source: src/common/Identifiers.ma @ 2599

Last change on this file since 2599 was 2599, checked in by tranquil, 7 years ago
  • map_opt and map on positive maps are now clean (erase empty subtrees)
  • minor changes to blocks
File size: 38.2 KB
Line 
1include "basics/types.ma".
2include "ASM/String.ma".
3include "utilities/binary/positive.ma".
4include "utilities/lists.ma".
5include "utilities/extralib.ma".
6include "common/Errors.ma".
7
8(* identifiers and their generators are tagged to differentiate them, and to
9   provide extra type checking. *)
10
11(* in common/PreIdentifiers.ma, via Errors.ma.
12inductive identifier (tag:String) : Type[0] ≝
13  an_identifier : Pos → identifier tag.
14*)
15
16record universe (tag:String) : Type[0] ≝
17{
18  next_identifier : Pos
19}.
20
21definition new_universe : ∀tag:String. universe tag ≝
22  λtag. mk_universe tag one.
23
24let rec fresh (tag:String) (u:universe tag) on u : identifier tag × (universe tag) ≝
25  let id ≝ next_identifier ? u in
26  〈an_identifier tag id, mk_universe tag (succ id)〉.
27
28
29let rec fresh_for_univ tag (id:identifier tag) (u:universe tag) on id : Prop ≝
30  match id with [ an_identifier p ⇒ p < next_identifier … u ].
31
32
33lemma fresh_is_fresh : ∀tag,id,u,u'.
34  〈id,u〉 = fresh tag u' →
35  fresh_for_univ tag id u.
36#tag * #id * #u * #u' #E whd in E:(???%); destruct //
37qed.
38
39lemma fresh_remains_fresh : ∀tag,id,id',u,u'.
40  fresh_for_univ tag id u →
41  〈id',u'〉 = fresh tag u →
42  fresh_for_univ tag id u'.
43#tag * #id * #id' * #u * #u' normalize #H #E destruct /2 by le_S/
44qed.
45
46lemma fresh_distinct : ∀tag,id,id',u,u'.
47  fresh_for_univ tag id u →
48  〈id',u'〉 = fresh tag u →
49  id ≠ id'.
50#tag * #id * #id' * #u * #u' normalize #H #E destruct % #E' destruct /2 by absurd/
51qed.
52
53
54let rec env_fresh_for_univ tag A (env:list (identifier tag × A)) (u:universe tag) on u : Prop ≝
55  All ? (λida. fresh_for_univ tag (\fst ida) u) env.
56
57lemma fresh_env_extend : ∀tag,A,env,u,u',id,a.
58  env_fresh_for_univ tag A env u →
59  〈id,u'〉 = fresh tag u →
60  env_fresh_for_univ tag A (〈id,a〉::env) u'.
61#tag #A #env * #u * #u' #id #a
62#H #E whd % [ @(fresh_is_fresh … E) | @(All_mp … H) * #id #a #H' /2 by fresh_remains_fresh/ ]
63qed.
64
65definition eq_identifier : ∀t. identifier t → identifier t → bool ≝
66  λt,l,r.
67  match l with
68  [ an_identifier l' ⇒
69    match r with
70    [ an_identifier r' ⇒
71      eqb l' r'
72    ]
73  ].
74
75lemma eq_identifier_elim : ∀P:bool → Type[0]. ∀t,x,y.
76  (x = y → P true) → (x ≠ y → P false) →
77  P (eq_identifier t x y).
78#P #t * #x * #y #T #F
79change with (P (eqb ??))
80@(eqb_elim x y P) [ /2 by / | * #H @F % #E destruct /2 by / ]
81qed.
82
83lemma eq_identifier_eq:
84  ∀tag: String.
85  ∀l.
86  ∀r.
87    eq_identifier tag l r = true → l = r.
88  #tag #l #r cases l cases r
89  #pos_l #pos_r
90  cases pos_l cases pos_r
91  [1:
92    #_ %
93  |2,3,4,7:
94    #p1_l normalize in ⊢ (% → ?);
95    #absurd destruct(absurd)
96  |5,9:
97    #p1_l #p1_r normalize in ⊢ (% → ?);
98    #relevant lapply (eqb_true_to_eq … relevant) #EQ >EQ %
99  |*:
100    #p_l #p_r normalize in ⊢ (% → ?);
101    #absurd destruct(absurd)
102  ]
103qed.
104
105axiom neq_identifier_neq:
106  ∀tag: String.
107  ∀l, r: identifier tag.
108    eq_identifier tag l r = false → (l = r → False).
109
110include "basics/deqsets.ma".
111definition Deq_identifier : String → DeqSet ≝ λtag.
112  mk_DeqSet (identifier tag) (eq_identifier tag) ?.
113#x#y @eq_identifier_elim /2 by conj/ * #H % [#ABS destruct(ABS) | #G elim (H G)]
114qed.
115
116unification hint 0 ≔ tag; D ≟ Deq_identifier tag
117(*-----------------------------------------------------*)⊢
118identifier tag ≡ carr D.
119
120definition word_of_identifier ≝
121  λt.
122  λl: identifier t.
123  match l with   
124  [ an_identifier l' ⇒ l'
125  ].
126
127lemma eq_identifier_refl : ∀tag,id. eq_identifier tag id id = true.
128#tag * #id whd in ⊢ (??%?); >eqb_n_n @refl
129qed.
130
131axiom eq_identifier_sym:
132  ∀tag: String.
133  ∀l  : identifier tag.
134  ∀r  : identifier tag.
135    eq_identifier tag l r = eq_identifier tag r l.
136
137lemma eq_identifier_false : ∀tag,x,y. x≠y → eq_identifier tag x y = false.
138#tag * #x * #y #NE normalize @not_eq_to_eqb_false /2 by not_to_not/
139qed.
140
141definition identifier_eq : ∀tag:String. ∀x,y:identifier tag. (x=y) + (x≠y).
142#tag * #x * #y lapply (refl ? (eqb x y)) cases (eqb x y) in ⊢ (???% → %);
143#E [ % | %2 ]
144lapply E @eqb_elim
145[ #H #_ >H @refl | 2,3: #_ #H destruct | #H #_ % #H' destruct /2 by absurd/ ]
146qed.
147
148definition identifier_of_nat : ∀tag:String. nat → identifier tag ≝
149  λtag,n. an_identifier tag (succ_pos_of_nat  n).
150
151
152(* States that all identifiers in an environment are distinct from one another. *)
153let rec distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : Prop ≝
154match l with
155[ nil ⇒ True
156| cons hd tl ⇒ All ? (λia. \fst hd ≠ \fst ia) tl ∧
157               distinct_env tag A tl
158].
159
160lemma distinct_env_append_l : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A l.
161#tag #A #l elim l
162[ //
163| * #id #a #tl #IH #r * #H1 #H2 % /2 by All_append_l/
164] qed.
165
166lemma distinct_env_append_r : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A r.
