source: src/common/Identifiers.ma @ 1908

Last change on this file since 1908 was 1908, checked in by fguidi, 8 years ago

notation fixup following last commit of matita
we shifted the levels of precedence from 50 to 60 up by 5

File size: 23.4 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   
83definition word_of_identifier ≝
84  λt.
85  λl: identifier t.
86  match l with   
87  [ an_identifier l' ⇒ l'
88  ].
89
90lemma eq_identifier_refl : ∀tag,id. eq_identifier tag id id = true.
91#tag * #id whd in ⊢ (??%?); >eqb_n_n @refl
92qed.
93
94axiom eq_identifier_sym:
95  ∀tag: String.
96  ∀l  : identifier tag.
97  ∀r  : identifier tag.
98    eq_identifier tag l r = eq_identifier tag r l.
99
100lemma eq_identifier_false : ∀tag,x,y. x≠y → eq_identifier tag x y = false.
101#tag * #x * #y #NE normalize @not_eq_to_eqb_false /2 by not_to_not/
102qed.
103
104definition identifier_eq : ∀tag:String. ∀x,y:identifier tag. (x=y) + (x≠y).
105#tag * #x * #y lapply (refl ? (eqb x y)) cases (eqb x y) in ⊢ (???% → %);
106#E [ % | %2 ]
107lapply E @eqb_elim
108[ #H #_ >H @refl | 2,3: #_ #H destruct | #H #_ % #H' destruct /2 by absurd/ ]
109qed.
110
111definition identifier_of_nat : ∀tag:String. nat → identifier tag ≝
112  λtag,n. an_identifier tag (succ_pos_of_nat  n).
113
114
115(* States that all identifiers in an environment are distinct from one another. *)
116let rec distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : Prop ≝
117match l with
118[ nil ⇒ True
119| cons hd tl ⇒ All ? (λia. \fst hd ≠ \fst ia) tl ∧
120               distinct_env tag A tl
121].
122
123lemma distinct_env_append_l : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A l.
124#tag #A #l elim l
125[ //
126| * #id #a #tl #IH #r * #H1 #H2 % /2 by All_append_l/
127] qed.
128
129lemma distinct_env_append_r : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A r.
130#tag #A #l elim l
131[ //
132| * #id #a #tl #IH #r * #H1 #H2 /2 by /
133] qed.
134
135(* check_distinct_env is quadratic - we could pay more attention when building maps to make sure that
136   the original environment was distinct. *)
137
138axiom DuplicateVariable : String.
139
140let 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) ≝
141match l return λl.res (All ?? l) with
142[ nil ⇒ OK ? I
143| cons hd tl ⇒
144    match identifier_eq tag id (\fst hd) with
145    [ inl _ ⇒ Error ? [MSG DuplicateVariable; CTX ? id]
146    | inr NE ⇒
147        do Htl ← check_member_env tag A id tl;
148        OK ? (conj ?? NE Htl)
149    ]
150].
151
152let rec check_distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : res (distinct_env tag A l) ≝
153match l return λl.res (distinct_env tag A l) with
154[ nil ⇒ OK ? I
155| cons hd tl ⇒
156    do Hhd ← check_member_env tag A (\fst hd) tl;
157    do Htl ← check_distinct_env tag A tl;
158    OK ? (conj ?? Hhd Htl)
159].
160
161
162
163
164(* Maps from identifiers to arbitrary types. *)
165
166include "common/PositiveMap.ma".
167
168inductive identifier_map (tag:String) (A:Type[0]) : Type[0] ≝
169  an_id_map : positive_map A → identifier_map tag A.
170 
171definition empty_map : ∀tag:String. ∀A. identifier_map tag A ≝
172  λtag,A. an_id_map tag A (pm_leaf A).
173
174let rec lookup tag A (m:identifier_map tag A) (l:identifier tag) on m : option A ≝
175  lookup_opt A (match l with [ an_identifier l' ⇒ l' ])
176               (match m with [ an_id_map m' ⇒ m' ]).
177
178definition lookup_def ≝
179λtag,A,m,l,d. match lookup tag A m l with [ None ⇒ d | Some x ⇒ x].
180
181let rec member tag A (m:identifier_map tag A) (l:identifier tag) on m : bool ≝
182  match lookup tag A m l with [ None ⇒ false | _ ⇒ true ].
