arguments maybe_Some _ !_ /.

(** * Union, intersection and difference *)
238
239
240
241
242
Instance option_union_with {A} : UnionWith A (option A) := λ f mx my,
  match mx, my with
  | Some x, Some y => f x y
  | Some x, None => Some x
  | None, Some y => Some y
Robbert Krebbers's avatar
Robbert Krebbers committed
243
244
245
  | None, None => None
  end.
Instance option_intersection_with {A} : IntersectionWith A (option A) :=
246
247
248
249
250
  λ f mx my, match mx, my with Some x, Some y => f x y | _, _ => None end.
Instance option_difference_with {A} : DifferenceWith A (option A) := λ f mx my,
  match mx, my with
  | Some x, Some y => f x y
  | Some x, None => Some x
Robbert Krebbers's avatar
Robbert Krebbers committed
251
252
253
  | None, _ => None
  end.
Instance option_union {A} : Union (option A) := union_with (λ x _, Some x).
254
255
256
257

Lemma option_union_Some {A} (mx my : option A) z :
  mx  my = Some z  mx = Some z  my = Some z.
Proof. destruct mx, my; naive_solver. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
258

259
260
261
262
Class DiagNone {A B C} (f : option A  option B  option C) :=
  diag_none : f None None = None.

Section union_intersection_difference.
Robbert Krebbers's avatar
Robbert Krebbers committed
263
  Context {A} (f : A  A  option A).
264
265
266
267
268
269
270
271

  Global Instance union_with_diag_none : DiagNone (union_with f).
  Proof. reflexivity. Qed.
  Global Instance intersection_with_diag_none : DiagNone (intersection_with f).
  Proof. reflexivity. Qed.
  Global Instance difference_with_diag_none : DiagNone (difference_with f).
  Proof. reflexivity. Qed.
  Global Instance union_with_left_id : LeftId (=) None (union_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
272
  Proof. by intros [?|]. Qed.
273
  Global Instance union_with_right_id : RightId (=) None (union_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
274
  Proof. by intros [?|]. Qed.
275
  Global Instance union_with_comm : Comm (=) f  Comm (=) (union_with f).
276
  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
277
  Global Instance intersection_with_left_ab : LeftAbsorb (=) None (intersection_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
278
  Proof. by intros [?|]. Qed.
279
  Global Instance intersection_with_right_ab : RightAbsorb (=) None (intersection_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
280
  Proof. by intros [?|]. Qed.
281
  Global Instance difference_with_comm : Comm (=) f  Comm (=) (intersection_with f).
282
  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
283
  Global Instance difference_with_right_id : RightId (=) None (difference_with f).
Robbert Krebbers's avatar
Robbert Krebbers committed
284
  Proof. by intros [?|]. Qed.
285
End union_intersection_difference.
Robbert Krebbers's avatar
Robbert Krebbers committed
286
287
288
289
290

(** * Tactics *)
Tactic Notation "case_option_guard" "as" ident(Hx) :=
  match goal with
  | H : appcontext C [@mguard option _ ?P ?dec] |- _ =>
291
292
    change (@mguard option _ P dec) with (λ A (f : P  option A),
      match @decide P dec with left H' => f H' | _ => None end) in *;
Robbert Krebbers's avatar
Robbert Krebbers committed
293
294
    destruct_decide (@decide P dec) as Hx
  | |- appcontext C [@mguard option _ ?P ?dec] =>
295
296
    change (@mguard option _ P dec) with (λ A (f : P  option A),
      match @decide P dec with left H' => f H' | _ => None end) in *;
Robbert Krebbers's avatar
Robbert Krebbers committed
297
298
299
300
301
    destruct_decide (@decide P dec) as Hx
  end.
Tactic Notation "case_option_guard" :=
  let H := fresh in case_option_guard as H.