167#tag #A #l elim l
168[ //
169| * #id #a #tl #IH #r * #H1 #H2 /2 by /
170] qed.
171
172(* check_distinct_env is quadratic - we could pay more attention when building maps to make sure that
173   the original environment was distinct. *)
174
175axiom DuplicateVariable : String.
176
177let rec check_member_env tag (A:Type[0]) (id:identifier tag) (l:list (identifier tag × A)) on l : res (All ? (λia. id ≠ \fst ia) l) ≝
178match l return λl.res (All ?? l) with
179[ nil ⇒ OK ? I
180| cons hd tl ⇒
181    match identifier_eq tag id (\fst hd) with
182    [ inl _ ⇒ Error ? [MSG DuplicateVariable; CTX ? id]
183    | inr NE ⇒
184        do Htl ← check_member_env tag A id tl;
185        OK ? (conj ?? NE Htl)
186    ]
187].
188
189let rec check_distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : res (distinct_env tag A l) ≝
190match l return λl.res (distinct_env tag A l) with
191[ nil ⇒ OK ? I
192| cons hd tl ⇒
193    do Hhd ← check_member_env tag A (\fst hd) tl;
194    do Htl ← check_distinct_env tag A tl;
195    OK ? (conj ?? Hhd Htl)
196].
197
198
199
200
201(* Maps from identifiers to arbitrary types. *)
202
203include "common/PositiveMap.ma".
204
205inductive identifier_map (tag:String) (A:Type[0]) : Type[0] ≝
206  an_id_map : positive_map A → identifier_map tag A.
207 
208definition empty_map : ∀tag:String. ∀A. identifier_map tag A ≝
209  λtag,A. an_id_map tag A (pm_leaf A).
210
211let rec lookup tag A (m:identifier_map tag A) (l:identifier tag) on m : option A ≝
212  lookup_opt A (match l with [ an_identifier l' ⇒ l' ])
213               (match m with [ an_id_map m' ⇒ m' ]).
214
215definition lookup_def ≝
216λtag,A,m,l,d. match lookup tag A m l with [ None ⇒ d | Some x ⇒ x].
217
218definition member ≝
219  λtag,A.λm:identifier_map tag A.λl:identifier tag.
220  match lookup tag A m l with [ None ⇒ false | _ ⇒ true ].
221
222interpretation "identifier map membership" 'mem a b = (member ?? b a).
223
224definition lookup_safe : ∀tag,A.∀m : identifier_map tag A.∀i.i∈m → A ≝
225λtag,A,m,i.
226match lookup … m i return λx.match x in option return λ_.bool with [ _ ⇒ ?] → ? with
227[ Some x ⇒ λ_.x
228| None ⇒ λprf.⊥
229]. @prf qed.
230
231lemma lookup_eq_safe : ∀tag,A,m,i,prf.lookup tag A m i = Some ? (lookup_safe tag A m i prf).
232#tag #A #m #i whd in match (i∈m);
233whd in match lookup_safe; normalize nodelta
234cases (lookup ????) normalize nodelta [*] // qed.
235
236(* Always adds the identifier to the map. *)
237let rec add tag A (m:identifier_map tag A) (l:identifier tag) (a:A) on m : identifier_map tag A ≝
238  an_id_map tag A (insert A (match l with [ an_identifier l' ⇒ l' ]) a
239                            (match m with [ an_id_map m' ⇒ m' ])).
240
241lemma lookup_add_hit : ∀tag,A,m,i,a.
242  lookup tag A (add tag A m i a) i = Some ? a.
243#tag #A * #m * #i #a
244@lookup_opt_insert_hit
245qed.
246
247lemma lookup_def_add_hit : ∀tag,A,m,i,a,d.
248  lookup_def tag A (add tag A m i a) i d = a.
249#tag #A * #m * #i #a #d
250@lookup_insert_hit
251qed.
252
253lemma lookup_add_miss : ∀tag,A,m,i,j,a.
254  i ≠ j →
255  lookup tag A (add tag A m j a) i = lookup tag A m i.
256#tag #A * #m * #i * #j #a #H
257@lookup_opt_insert_miss /2 by not_to_not/
258qed.
259
260axiom lookup_def_add_miss : ∀tag,A,m,i,j,a,d.
261  i ≠ j →
262  lookup_def tag A (add tag A m j a) i d = lookup_def tag A m i d.
263
264lemma lookup_add_oblivious : ∀tag,A,m,i,j,a.
265  (lookup tag A m i ≠ None ?) →
266  lookup tag A (add tag A m j a) i ≠ None ?.
267#tag #A #m #i #j #a #H
268cases (identifier_eq ? i j)
269[ #E >E >lookup_add_hit % #N destruct
270| #NE >lookup_add_miss //
271] qed.
272
273lemma lookup_add_cases : ∀tag,A,m,i,j,a,v.
274  lookup tag A (add tag A m i a) j = Some ? v →
275  (i=j ∧ v = a) ∨ lookup tag A m j = Some ? v.
276#tag #A #m #i #j #a #v
277cases (identifier_eq ? i j)
278[ #E >E >lookup_add_hit #H %1 destruct % //
279| #NE >lookup_add_miss /2 by or_intror, sym_not_eq/
280] qed.
281
282(* Extract every identifier, value pair from the map. *)
283definition elements : ∀tag,A. identifier_map tag A → list (identifier tag × A) ≝
284λtag,A,m.
285  fold ?? (λl,a,el. 〈an_identifier tag l, a〉::el)
286          (match m with [ an_id_map m' ⇒ m' ]) [ ].
287
288(* Test a predicate on all of the entries in a map.  The predicate is given a
289   proof that the entry appears in the map. *)
290
291definition idmap_all : ∀tag,A. ∀m:identifier_map tag A. (∀id,a. lookup tag A m id = Some A a → bool) → bool ≝
292λtag,A,m,f. pm_all A (match m with [ an_id_map m' ⇒ m' ])
293                     (λp,a,H. f (an_identifier tag p) a ?).
294cases m in H ⊢ %; #m' normalize //
295qed.
296
297inductive idmap_pred_graph : ∀tag,A,m,id,a,L. ∀f:(∀id,a. lookup tag A m id = Some A a → bool). bool → Prop ≝
298| idmappg : ∀tag,A,m,id,a,L,f. idmap_pred_graph tag A m id a L f (f id a L).
299
300lemma idmap_pred_irr : ∀tag,A,m,id,a,L,L'. ∀f:(∀id,a. lookup tag A m id = Some A a → bool).
301  f id a L = f id a L'.
302#tag #A #m #id #a #L #L' #f
303cut (idmap_pred_graph tag A m id a L f (f id a L)) [ % ]
304cases (f id a L) #H
305cut (idmap_pred_graph tag A m id a L' f ?) [ 2,5: @H | 1,4: skip ] * //
306qed.
307
308lemma idmap_all_ok : ∀tag,A,m,f.
309  bool_to_Prop (idmap_all tag A m f) ↔ (∀id,a,H. f id a H).