183
184(* Always adds the identifier to the map. *)
185let rec add tag A (m:identifier_map tag A) (l:identifier tag) (a:A) on m : identifier_map tag A ≝
186  an_id_map tag A (insert A (match l with [ an_identifier l' ⇒ l' ]) a
187                            (match m with [ an_id_map m' ⇒ m' ])).
188
189lemma lookup_add_hit : ∀tag,A,m,i,a.
190  lookup tag A (add tag A m i a) i = Some ? a.
191#tag #A * #m * #i #a
192@lookup_opt_insert_hit
193qed.
194
195lemma lookup_def_add_hit : ∀tag,A,m,i,a,d.
196  lookup_def tag A (add tag A m i a) i d = a.
197#tag #A * #m * #i #a #d
198@lookup_insert_hit
199qed.
200
201lemma lookup_add_miss : ∀tag,A,m,i,j,a.
202  i ≠ j →
203  lookup tag A (add tag A m j a) i = lookup tag A m i.
204#tag #A * #m * #i * #j #a #H
205@lookup_opt_insert_miss /2 by not_to_not/
206qed.
207
208axiom lookup_def_add_miss : ∀tag,A,m,i,j,a,d.
209  i ≠ j →
210  lookup_def tag A (add tag A m j a) i d = lookup_def tag A m i d.
211
212lemma lookup_add_oblivious : ∀tag,A,m,i,j,a.
213  (lookup tag A m i ≠ None ?) →
214  lookup tag A (add tag A m j a) i ≠ None ?.
215#tag #A #m #i #j #a #H
216cases (identifier_eq ? i j)
217[ #E >E >lookup_add_hit % #N destruct
218| #NE >lookup_add_miss //
219] qed.
220
221lemma lookup_add_cases : ∀tag,A,m,i,j,a,v.
222  lookup tag A (add tag A m i a) j = Some ? v →
223  (i=j ∧ v = a) ∨ lookup tag A m j = Some ? v.
224#tag #A #m #i #j #a #v
225cases (identifier_eq ? i j)
226[ #E >E >lookup_add_hit #H %1 destruct % //
227| #NE >lookup_add_miss /2 by or_intror, sym_not_eq/
228] qed.
229
230(* Extract every identifier, value pair from the map. *)
231definition elements : ∀tag,A. identifier_map tag A → list (identifier tag × A) ≝
232λtag,A,m.
233  fold ?? (λl,a,el. 〈an_identifier tag l, a〉::el)
234          (match m with [ an_id_map m' ⇒ m' ]) [ ].
235
236axiom MissingId : String.
237
238(* Only updates an existing entry; fails with an error otherwise. *)
239definition update : ∀tag,A. identifier_map tag A → identifier tag → A → res (identifier_map tag A) ≝
240λtag,A,m,l,a.
241  match update A (match l with [ an_identifier l' ⇒ l' ]) a
242                 (match m with [ an_id_map m' ⇒ m' ]) with
243  [ None ⇒ Error ? ([MSG MissingId; CTX tag l]) (* missing identifier *)
244  | Some m' ⇒ OK ? (an_id_map tag A m')
245  ].
246
247definition foldi:
248  ∀A, B: Type[0].
249  ∀tag: String.
250  (identifier tag -> A -> B -> B) -> identifier_map tag A -> B -> B ≝
251λA,B,tag,f,m,b.
252  match m with
253  [ an_id_map m' ⇒ fold A B (λbv. f (an_identifier ? bv)) m' b ].
254
255(* A predicate that an identifier is in a map, and a failure-avoiding lookup
256   and update using it. *)
257
258definition present : ∀tag,A. identifier_map tag A → identifier tag → Prop ≝
259λtag,A,m,i. lookup … m i ≠ None ?.
260
261lemma member_present : ∀tag,A,m,id.
262  member tag A m id = true → present tag A m id.
263#tag #A * #m #id normalize cases (lookup_opt A ??) normalize
264[ #E destruct
265| #x #E % #E' destruct
266] qed.
267
268include "ASM/Util.ma".
269
270definition lookup_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A ≝
271λtag,A,m,id. match lookup ?? m id return λx. x ≠ None ? → ? with [ Some a ⇒ λ_. a | None ⇒ λH.⊥ ].
272cases H #H'  cases (H' (refl ??)) qed.
273
274lemma lookup_lookup_present : ∀tag,A,m,id,p.