302
303
Lemma option_guard_True {A} P `{Decision P} (mx : option A) :
  P  guard P; mx = mx.
Robbert Krebbers's avatar
Robbert Krebbers committed
304
Proof. intros. by case_option_guard. Qed.
305
306
Lemma option_guard_False {A} P `{Decision P} (mx : option A) :
  ¬P  guard P; mx = None.
Robbert Krebbers's avatar
Robbert Krebbers committed
307
Proof. intros. by case_option_guard. Qed.
308
309
Lemma option_guard_iff {A} P Q `{Decision P, Decision Q} (mx : option A) :
  (P  Q)  guard P; mx = guard Q; mx.
Robbert Krebbers's avatar
Robbert Krebbers committed
310
311
Proof. intros [??]. repeat case_option_guard; intuition. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
312
Tactic Notation "simpl_option" "by" tactic3(tac) :=
313
  let assert_Some_None A mx H := first
Robbert Krebbers's avatar
Robbert Krebbers committed
314
    [ let x := fresh in evar (x:A); let x' := eval unfold x in x in clear x;
315
316
      assert (mx = Some x') as H by tac
    | assert (mx = None) as H by tac ]
Robbert Krebbers's avatar
Robbert Krebbers committed
317
318
319
320
  in repeat match goal with
  | H : appcontext [@mret _ _ ?A] |- _ =>
     change (@mret _ _ A) with (@Some A) in H
  | |- appcontext [@mret _ _ ?A] => change (@mret _ _ A) with (@Some A)
321
322
323
324
  | H : context [mbind (M:=option) (A:=?A) ?f ?mx] |- _ =>
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx in H; clear Hx
  | H : context [fmap (M:=option) (A:=?A) ?f ?mx] |- _ =>
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx in H; clear Hx
325
  | H : context [from_option (A:=?A) _ _ ?mx] |- _ =>
326
327
328
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx in H; clear Hx
  | H : context [ match ?mx with _ => _ end ] |- _ =>
    match type of mx with
Robbert Krebbers's avatar
Robbert Krebbers committed
329
    | option ?A =>
330
      let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx in H; clear Hx
Robbert Krebbers's avatar
Robbert Krebbers committed
331
    end
332
333
334
335
  | |- context [mbind (M:=option) (A:=?A) ?f ?mx] =>
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx; clear Hx
  | |- context [fmap (M:=option) (A:=?A) ?f ?mx] =>
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx; clear Hx
336
  | |- context [from_option (A:=?A) _ _ ?mx] =>
337
338
339
    let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx; clear Hx
  | |- context [ match ?mx with _ => _ end ] =>
    match type of mx with
Robbert Krebbers's avatar
Robbert Krebbers committed
340
    | option ?A =>
341
      let Hx := fresh in assert_Some_None A mx Hx; rewrite Hx; clear Hx
Robbert Krebbers's avatar
Robbert Krebbers committed
342
343
344
345
346
347
348
349
350
    end
  | H : context [decide _] |- _ => rewrite decide_True in H by tac
  | H : context [decide _] |- _ => rewrite decide_False in H by tac
  | H : context [mguard _ _] |- _ => rewrite option_guard_False in H by tac
  | H : context [mguard _ _] |- _ => rewrite option_guard_True in H by tac
  | _ => rewrite decide_True by tac
  | _ => rewrite decide_False by tac
  | _ => rewrite option_guard_True by tac
  | _ => rewrite option_guard_False by tac
351
352
353
354
  | H : context [None  _] |- _ => rewrite (left_id_L None ()) in H
  | H : context [_  None] |- _ => rewrite (right_id_L None ()) in H
  | |- context [None  _] => rewrite (left_id_L None ())
  | |- context [_  None] => rewrite (right_id_L None ())
Robbert Krebbers's avatar
Robbert Krebbers committed
355
  end.
356
Tactic Notation "simplify_option_eq" "by" tactic3(tac) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
357
  repeat match goal with
358
  | _ => progress simplify_eq/=
Robbert Krebbers's avatar
Robbert Krebbers committed
359
  | _ => progress simpl_option by tac
Robbert Krebbers's avatar
Robbert Krebbers committed
360
361
362
363
364
  | _ : maybe _ ?x = Some _ |- _ => is_var x; destruct x
  | _ : maybe2 _ ?x = Some _ |- _ => is_var x; destruct x
  | _ : maybe3 _ ?x = Some _ |- _ => is_var x; destruct x
  | _ : maybe4 _ ?x = Some _ |- _ => is_var x; destruct x
  | H : _  _ = Some _ |- _ => apply option_union_Some in H; destruct H
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
  | H : mbind (M:=option) ?f ?mx = ?my |- _ =>
    match mx with Some _ => fail 1 | None => fail 1 | _ => idtac end;
    match my with Some _ => idtac | None => idtac | _ => fail 1 end;
    let x := fresh in destruct mx as [x|] eqn:?;
      [change (f x = my) in H|change (None = my) in H]
  | H : ?my = mbind (M:=option) ?f ?mx |- _ =>
    match mx with Some _ => fail 1 | None => fail 1 | _ => idtac end;
    match my with Some _ => idtac | None => idtac | _ => fail 1 end;
    let x := fresh in destruct mx as [x|] eqn:?;
      [change (my = f x) in H|change (my = None) in H]
  | H : fmap (M:=option) ?f ?mx = ?my |- _ =>
    match mx with Some _ => fail 1 | None => fail 1 | _ => idtac end;
    match my with Some _ => idtac | None => idtac | _ => fail 1 end;
    let x := fresh in destruct mx as [x|] eqn:?;
      [change (Some (f x) = my) in H|change (None = my) in H]
  | H : ?my = fmap (M:=option) ?f ?mx |- _ =>
    match mx with Some _ => fail 1 | None => fail 1 | _ => idtac end;
    match my with Some _ => idtac | None => idtac | _ => fail 1 end;
    let x := fresh in destruct mx as [x|] eqn:?;
      [change (my = Some (f x)) in H|change (my = None) in H]
Robbert Krebbers's avatar
Robbert Krebbers committed
385
386
387
  | _ => progress case_decide
  | _ => progress case_option_guard
  end.
388
Tactic Notation "simplify_option_eq" := simplify_option_eq by eauto.