310#tag #A * #m #f
311whd in match (idmap_all ????); @(iff_trans … (pm_all_ok …)) %
312[ #H * #id #a #PR lapply (H id a PR) #X @eq_true_to_b <X @idmap_pred_irr
313| #H #p #a #PR @H
314] qed.
315
316
317axiom MissingId : String.
318
319(* Only updates an existing entry; fails with an error otherwise. *)
320definition update : ∀tag,A. identifier_map tag A → identifier tag → A → res (identifier_map tag A) ≝
321λtag,A,m,l,a.
322  match update A (match l with [ an_identifier l' ⇒ l' ]) a
323                 (match m with [ an_id_map m' ⇒ m' ]) with
324  [ None ⇒ Error ? ([MSG MissingId; CTX tag l]) (* missing identifier *)
325  | Some m' ⇒ OK ? (an_id_map tag A m')
326  ].
327
328(* Remove an entry from a map (and leave it equivalent, otherwise) *)
329definition remove : ∀tag,A. identifier_map tag A → identifier tag → identifier_map tag A ≝
330λtag,A,m,id. an_id_map tag A (pm_set A (match id with [ an_identifier p ⇒ p ]) (None ?)
331                                       (match m with [ an_id_map m ⇒ m ])).
332
333lemma lookup_remove_hit : ∀tag,A,m,id.
334  lookup tag A (remove tag A m id) id = None ?.
335#tag #A * #m * #id @lookup_opt_pm_set_hit
336qed.
337
338lemma lookup_remove_miss : ∀tag,A,m,id,id'.
339  id ≠ id' →
340  lookup tag A (remove tag A m id') id = lookup tag A m id.
341#tag #A * #m * #id * #id' #NE
342@lookup_opt_pm_set_miss
343/2 by not_to_not/
344qed.
345
346(* Fold over the entries in a map.  There are some lemmas to help reason about
347   this near the bottom of the file (they require sets). *)
348
349definition foldi:
350  ∀A, B: Type[0].
351  ∀tag: String.
352  (identifier tag -> A -> B -> B) -> identifier_map tag A -> B -> B ≝
353λA,B,tag,f,m,b.
354  match m with
355  [ an_id_map m' ⇒ fold A B (λbv. f (an_identifier ? bv)) m' b ].
356
357(* An informative, dependently-typed, fold. *)
358
359definition fold_inf:
360  ∀A, B: Type[0].
361  ∀tag: String.
362  ∀m:identifier_map tag A.
363  (∀id:identifier tag. ∀a:A. lookup … m id = Some A a → B → B) → B → B ≝
364λA,B,tag,m.
365  match m return λm. (∀id:identifier tag. ∀a:A. lookup … m id = Some A a → B → B) → B → B with
366  [ an_id_map m' ⇒ λf,b. pm_fold_inf A B m' (λbv,a,H. f (an_identifier ? bv) a H) b ].
367
368(* Find one element of a map that satisfies a predicate *)
369definition find : ∀tag,A. identifier_map tag A → (identifier tag → A → bool) →
370  option (identifier tag × A) ≝
371λtag,A,m,p.
372  match m with [ an_id_map m' ⇒
373    option_map … (λx. 〈an_identifier tag (\fst x), \snd x〉)
374      (pm_find … m' (λid. p (an_identifier tag id))) ].
375
376lemma find_lookup : ∀tag,A,m,p,id,a.
377  find tag A m p = Some ? 〈id,a〉 →
378  lookup … m id = Some ? a.
379#tag #A * #m #p * #id #a #FIND
380@(pm_find_lookup A (λid. p (an_identifier tag id)) id a m)
381whd in FIND:(??%?); cases (pm_find ???) in FIND ⊢ %;
382[ normalize #E destruct
383| * #id' #a' normalize #E destruct %
384] qed.
385
386lemma find_none : ∀tag,A,m,p,id,a.
387  find tag A m p = None ? →
388  lookup … m id = Some ? a →
389  ¬ p id a.
390#tag #A * #m #p * #id #a #FIND
391@(pm_find_none A m (λid. p (an_identifier tag id)))
392whd in FIND:(??%?); cases (pm_find ???) in FIND ⊢ %;
393[ normalize #E destruct %
394| * #id' #a' normalize #E destruct
395] qed.
396
397
398lemma find_predicate : ∀tag,A,m,p,id,a.
399  find tag A m p = Some ? 〈id,a〉 →
400  p id a.
401#tag #A * #m #p * #id #a #FIND whd in FIND:(??%?);
402@(pm_find_predicate A m (λid. p (an_identifier tag id)) id a)
403cases (pm_find ???) in FIND ⊢ %;
404[ normalize #E destruct
405| * #id' #a' normalize #E destruct %
406] qed.
407
408(* A predicate that an identifier is in a map, and a failure-avoiding lookup
409   and update using it. *)
410
411definition present : ∀tag,A. identifier_map tag A → identifier tag → Prop ≝
412λtag,A,m,i. lookup … m i ≠ None ?.
413
414lemma member_present : ∀tag,A,m,id.
415  member tag A m id = true → present tag A m id.
416#tag #A * #m #id normalize cases (lookup_opt A ??) normalize
417[ #E destruct
418| #x #E % #E' destruct
419] qed.
420
421lemma present_member : ∀tag,A,m,id.
422  present tag A m id → member tag A m id.
423#tag #A #m #id whd in ⊢ (% → ?%); cases (lookup ????) // * #H cases (H (refl ??))
424qed.
425
426definition lookup_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A ≝
427λtag,A,m,id. match lookup ?? m id return λx. x ≠ None ? → ? with [ Some a ⇒ λ_. a | None ⇒ λH.⊥ ].
428cases H #H'  cases (H' (refl ??)) qed.
429
430lemma lookup_lookup_present : ∀tag,A,m,id,p.
431  lookup tag A m id = Some ? (lookup_present tag A m id p).
432#tag #A #m #id #p
433whd in p ⊢ (???(??%));
434cases (lookup tag A m id) in p ⊢ %;
435[ * #H @⊥ @H @refl
436| #a #H @refl
437] qed.
438
439lemma lookup_is_present : ∀tag,T,m,id,t.
440  lookup tag T m id = Some T t →
441  present ?? m id.
442#tag #T #m #id #t #L normalize >L % #E destruct
443qed.
444
445lemma lookup_present_eq : ∀tag,T,m,id,t.
446  lookup tag T m id = Some T t →
447  ∀H. lookup_present tag T m id H = t.
448#tag #T #m #id #t #L #H
449lapply (lookup_lookup_present … H) >L #E destruct %
450qed.
451
452
453definition update_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A → identifier_map tag A ≝
454λtag,A,m,l,p,a.
455  let l' ≝ match l with [ an_identifier l' ⇒ l' ] in
456  let m' ≝ match m with [ an_id_map m' ⇒ m' ] in
457  let u' ≝ update A l' a m' in
458  match u' return λx. update ???? = x → ? with
459  [ None ⇒ λE.⊥
460  | Some m' ⇒ λ_. an_id_map tag A m'
461  ] (refl ? u').