275  lookup tag A m id = Some ? (lookup_present tag A m id p).
276#tag #A #m #id #p
277whd in p ⊢ (???(??%));
278cases (lookup tag A m id) in p ⊢ %;
279[ * #H @⊥ @H @refl
280| #a #H @refl
281] qed.
282
283definition update_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A → identifier_map tag A ≝
284λtag,A,m,l,p,a.
285  let l' ≝ match l with [ an_identifier l' ⇒ l' ] in
286  let m' ≝ match m with [ an_id_map m' ⇒ m' ] in
287  let u' ≝ update A l' a m' in
288  match u' return λx. update ???? = x → ? with
289  [ None ⇒ λE.⊥
290  | Some m' ⇒ λ_. an_id_map tag A m'
291  ] (refl ? u').
292cases l in p E; cases m; -l' -m' #m' #l'
293whd in ⊢ (% → ?);
294 whd in ⊢ (?(??(???%%)?) → ??(??%?%)? → ?);
295#NL #U cases NL #H @H @(update_fail … U)
296qed.
297
298lemma update_still_present : ∀tag,A,m,id,a,id'.
299  ∀H:present tag A m id.
300  ∀H':present tag A m id'.
301  present tag A (update_present tag A m id' H' a) id.
302#tag #A * #m * #id #a * #id' #H #H'
303whd whd in ⊢ (?(??(???(%??????)?)?)); normalize nodelta
304cases (identifier_eq ? (an_identifier tag id) (an_identifier tag id'))
305[ #E >E @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); >(update_lookup_opt_same ????? U)
306  % #E' destruct
307| #NE @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); whd in ⊢ (?(??(??%%)?));
308  <(update_lookup_opt_other ????? U id) [ @H | % #E cases NE >E #H @H @refl ]
309] qed.
310
311
312let rec fresh_for_map tag A (id:identifier tag) (m:identifier_map tag A) on id : Prop ≝
313  lookup … m id = None A.
314
315lemma fresh_for_empty_map : ∀tag,A,id.
316  fresh_for_map tag A id (empty_map tag A).
317#tag #A * #id //
318qed.
319
320definition fresh_map_for_univ ≝
321λtag,A. λm:identifier_map tag A. λu:universe tag.
322  ∀id. present tag A m id → fresh_for_univ tag id u.
323
324lemma fresh_fresh_for_map : ∀tag,A,m,id,u,u'.
325  fresh_map_for_univ tag A m u →
326  〈id,u'〉 = fresh tag u →
327  fresh_for_map tag A id m.
328#tag #A * #m * #id * #u * #u' whd in ⊢ (% → ???% → %);
329#FMU #E destruct lapply (FMU (an_identifier tag u)) whd in ⊢ ((% → %) → ?);
330generalize in ⊢ ((?(??%?) → ?) → ??%?); *
331[ // | #a #H @False_ind lapply (H ?) /2 by absurd/ % #E destruct
332qed.
333
334lemma fresh_map_preserved : ∀tag,A,m,u,u',id.
335  fresh_map_for_univ tag A m u →
336  〈id,u'〉 = fresh tag u →
337  fresh_map_for_univ tag A m u'.
338#tag #A #m #u * #u' #id whd in ⊢ (% → ? → %); #H #E
339#id' #PR @(fresh_remains_fresh … E) @H //
340qed.
341
342lemma fresh_map_add : ∀tag,A,m,u,id,a.
343  fresh_map_for_univ tag A m u →
344  fresh_for_univ tag id u →
345  fresh_map_for_univ tag A (add tag A m id a) u.
346#tag #A * #m #u #id #a #Hm #Hi
347#id' #PR cases (identifier_eq tag id' id)
348[ #E >E @Hi
349| #NE @Hm whd in PR;
350  change with (add tag A (an_id_map tag A m) id a) in PR:(?(??(???%?)?));
351  >lookup_add_miss in PR; //
352] qed.
353
354lemma present_not_fresh : ∀tag,A,m,id,id'.
355  present tag A m id →
356  fresh_for_map tag A id' m →
357  id ≠ id'.
358#tag #A #m #id * #id' whd in ⊢ (% → % → ?);
359* #NE #E % #E' destruct @(NE E)
360qed.
361
362lemma fresh_for_map_add : ∀tag,A,id,m,id',a.