462cases l in p E; cases m; -l' -m' #m' #l'
463whd in ⊢ (% → ?);
464 whd in ⊢ (?(??(???%%)?) → ??(??%?%)? → ?);
465#NL #U cases NL #H @H @(update_fail … U)
466qed.
467
468lemma update_still_present : ∀tag,A,m,id,a,id'.
469  ∀H:present tag A m id.
470  ∀H':present tag A m id'.
471  present tag A (update_present tag A m id' H' a) id.
472#tag #A * #m * #id #a * #id' #H #H'
473whd whd in ⊢ (?(??(???(%??????)?)?)); normalize nodelta
474cases (identifier_eq ? (an_identifier tag id) (an_identifier tag id'))
475[ #E >E @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); >(update_lookup_opt_same ????? U)
476  % #E' destruct
477| #NE @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); whd in ⊢ (?(??(??%%)?));
478  <(update_lookup_opt_other ????? U id) [ @H | % #E cases NE >E #H @H @refl ]
479] qed.
480
481lemma lookup_present_add_hit:
482  ∀tag, A, map, k, v, k_pres.
483    lookup_present tag A (add … map k v) k k_pres = v.
484  #tag #a #map #k #v #k_pres
485  lapply (lookup_lookup_present … (add … map k v) … k_pres)
486  >lookup_add_hit #Some_assm destruct(Some_assm)
487  <e0 %
488qed.
489
490lemma lookup_present_add_miss:
491  ∀tag, A, map, k, k', v, k_pres', k_pres''.
492    k' ≠ k →
493      lookup_present tag A (add … map k v) k' k_pres' = lookup_present tag A map k' k_pres''.
494  #tag #A #map #k #k' #v #k_pres' #k_pres'' #neq_assm
495  lapply (lookup_lookup_present … (add … map k v) ? k_pres')
496  >lookup_add_miss try assumption
497  #Some_assm
498  lapply (lookup_lookup_present … map k') >Some_assm #Some_assm'
499  lapply (Some_assm' k_pres'') #Some_assm'' destruct assumption
500qed.
501
502lemma present_add_present:
503  ∀tag, a, map, k, k', v.
504    k' ≠ k →
505      present tag a (add tag a map k v) k' →
506        present tag a map k'.
507  #tag #a #map #k #k' #v #neq_hyp #present_hyp
508  whd in match present; normalize nodelta
509  whd in match present in present_hyp; normalize nodelta in present_hyp;
510  cases (not_None_to_Some a … present_hyp) #v' #Some_eq_hyp
511  lapply (lookup_add_cases tag ?????? Some_eq_hyp) *
512  [1:
513    * #k_eq_hyp @⊥ /2 by absurd/
514  |2:
515    #Some_eq_hyp' /2 by /
516  ]
517qed.
518
519lemma present_add_hit:
520  ∀tag, a, map, k, v.
521    present tag a (add tag a map k v) k.
522  #tag #a #map #k #v
523  whd >lookup_add_hit
524  % #absurd destruct
525qed.
526
527lemma present_add_miss:
528  ∀tag, a, map, k, k', v.
529    present tag a map k' → present tag a (add tag a map k v) k'.
530  #tag #a #map #k #k' #v #present_assm
531  whd @lookup_add_oblivious assumption
532qed.
533
534lemma present_add_cases: ∀tag,A,map,k,v,k'.
535  present tag A (add tag A map k v) k' →
536  k = k' ∨ (k ≠ k' ∧ present tag A map k').
537#tag #A #map #k #v #k' normalize
538cases (identifier_eq ? k k')
539[ #E /2 by or_introl/
540| #NE >lookup_add_miss /3 by or_intror, conj, absurd, nmk/
541] qed.
542
543
544let rec fresh_for_map tag A (id:identifier tag) (m:identifier_map tag A) on id : Prop ≝
545  lookup … m id = None A.
546
547lemma fresh_for_empty_map : ∀tag,A,id.
548  fresh_for_map tag A id (empty_map tag A).
549#tag #A * #id //
550qed.
551
552definition fresh_map_for_univ ≝
553λtag,A. λm:identifier_map tag A. λu:universe tag.
554  ∀id. present tag A m id → fresh_for_univ tag id u.
555
556lemma fresh_fresh_for_map : ∀tag,A,m,id,u,u'.
557  fresh_map_for_univ tag A m u →
558  〈id,u'〉 = fresh tag u →
559  fresh_for_map tag A id m.
560#tag #A * #m * #id * #u * #u' whd in ⊢ (% → ???% → %);
561#FMU #E destruct lapply (FMU (an_identifier tag u)) whd in ⊢ ((% → %) → ?);
562generalize in ⊢ ((?(??%?) → ?) → ??%?); *
563[ // | #a #H @False_ind lapply (H ?) /2 by absurd/ % #E destruct
564qed.
565
566lemma fresh_map_preserved : ∀tag,A,m,u,u',id.
567  fresh_map_for_univ tag A m u →
568  〈id,u'〉 = fresh tag u →
569  fresh_map_for_univ tag A m u'.
570#tag #A #m #u * #u' #id whd in ⊢ (% → ? → %); #H #E
571#id' #PR @(fresh_remains_fresh … E) @H //
572qed.
573
574lemma fresh_map_add : ∀tag,A,m,u,id,a.
575  fresh_map_for_univ tag A m u →
576  fresh_for_univ tag id u →
577  fresh_map_for_univ tag A (add tag A m id a) u.
578#tag #A * #m #u #id #a #Hm #Hi
579#id' #PR cases (identifier_eq tag id' id)
580[ #E >E @Hi
581| #NE @Hm whd in PR;
582  change with (add tag A (an_id_map tag A m) id a) in PR:(?(??(???%?)?));
583  >lookup_add_miss in PR; //
584] qed.
585
586lemma present_not_fresh : ∀tag,A,m,id,id'.
587  present tag A m id →
588  fresh_for_map tag A id' m →
589  id ≠ id'.
590#tag #A #m #id * #id' whd in ⊢ (% → % → ?);
591* #NE #E % #E' destruct @(NE E)
592qed.
593
594lemma fresh_for_map_add : ∀tag,A,id,m,id',a.
595  id ≠ id' →
596  fresh_for_map tag A id m →
597  fresh_for_map tag A id (add tag A m id' a).
598#tag #A * #id #m #id' #a #NE #F
599whd >lookup_add_miss //
600qed.
601
602(* Extending the domain of a map (without necessarily preserving contents). *)
603
604definition extends_domain : ∀tag,A. identifier_map tag A → identifier_map tag A → Prop ≝
605λtag,A,m1,m2. ∀l. present ?? m1 l → present ?? m2 l.
606
607lemma extends_dom_trans : ∀tag,A,m1,m2,m3.
608  extends_domain tag A m1 m2 → extends_domain tag A m2 m3 → extends_domain tag A m1 m3.
609#tag #A #m1 #m2 #m3 #H1 #H2 #l #P1 @H2 @H1 @P1 qed.
610
611
612(* Sets *)
613
614definition identifier_set ≝ λtag.identifier_map tag unit.
615
616definition empty_set : ∀tag.identifier_set tag ≝ λtag.empty_map ….
617
618
619definition add_set : ∀tag.identifier_set tag → identifier tag → identifier_set tag ≝
620  λtag,s,i.add … s i it.
621
622definition singleton_set : ∀tag:String. identifier tag → identifier_set tag ≝
623λtag,i. add_set tag (empty_set tag) i.
624
625let rec union_set (tag:String) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_set tag ≝
626  an_id_map tag unit (merge … (λo,o'.match o with [Some _ ⇒ Some ? it | None ⇒ !_ o'; return it])
627    (match s with [ an_id_map s0 ⇒ s0 ])
628    (match s' with [ an_id_map s1 ⇒ s1 ])).
629
630
631(* set minus is generalised to maps *)
632let rec minus_set (tag:String) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_map tag A ≝
633  an_id_map tag A (merge A B A (λo,o'.match o' with [None ⇒ o | Some _ ⇒ None ?])
634    (match s with [ an_id_map s0 ⇒ s0 ])
635    (match s' with [ an_id_map s1 ⇒ s1 ])).
636
637notation "a ∖ b" left associative with precedence 55 for @{'setminus $a $b}.
638
639interpretation "identifier set union" 'union a b = (union_set ??? a b).
640notation "∅" non associative with precedence 90 for @{ 'empty }.
641interpretation "empty identifier set" 'empty = (empty_set ?).
642interpretation "singleton identifier set" 'singl a = (add_set ? (empty_set ?) a).
643interpretation "identifier map difference" 'setminus a b = (minus_set ??? a b).
644
645definition IdentifierSet : String → Setoid ≝ λtag.
646  mk_Setoid (identifier_set tag) (λs,s'.∀i.i ∈ s = (i ∈ s')) ???.
647  // qed.
648
649unification hint 0 ≔ tag;
650S ≟ IdentifierSet tag
651(*-----------------------------*)⊢
652identifier_set tag ≡ std_supp S.
653unification hint 0 ≔ tag;
654S ≟ IdentifierSet tag
655(*-----------------------------*)⊢
656identifier_map tag unit ≡ std_supp S.
657
658lemma mem_set_add : ∀tag,A.∀i,j : identifier tag.∀s,x.
659  i ∈ add ? A s j x = (eq_identifier ? i j ∨ i ∈ s).
660#tag #A *#i *#j *#s #x normalize
661@(eqb_elim i j)
662[#EQ destruct
663  >(lookup_opt_insert_hit A x j)
664|#NEQ >(lookup_opt_insert_miss … s NEQ)
665] elim (lookup_opt  A j s) normalize // qed.
666
667lemma mem_set_add_id : ∀tag,A,i,s,x.bool_to_Prop (i ∈ add tag A s i x).
668#tag #A #i #s #x >mem_set_add
669@eq_identifier_elim [#_ %| #ABS elim (absurd … (refl ? i) ABS)] qed.
670
671lemma in_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
672  if i ∈ m then (∃s.lookup … m i = Some ? s) else (lookup … m i = None ?).
673#tag #A * #m * #i normalize
674elim (lookup_opt A i m) normalize
675[ % | #x %{x} % ]
676qed.
677(*
678lemma union_empty_l : ∀tag.∀s:identifier_set tag. ∅ ∪ s = s.
679#tag * normalize #m >map_opt_id_eq_ext // * %
680qed.
681
682lemma union_empty_r : ∀tag.∀s:identifier_set tag. s ∪ ∅ = s.
683#tag * * [//] *[2: *] #l#r normalize
684>map_opt_id_eq_ext [1,3: >map_opt_id_eq_ext [2,4: *] |*: *] //
685qed.
686
687lemma minus_empty_l : ∀tag,A.∀s:identifier_map tag A. ∅ ∖ s ≅ ∅.
688#tag #A * * [//] *[2:#x]#l#r * * normalize [1,4://]
689#p >lookup_opt_map elim (lookup_opt ???) normalize //
690qed.
691
692lemma minus_empty_r : ∀tag,A.∀s:identifier_map tag A. s ∖ ∅ = s.
693#tag #A * * [//] *[2:#x]#l#r normalize
694>map_opt_id >map_opt_id //
695qed.
696*)
697lemma mem_set_union : ∀tag.∀i : identifier tag.∀s,s' : identifier_set tag.
698  i ∈ (s ∪ s') = (i ∈ s ∨ i ∈ s').
699#tag * #i * #s * #s' normalize
700>lookup_opt_merge [2: @refl]
701elim (lookup_opt ???)
702elim (lookup_opt ???)
703normalize // qed.
704
705lemma mem_set_minus : ∀tag,A,B.∀i : identifier tag.∀s : identifier_map tag A.
706  ∀s' : identifier_map tag B.
707  i ∈ (s ∖ s') = (i ∈ s ∧ ¬ i ∈ s').
708#tag #A #B * #i * #s * #s' normalize
709>lookup_opt_merge [2: @refl]
710elim (lookup_opt ???)
711elim (lookup_opt ???)
712normalize // qed.
713
714lemma set_eq_ext_node : ∀tag.∀o,o',l,l',r,r'.
715  an_id_map tag ? (pm_node ? o l r) ≅ an_id_map … (pm_node ? o' l' r') →
716    o = o' ∧ an_id_map tag ? l ≅ an_id_map … l' ∧ an_id_map tag ? r ≅ an_id_map … r'.
717#tag#o#o'#l#l'#r#r'#H
718%[
719%[ lapply (H (an_identifier ? one))
720   elim o [2: *] elim o' [2,4: *] normalize // #EQ destruct
721 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
722]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
723]
724qed.
725
726lemma set_eq_ext_leaf : ∀tag,A.∀o,l,r.
727  (∀i.i∈an_id_map tag A (pm_node ? o l r) = false) →
728    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
729#tag#A#o#l#r#H
730%[
731%[ lapply (H (an_identifier ? one))
732   elim o [2: #a] normalize // #EQ destruct
733 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
734]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
735]
736qed.
737
738
739definition id_map_size : ∀tag : String.∀A. identifier_map tag A → ℕ ≝
740  λtag,A,s.match s with [an_id_map p ⇒ |p|].
741
742interpretation "identifier map domain size" 'card s = (id_map_size ?? s).
743
744lemma set_eq_ext_empty_to_card : ∀tag,A.∀s : identifier_map tag A. (∀i.i∈s = false) → |s| = 0.
745#tag#A * #s elim s [//]
746#o#l#r normalize in ⊢((?→%)→(?→%)→?); #Hil #Hir #H
747elim (set_eq_ext_leaf … H) * #EQ destruct #Hl #Hr normalize
748>(Hil Hl) >(Hir Hr) // qed.