363  id ≠ id' →
364  fresh_for_map tag A id m →
365  fresh_for_map tag A id (add tag A m id' a).
366#tag #A * #id #m #id' #a #NE #F
367whd >lookup_add_miss //
368qed.
369
370
371(* Sets *)
372
373definition identifier_set ≝ λtag.identifier_map tag unit.
374
375definition empty_set : ∀tag.identifier_set tag ≝ λtag.empty_map ….
376
377
378definition add_set : ∀tag.identifier_set tag → identifier tag → identifier_set tag ≝
379  λtag,s,i.add … s i it.
380
381definition singleton_set : ∀tag:String. identifier tag → identifier_set tag ≝
382λtag,i. add_set tag (empty_set tag) i.
383
384(* mem set is generalised to all maps *)
385let rec mem_set (tag:String) A (s:identifier_map tag A) (i:identifier tag) on s : bool ≝
386  match lookup … s i with
387  [ None ⇒ false
388  | Some _ ⇒ true
389  ].
390 
391let rec union_set (tag:String) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_set tag ≝
392  an_id_map tag unit (merge … (λo,o'.match o with [Some _ ⇒ Some ? it | None ⇒ !_ o'; return it])
393    (match s with [ an_id_map s0 ⇒ s0 ])
394    (match s' with [ an_id_map s1 ⇒ s1 ])).
395
396
397(* set minus is generalised to maps *)
398let rec minus_set (tag:String) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_map tag A ≝
399  an_id_map tag A (merge A B A (λo,o'.match o' with [None ⇒ o | Some _ ⇒ None ?])
400    (match s with [ an_id_map s0 ⇒ s0 ])
401    (match s' with [ an_id_map s1 ⇒ s1 ])).
402
403notation "a ∖ b" left associative with precedence 55 for @{'setminus $a $b}.
404
405interpretation "identifier set union" 'union a b = (union_set ??? a b).
406notation "∅" non associative with precedence 90 for @{ 'empty }.
407interpretation "empty identifier set" 'empty = (empty_set ?).
408interpretation "singleton identifier set" 'singl a = (add_set ? (empty_set ?) a).
409interpretation "identifier set membership" 'mem a b = (mem_set ?? b a).
410interpretation "identifier map difference" 'setminus a b = (minus_set ??? a b).
411
412definition IdentifierSet : String → Setoid ≝ λtag.
413  mk_Setoid (identifier_set tag) (λs,s'.∀i.i ∈ s = (i ∈ s')) ???.
414  // qed.
415
416unification hint 0 ≔ tag;
417S ≟ IdentifierSet tag
418(*-----------------------------*)⊢
419identifier_set tag ≡ std_supp S.
420unification hint 0 ≔ tag;
421S ≟ IdentifierSet tag
422(*-----------------------------*)⊢
423identifier_map tag unit ≡ std_supp S.
424
425lemma mem_set_add : ∀tag,A.∀i,j : identifier tag.∀s,x.
426  i ∈ add ? A s j x = (eq_identifier ? i j ∨ i ∈ s).
427#tag #A *#i *#j *#s #x normalize
428@(eqb_elim i j)
429[#EQ destruct
430  >(lookup_opt_insert_hit A x j)
431|#NEQ >(lookup_opt_insert_miss … s NEQ)
432] elim (lookup_opt  A j s) normalize // qed.
433
434lemma mem_set_add_id : ∀tag,A,i,s,x.bool_to_Prop (i ∈ add tag A s i x).
435#tag #A #i #s #x >mem_set_add
436@eq_identifier_elim [#_ %| #ABS elim (absurd … (refl ? i) ABS)] qed.
437
438lemma in_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
439  if i ∈ m then (∃s.lookup … m i = Some ? s) else (lookup … m i = None ?).
440#tag #A * #m * #i normalize
441elim (lookup_opt A i m) normalize
442[ % | #x %{x} % ]
443qed.
444
445lemma union_empty_l : ∀tag.∀s:identifier_set tag. ∅ ∪ s = s.
446#tag * normalize #m >map_opt_id_eq_ext // * %
447qed.
448
449lemma union_empty_r : ∀tag.∀s:identifier_set tag. s ∪ ∅ = s.
450#tag * * [//] *[2: *] #l#r normalize
451>map_opt_id_eq_ext [1,3: >map_opt_id_eq_ext [2,4: *] |*: *] //
452qed.