749
750lemma set_eq_ext_to_card : ∀tag.∀s,s' : identifier_set tag. s ≅ s' → |s| = |s'|.
751#tag *#s elim s
752[** [//] #o#l#r #H
753  >(set_eq_ext_empty_to_card … (std_symm … H)) //
754| #o#l#r normalize in ⊢((?→?→??%?)→(?→?→??%?)→?);
755  #Hil #Hir **
756  [#H @(set_eq_ext_empty_to_card … H)]
757  #o'#l'#r' #H elim (set_eq_ext_node … H) * #EQ destruct(EQ) #Hl #Hr
758  normalize >(Hil ? Hl) >(Hir ? Hr) //
759] qed.
760
761lemma add_size: ∀tag,A,s,i,x.
762  |add tag A s i x| = (if i ∈ s then 0 else 1) + |s|.
763#tag #A *#s *#i #x
764lapply (insert_size ? i x s)
765lapply (refl ? (lookup_opt ? i s))
766generalize in ⊢ (???%→?); * [2: #x']
767normalize #EQ >EQ normalize //
768qed.
769
770lemma mem_set_O_lt_card : ∀tag,A.∀i.∀s : identifier_map tag A. i ∈ s → |s| > 0.
771#tag #A * #i * #s normalize #H
772@(lookup_opt_O_lt_size … i)
773% #EQ >EQ in H; normalize *
774qed.
775
776(* NB: no control on values if applied to maps *)
777definition set_subset ≝ λtag,A,B.λs : identifier_map tag A.
778  λs' : identifier_map tag B. ∀i.i ∈ s → (bool_to_Prop (i ∈ s')).
779
780interpretation "identifier set subset" 'subseteq s s' = (set_subset ??? s s').
781
782lemma add_subset :
783  ∀tag,A,B.∀i : identifier tag.∀x.∀s : identifier_map ? A.∀s' : identifier_map ? B.
784    i ∈ s' → s ⊆ s' → add … s i x ⊆ s'.
785#tag#A#B#i#x#s#s' #H #G #j
786>mem_set_add
787@eq_identifier_elim #H' [* >H' @H | #js @(G ? js)]
788qed.
789
790definition set_forall : ∀tag,A.(identifier tag → Prop) →
791  identifier_map tag A → Prop ≝ λtag,A,P,m.∀i. i ∈ m → P i.
792 
793lemma set_forall_add : ∀tag,P,m,i.set_forall tag ? P m → P i →
794  set_forall tag ? P (add_set ? m i).
795#tag#P#m#i#Pm#Pi#j
796>mem_set_add
797@eq_identifier_elim
798[#EQ destruct(EQ) #_ @Pi
799|#_ @Pm
800]
801qed.
802
803include "utilities/proper.ma".
804
805lemma minus_subset : ∀tag,A,B.minus_set tag A B ⊨ set_subset … ++> set_subset … -+> set_subset ….
806#tag#A#B#s#s' #H #s'' #s''' #G #i
807>mem_set_minus >mem_set_minus
808#H' elim (andb_Prop_true … H') -H' #is #nis''
809>(H … is)
810elim (true_or_false_Prop (i∈s'''))
811[ #is''' >(G … is''') in nis''; *
812| #nis''' >nis''' %
813]
814qed.
815
816lemma subset_node : ∀tag,A,B.∀o,o',l,l',r,r'.
817  an_id_map tag A (pm_node ? o l r) ⊆ an_id_map tag B (pm_node ? o' l' r') →
818    opt_All ? (λ_.o' ≠ None ?) o ∧ an_id_map tag ? l ⊆ an_id_map tag  ? l' ∧
819      an_id_map tag ? r ⊆ an_id_map tag ? r'.
820#tag#A#B#o#o'#l#l'#r#r'#H
821%[%
822  [ lapply (H (an_identifier ? (one))) elim o [2: #a] elim o' [2:#b]
823    normalize // [#_ % #ABS destruct(ABS) | #G lapply (G I) *]
824  | *#p lapply (H (an_identifier ? (p0 p)))
825  ]
826 | *#p lapply (H (an_identifier ? (p1 p)))
827] #H @H
828qed.
829
830lemma subset_leaf : ∀tag,A.∀o,l,r.
831  an_id_map tag A (pm_node ? o l r) ⊆ ∅ →
832    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
833#tag#A#o#l#r#H
834%[
835%[ lapply (H (an_identifier ? one))
836   elim o [2: #a] normalize // #EQ lapply(EQ I) *
837 | *#p lapply (H (an_identifier ? (p0 p)))
838 ]
839|  *#p lapply (H (an_identifier ? (p1 p)))
840] normalize elim (lookup_opt ? p ?) normalize
841// #a #H lapply (H I) *
842qed.
843
844lemma subset_card : ∀tag,A,B.∀s : identifier_map tag A.∀s' : identifier_map tag B.
845  s ⊆ s' → |s| ≤ |s'|.
846#tag #A #B *#s elim s
847[ //
848| #o#l#r #Hil #Hir **
849  [ #H elim (subset_leaf … H) * #EQ >EQ #Hl #Hr
850    lapply (set_eq_ext_empty_to_card … Hl)
851    lapply (set_eq_ext_empty_to_card … Hr)
852    normalize //
853  | #o' #l' #r' #H elim (subset_node … H) *
854    elim o [2: #a] elim o' [2,4: #a']
855    [3: #G normalize in G; elim(absurd ? (refl ??) G)
856    |*: #_ #Hl #Hr lapply (Hil ? Hl) lapply (Hir ? Hr)
857      normalize #H1 #H2
858      [@le_S_S | @(transitive_le … (|l'|+|r'|)) [2: / by /]]
859      @le_plus assumption
860    ]
861  ]
862]
863qed.
864
865lemma mem_set_empty : ∀tag,A.∀i: identifier tag. i∈empty_map tag A = false.
866#tag #A * #i normalize %
867qed.
868
869lemma mem_set_singl_to_eq : ∀tag.∀i,j : identifier tag.i∈{(j)} → i = j.
870#tag
871#i #j >mem_set_add >mem_set_empty
872#H elim (orb_true_l … H) -H
873[@eq_identifier_elim [//] #_] #EQ destruct
874qed.
875
876lemma subset_add_set : ∀tag,i,s.s ⊆ add_set tag s i.
877#tag#i#s#j #H >mem_set_add >H
878>commutative_orb %
879qed.
880
881lemma add_set_monotonic : ∀tag,i,s,s'.s ⊆ s' → add_set tag s i ⊆ add_set tag s' i.
882#tag#i#s#s' #H #j >mem_set_add >mem_set_add
883@orb_elim elim (eq_identifier ???)
884whd lapply (H j) /2 by /
885qed.
886
887lemma transitive_subset : ∀tag,A.transitive ? (set_subset tag A A).
888#tag#A#s#s'#s''#H#G#i #is
889@(G … (H … is))
890qed.
891
892definition set_from_list : ∀tag.list (identifier tag) → identifier_map tag unit ≝
893  λtag.foldl … (add_set ?) ∅.
894
895coercion id_set_from_list : ∀tag.∀l : list (identifier tag).identifier_map tag unit ≝
896  set_from_list on _l : list (identifier ?) to identifier_map ? unit.
897
898lemma mem_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
899i∈m → lookup … m i ≠ None ?.
900#tag#A * #m #i
901whd in match (i∈?);
902elim (lookup ????) normalize [2: #x]
903* % #EQ destruct(EQ)
904qed.
905
906
907
908lemma mem_list_as_set : ∀tag.∀l : list (identifier tag).
909  ∀i.i ∈ l → In ? l i.
910#tag #l @(list_elim_left … l)
911[ #i *
912| #t #h #Hi  #i
913  whd in ⊢ (?(???%?)→?);
914  >foldl_append
915  whd in ⊢ (?(???%?)→?);
916  >mem_set_add
917  @eq_identifier_elim
918  [ #EQi destruct(EQi)
919    #_ @Exists_append_r % %
920  | #_ #H @Exists_append_l @Hi assumption
921  ]
922]
923qed.
924
925lemma list_as_set_mem : ∀tag.∀l : list (identifier tag).
926  ∀i.In ? l i → i ∈ l.
927#tag #l @(list_elim_left … l)
928[ #i *
929| #t #h #Hi #i #H
930  whd in ⊢ (?(???%?));
931  >foldl_append
932  whd in ⊢ (?(???%?));
933  elim (Exists_append … H) -H
934  [ #H >mem_set_add
935    @eq_identifier_elim [//] #_ normalize
936    @Hi @H
937  | * [2: *] #EQi destruct(EQi) >mem_set_add_id %
938  ]
939]
940qed.
941
942lemma list_as_set_All : ∀tag,P.∀ l : list (identifier tag).
943  (∀i.i ∈ l → P i) → All ? P l.
944#tag #P #l @(list_elim_left … l)
945[ #_ %
946| #x #l' #Hi
947  whd in match (set_from_list … (l'@[x]));
948  >foldl_append
949  #H @All_append
950  [ @Hi #i #G @H
951    whd in ⊢ (?(???%?));
952    >mem_set_add @orb_Prop_r @G
953  | % [2: %]
954    @H
955    whd in ⊢ (?(???%?));
956    @mem_set_add_id
957  ]
958]
959qed.
960
961lemma All_list_as_set : ∀tag,P.∀ l : list (identifier tag).
962  All ? P l → ∀i.i ∈ l → P i.
963#tag #P #l @(list_elim_left … l)
964[ * #i *
965| #x #l' #Hi #H
966  lapply (All_append_l … H)
967  lapply (All_append_r … H)
968  * #Px * #Pl' #i
969  whd in match (set_from_list … (l'@[x]));
970  >foldl_append
971  >mem_set_add
972  @eq_identifier_elim
973  [ #EQx >EQx #_ @Px
974  | #_ whd in match (?∨?); @Hi @Pl'
975  ]
976]
977qed. 
978
979lemma map_mem_prop :
980  ∀tag,A.∀m : identifier_map tag A.∀i.
981  lookup ?? m i ≠ None ? → i ∈ m.
982#p #globals #m #i
983lapply (in_map_domain … m i)
984cases (i∈m)
985[ * #x #_ #_ %
986| #EQ >EQ * #ABS @ABS %
987] qed.
988
989
990(* Attempt to choose an entry in the map/set, and if successful return the entry
991   and the map/set without it. *)
992
993definition choose : ∀tag,A. identifier_map tag A → option (identifier tag × A × (identifier_map tag A)) ≝
994λtag,A,m.
995  match pm_choose A (match m with [ an_id_map m' ⇒ m' ]) with
996  [ None ⇒ None ?
997  | Some x ⇒ Some ? 〈〈an_identifier tag (\fst (\fst x)), \snd (\fst x)〉, an_id_map tag A (\snd x)〉
998  ].
999
1000lemma choose_empty : ∀tag,A,m.
1001  choose tag A m = None ? ↔ ∀id. lookup tag A m id = None ?.
1002#tag #A * #m lapply (pm_choose_empty A m) * #H1 #H2 %
1003[ normalize #C * @H1 cases (pm_choose A m) in C ⊢ %; [ // | normalize #x #E destruct ]
1004| normalize #L lapply (pm_choose_empty A m) cases (pm_choose A m)
1005  [ * #H1 #H2 normalize // | #x * #_ #H lapply (H ?) [ #p @(L (an_identifier ? p)) | #E destruct ] ]
1006] qed.
1007
1008lemma choose_some : ∀tag,A,m,id,a,m'.
1009  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
1010  lookup tag A m id = Some A a ∧
1011  lookup tag A m' id = None A ∧
1012  (∀id'. id = id' ∨ lookup tag A m id' = lookup tag A m' id').
1013#tag #A * #m * #id #a * #m' #C
1014lapply (pm_choose_some A m id a m' ?)
1015[ whd in C:(??%?); cases (pm_choose A m) in C ⊢ %; normalize [ #E destruct | * * #x #y #z #E destruct % ] ]
1016* * * #L1 #L2 #L3 #_
1017% [ % [ @L1 | @L2 ] | * #id' cases (L3 id') [ /2 by or_introl/ | #L4 %2 @L4 ] ]
1018qed.
1019
1020lemma choose_some_subset : ∀tag,A,m,id,a,m'.
1021  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
1022  m' ⊆ m.
1023#tag #A #m #id #a #m' #C
1024cases (choose_some … m' C) * #L1 #L2 #L3
1025#id' whd in ⊢ (?% → ?%);
1026cases (L3 id')
1027[ #E destruct >L2 *
1028| #L4 >L4 //
1029] qed.
1030
1031lemma choose_some_card : ∀tag,A,m,id,a,m'.
1032  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
1033  |m| = S (|m'|).
1034#tag #A * #m * #id #a * #m' #C
1035lapply (pm_choose_some A m id a m' ?)
1036[ whd in C:(??%?); cases (pm_choose A m) in C ⊢ %; normalize [ #E destruct | * * #x #y #z #E destruct % ] ]
1037* #_ #H @H
1038qed.
1039
1040(* Remove an element from a map/set, returning the element and a new map/set. *)
1041
1042definition try_remove : ∀tag,A. identifier_map tag A → identifier tag → option (A × (identifier_map tag A)) ≝
1043λtag,A,m,id.
1044  match pm_try_remove A (match id with [ an_identifier id' ⇒ id']) (match m with [ an_id_map m' ⇒ m' ]) with
1045  [ None ⇒ None ?
1046  | Some x ⇒ Some ? 〈\fst x, an_id_map tag A (\snd x)〉
1047  ].
1048
1049lemma try_remove_empty : ∀tag,A,m,id.
1050  try_remove tag A m id = None ? ↔ lookup tag A m id = None ?.