453
454lemma minus_empty_l : ∀tag,A.∀s:identifier_map tag A. ∅ ∖ s ≅ ∅.
455#tag #A * * [//] *[2:#x]#l#r * * normalize [1,4://]
456#p >lookup_opt_map elim (lookup_opt ???) normalize //
457qed.
458
459lemma minus_empty_r : ∀tag,A.∀s:identifier_map tag A. s ∖ ∅ = s.
460#tag #A * * [//] *[2:#x]#l#r normalize
461>map_opt_id >map_opt_id //
462qed.
463
464lemma mem_set_union : ∀tag.∀i : identifier tag.∀s,s' : identifier_set tag.
465  i ∈ (s ∪ s') = (i ∈ s ∨ i ∈ s').
466#tag * #i * #s * #s' normalize
467>lookup_opt_merge [2: @refl]
468elim (lookup_opt ???)
469elim (lookup_opt ???)
470normalize // qed.
471
472lemma mem_set_minus : ∀tag,A,B.∀i : identifier tag.∀s : identifier_map tag A.
473  ∀s' : identifier_map tag B.
474  i ∈ (s ∖ s') = (i ∈ s ∧ ¬ i ∈ s').
475#tag #A #B * #i * #s * #s' normalize
476>lookup_opt_merge [2: @refl]
477elim (lookup_opt ???)
478elim (lookup_opt ???)
479normalize // qed.
480
481lemma set_eq_ext_node : ∀tag.∀o,o',l,l',r,r'.
482  an_id_map tag ? (pm_node ? o l r) ≅ an_id_map … (pm_node ? o' l' r') →
483    o = o' ∧ an_id_map tag ? l ≅ an_id_map … l' ∧ an_id_map tag ? r ≅ an_id_map … r'.
484#tag#o#o'#l#l'#r#r'#H
485%[
486%[ lapply (H (an_identifier ? one))
487   elim o [2: *] elim o' [2,4: *] normalize // #EQ destruct
488 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
489]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
490]
491qed.
492
493lemma set_eq_ext_leaf : ∀tag,A.∀o,l,r.
494  (∀i.i∈an_id_map tag A (pm_node ? o l r) = false) →
495    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
496#tag#A#o#l#r#H
497%[
498%[ lapply (H (an_identifier ? one))
499   elim o [2: #a] normalize // #EQ destruct
500 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
501]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
502]
503qed.
504
505
506definition id_map_size : ∀tag : String.∀A. identifier_map tag A → ℕ ≝
507  λtag,A,s.match s with [an_id_map p ⇒ |p|].
508
509interpretation "identifier map domain size" 'norm s = (id_map_size ?? s).
510
511lemma set_eq_ext_empty_to_card : ∀tag,A.∀s : identifier_map tag A. (∀i.i∈s = false) → |s| = 0.
512#tag#A * #s elim s [//]
513#o#l#r normalize in ⊢((?→%)→(?→%)→?); #Hil #Hir #H
514elim (set_eq_ext_leaf … H) * #EQ destruct #Hl #Hr normalize
515>(Hil Hl) >(Hir Hr) // qed.
516
517lemma set_eq_ext_to_card : ∀tag.∀s,s' : identifier_set tag. s ≅ s' → |s| = |s'|.
518#tag *#s elim s
519[** [//] #o#l#r #H
520  >(set_eq_ext_empty_to_card … (std_symm … H)) //
521| #o#l#r normalize in ⊢((?→?→??%?)→(?→?→??%?)→?);
522  #Hil #Hir **
523  [#H @(set_eq_ext_empty_to_card … H)]
524  #o'#l'#r' #H elim (set_eq_ext_node … H) * #EQ destruct(EQ) #Hl #Hr
525  normalize >(Hil ? Hl) >(Hir ? Hr) //
526] qed.
527
528lemma add_size: ∀tag,A,s,i,x.
529  |add tag A s i x| = (if i ∈ s then 0 else 1) + |s|.
530#tag #A *#s *#i #x
531lapply (insert_size ? i x s)
532lapply (refl ? (lookup_opt ? i s))
533generalize in ⊢ (???%→?); * [2: #x']
534normalize #EQ >EQ normalize //
535qed.
536
537lemma mem_set_O_lt_card : ∀tag,A.∀i.∀s : identifier_map tag A. i ∈ s → |s| > 0.