1051#tag #A * #m * #id lapply (pm_try_remove_none A id m) * #H1 #H2 %
1052[ normalize #C @H1 cases (pm_try_remove A id m) in C ⊢ %; [ // | normalize #x #E destruct ]
1053| normalize #L >H2 //
1054] qed.
1055
1056lemma try_remove_some : ∀tag,A,m,id,a,m'.
1057  try_remove tag A m id = Some ? 〈a,m'〉 →
1058  lookup tag A m id = Some A a ∧
1059  lookup tag A m' id = None A ∧
1060  (∀id'. id = id' ∨ lookup tag A m id' = lookup tag A m' id').
1061#tag #A * #m * #id #a * #m' #C
1062lapply (pm_try_remove_some A id m a m' ?)
1063[ whd in C:(??%?); cases (pm_try_remove A id m) in C ⊢ %; normalize [ #E destruct | * #x #y #E destruct % ] ]
1064* * * #L1 #L2 #L3 #_
1065% [ % [ @L1 | @L2 ] | * #id' cases (L3 id') [ /2 by or_introl/ | #L4 %2 @L4 ] ]
1066qed.
1067
1068lemma try_remove_some_card : ∀tag,A,m,id,a,m'.
1069  try_remove tag A m id = Some ? 〈a,m'〉 →
1070  |m| = S (|m'|).
1071#tag #A * #m * #id #a * #m' #C
1072lapply (pm_try_remove_some A id m a m' ?)
1073[ whd in C:(??%?); cases (pm_try_remove A id m) in C ⊢ %; normalize [ #E destruct | * #x #y #E destruct % ] ]
1074* #_ #H @H
1075qed.
1076
1077lemma try_remove_this : ∀tag,A,m,id,a.
1078  lookup tag A m id = Some A a →
1079  ∃m'. try_remove tag A m id = Some ? 〈a,m'〉.
1080#tag #A * #m * #id #a #L
1081cases (pm_try_remove_some' A id m a L)
1082#m' #R %{(an_id_map tag A m')} whd in ⊢ (??%?); >R %
1083qed.
1084 
1085(* Link a map with the set consisting of its domain. *)
1086
1087definition id_set_of_map : ∀tag,A. identifier_map tag A → identifier_set tag ≝
1088λtag,A,m. an_id_map tag unit (map … (λ_. it) (match m with [ an_id_map m' ⇒ m'])).
1089
1090lemma id_set_of_map_subset : ∀tag,A,m.
1091  id_set_of_map tag A m ⊆ m.
1092#tag #A * #m * #id normalize
1093>lookup_opt_map normalize cases (lookup_opt ???) //
1094qed.
1095
1096lemma id_set_of_map_present : ∀tag,A,m,id.
1097  present tag A m id ↔ present tag unit (id_set_of_map … m) id.
1098#tag #A * #m * #id %
1099normalize @not_to_not
1100>lookup_opt_map cases (lookup_opt ???) normalize //
1101#a #E destruct
1102qed.
1103
1104lemma id_set_of_map_card : ∀tag,A,m.
1105  |m| = |id_set_of_map tag A m|.
1106#tag #A * #m whd in ⊢ (??%%); >map_size //
1107qed.
1108
1109
1110(* Transforming a list into a set. *)
1111
1112definition set_of_list : ∀tag. list (identifier tag) → identifier_set tag ≝
1113λtag,l. foldl ?? (λs,id. add_set tag s id) ∅ l.
1114
1115lemma fold_add_set_monotone : ∀tag,l,s,id.
1116  present tag unit s id →
1117  present tag unit (foldl ?? (λs,id. add_set tag s id) s l) id.
1118#tag #l elim l
1119[ //
1120| #h #t #IH #s #id #PR
1121  whd in ⊢ (???%?); @IH
1122  @lookup_add_oblivious
1123  @PR
1124] qed.
1125
1126lemma in_set_of_list : ∀tag,l,id.
1127  Exists ? (λid'. id' = id) l →
1128  present ?? (set_of_list tag l) id.
1129#tag #l #id whd in match (set_of_list ??); generalize in match ∅; elim l
1130[ #s *
1131| #id' #tl #IH #s *
1132  [ #E whd in ⊢ (???%?); destruct
1133    @fold_add_set_monotone //
1134  | @IH
1135  ]
1136] qed.
1137
1138lemma in_set_of_list' : ∀tag,l,id.
1139  present ?? (set_of_list tag l) id →
1140  Exists ? (λid'. id = id') l.
1141#tag #l #id whd in match (set_of_list ??);
1142cut (¬present ?? ∅ id) [ /3 by refl, absurd, nmk/ ]
1143generalize in match ∅;
1144elim l
1145[ #s #F #T @⊥ @(absurd … T F)
1146| #id' #tl #IH #s #F #PR whd in PR:(???%?);
1147  cases (identifier_eq … id id')
1148  [ #E destruct /2 by or_introl/
1149  | #NE %2 @(IH … PR)
1150    @(not_to_not … F) /2 by present_add_present/
1151  ]
1152] qed.
1153
1154
1155(* Returns the domain of a map as the canonical set (one made only from the
1156   empty set and addition. *)
1157definition domain_of_map : ∀tag,A. identifier_map tag A → identifier_set tag ≝
1158λtag,A,m. an_id_map tag unit (domain_of_pm A (match m with [ an_id_map m ⇒ m ])).
1159lemma domain_of_map_present : ∀tag,A,m,id.
1160  present tag A m id ↔ present tag unit (domain_of_map … m) id.
1161#tag #A * #m * #p @domain_of_pm_present
1162qed.
1163
1164(* Some lemmas for reasoning about folds. *)
1165
1166lemma foldi_ind : ∀A,B,tag,f,m,b. ∀P:B → identifier_set tag → Prop.
1167  P b (empty_set …) →
1168  (∀k,ks,a,b. ¬present ?? ks k → lookup ?? m k = Some ? a → P b ks → P (f k a b) (add_set tag ks k)) →
1169  P (foldi A B tag f m b) (domain_of_map … m).
1170#A #B #tag #f * #m #b #P #P0 #STEP
1171whd in match (foldi ??????); change with (an_id_map ?? (domain_of_pm A m)) in match (domain_of_map ???);
1172@pm_fold_ind
1173[ @P0
1174| #p #ps #a #b0 #FR #L #pre @(STEP (an_identifier tag p) (an_id_map tag unit ps))
1175  [ normalize >FR /3 by absurd, nmk/
1176  | @L
1177  | @pre
1178  ]
1179] qed.
1180
1181lemma foldi_ind' : ∀A,B,tag,f,m,b,b'. ∀P:B → identifier_set tag → Prop.
1182  P b (empty_set …) →
1183  (∀k,ks,a,b. ¬present ?? ks k → lookup ?? m k = Some ? a → P b ks → P (f k a b) (add_set tag ks k)) →
1184  foldi A B tag f m b = b' →
1185  P b' (domain_of_map … m).
1186#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11  destruct @foldi_ind /2 by /
1187qed.
1188
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