538#tag #A * #i * #s normalize #H
539@(lookup_opt_O_lt_size … i)
540% #EQ >EQ in H; normalize *
541qed.
542
543(* NB: no control on values if applied to maps *)
544definition set_subset ≝ λtag,A,B.λs : identifier_map tag A.
545  λs' : identifier_map tag B. ∀i.i ∈ s → (bool_to_Prop (i ∈ s')).
546
547interpretation "identifier set subset" 'subseteq s s' = (set_subset ??? s s').
548
549lemma add_subset :
550  ∀tag,A,B.∀i : identifier tag.∀x.∀s : identifier_map ? A.∀s' : identifier_map ? B.
551    i ∈ s' → s ⊆ s' → add … s i x ⊆ s'.
552#tag#A#B#i#x#s#s' #H #G #j
553>mem_set_add
554@eq_identifier_elim #H' [* >H' @H | #js @(G ? js)]
555qed.
556
557definition set_forall : ∀tag,A.(identifier tag → Prop) →
558  identifier_map tag A → Prop ≝ λtag,A,P,m.∀i. i ∈ m → P i.
559 
560lemma set_forall_add : ∀tag,P,m,i.set_forall tag ? P m → P i →
561  set_forall tag ? P (add_set ? m i).
562#tag#P#m#i#Pm#Pi#j
563>mem_set_add
564@eq_identifier_elim
565[#EQ destruct(EQ) #_ @Pi
566|#_ @Pm
567]
568qed.
569
570include "utilities/proper.ma".
571
572lemma minus_subset : ∀tag,A,B.minus_set tag A B ⊨ set_subset … ++> set_subset … -+> set_subset ….
573#tag#A#B#s#s' #H #s'' #s''' #G #i
574>mem_set_minus >mem_set_minus
575#H' elim (andb_Prop_true … H') -H' #is #nis''
576>(H … is)
577elim (true_or_false_Prop (i∈s'''))
578[ #is''' >(G … is''') in nis''; *
579| #nis''' >nis''' %
580]
581qed.
582
583lemma subset_node : ∀tag,A,B.∀o,o',l,l',r,r'.
584  an_id_map tag A (pm_node ? o l r) ⊆ an_id_map tag B (pm_node ? o' l' r') →
585    opt_All ? (λ_.o' ≠ None ?) o ∧ an_id_map tag ? l ⊆ an_id_map tag  ? l' ∧
586      an_id_map tag ? r ⊆ an_id_map tag ? r'.
587#tag#A#B#o#o'#l#l'#r#r'#H
588%[%
589  [ lapply (H (an_identifier ? (one))) elim o [2: #a] elim o' [2:#b]
590    normalize // [#_ % #ABS destruct(ABS) | #G lapply (G I) *]
591  | *#p lapply (H (an_identifier ? (p0 p)))
592  ]
593 | *#p lapply (H (an_identifier ? (p1 p)))
594] #H @H
595qed.
596
597lemma subset_leaf : ∀tag,A.∀o,l,r.
598  an_id_map tag A (pm_node ? o l r) ⊆ ∅ →
599    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
600#tag#A#o#l#r#H
601%[
602%[ lapply (H (an_identifier ? one))
603   elim o [2: #a] normalize // #EQ lapply(EQ I) *
604 | *#p lapply (H (an_identifier ? (p0 p)))
605 ]
606|  *#p lapply (H (an_identifier ? (p1 p)))
607] normalize elim (lookup_opt ? p ?) normalize
608// #a #H lapply (H I) *
609qed.
610
611lemma subset_card : ∀tag,A,B.∀s : identifier_map tag A.∀s' : identifier_map tag B.
612  s ⊆ s' → |s| ≤ |s'|.
613#tag #A #B *#s elim s
614[ //
615| #o#l#r #Hil #Hir **
616  [ #H elim (subset_leaf … H) * #EQ >EQ #Hl #Hr
617    lapply (set_eq_ext_empty_to_card … Hl)
618    lapply (set_eq_ext_empty_to_card … Hr)
619    normalize //
620  | #o' #l' #r' #H elim (subset_node … H) *
621    elim o [2: #a] elim o' [2,4: #a']
622    [3: #G normalize in G; elim(absurd ? (refl ??) G)
623    |*: #_ #Hl #Hr lapply (Hil ? Hl) lapply (Hir ? Hr)
624      normalize #H1 #H2
625      [@le_S_S | @(transitive_le … (|l'|+|r'|)) [2: / by /]]
626      @le_plus assumption
627    ]
628  ]
629]
630qed.
631
632lemma mem_set_empty : ∀tag.∀i: identifier tag. i∈∅ = false.
633#tag * #i normalize %
634qed.
635
636lemma mem_set_singl_to_eq : ∀tag.∀i,j : identifier tag.i∈{(j)} → i = j.
637#tag
638#i #j >mem_set_add >mem_set_empty
639#H elim (orb_true_l … H) -H
640[@eq_identifier_elim [//] #_] #EQ destruct
641qed.
642
643lemma subset_add_set : ∀tag,i,s.s ⊆ add_set tag s i.
644#tag#i#s#j #H >mem_set_add >H
645>commutative_orb %
646qed.
647
648lemma add_set_monotonic : ∀tag,i,s,s'.s ⊆ s' → add_set tag s i ⊆ add_set tag s' i.
649#tag#i#s#s' #H #j >mem_set_add >mem_set_add
650@orb_elim elim (eq_identifier ???)
651whd lapply (H j) /2 by /
652qed.
653
654lemma transitive_subset : ∀tag,A.transitive ? (set_subset tag A A).
655#tag#A#s#s'#s''#H#G#i #is
656@(G … (H … is))
657qed.
658
659definition set_from_list : ∀tag.list (identifier tag) → identifier_map tag unit ≝
660  λtag.foldl … (add_set ?) ∅.
661
662coercion id_set_from_list : ∀tag.∀l : list (identifier tag).identifier_map tag unit ≝
663  set_from_list on _l : list (identifier ?) to identifier_map ? unit.
664
665lemma mem_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
666i∈m → lookup … m i ≠ None ?.
667#tag#A * #m #i
668whd in match (i∈?);
669elim (lookup ????) normalize [2: #x]
670* % #EQ destruct(EQ)
671qed.
672
673
674
675lemma mem_list_as_set : ∀tag.∀l : list (identifier tag).
676  ∀i.i ∈ l → In ? l i.
677#tag #l @(list_elim_left … l)
678[ #i *
679| #t #h #Hi  #i
680  whd in ⊢ (?(???%?)→?);
681  >foldl_append
682  whd in ⊢ (?(???%?)→?);
683  >mem_set_add
684  @eq_identifier_elim
685  [ #EQi destruct(EQi)
686    #_ @Exists_append_r % %
687  | #_ #H @Exists_append_l @Hi assumption
688  ]
689]
690qed.
691
692lemma list_as_set_mem : ∀tag.∀l : list (identifier tag).
693  ∀i.In ? l i → i ∈ l.
694#tag #l @(list_elim_left … l)
695[ #i *
696| #t #h #Hi #i #H
697  whd in ⊢ (?(???%?));
698  >foldl_append
699  whd in ⊢ (?(???%?));
700  elim (Exists_append … H) -H
701  [ #H >mem_set_add
702    @eq_identifier_elim [//] #_ normalize
703    @Hi @H
704  | * [2: *] #EQi destruct(EQi) >mem_set_add_id %
705  ]
706]
707qed.
708
709lemma list_as_set_All : ∀tag,P.∀ l : list (identifier tag).
710  (∀i.i ∈ l → P i) → All ? P l.
711#tag #P #l @(list_elim_left … l)
712[ #_ %
713| #x #l' #Hi
714  whd in match (l'@[x] : identifier_map tag unit);
715  >foldl_append
716  #H @All_append
717  [ @Hi #i #G @H
718    whd in ⊢ (?(???%?));
719    >mem_set_add @orb_Prop_r @G
720  | % [2: %]
721    @H
722    whd in ⊢ (?(???%?));
723    @mem_set_add_id
724  ]
725]
726qed.
727
728lemma All_list_as_set : ∀tag,P.∀ l : list (identifier tag).
729  All ? P l → ∀i.i ∈ l → P i.
730#tag #P #l @(list_elim_left … l)
731[ * #i *
732| #x #l' #Hi #H
733  lapply (All_append_l … H)
734  lapply (All_append_r … H)
735  * #Px * #Pl' #i
736  whd in match (l'@[x] : identifier_map ??);
737  >foldl_append
738  >mem_set_add
739  @eq_identifier_elim
740  [ #EQx >EQx #_ @Px
741  | #_ whd in match (?∨?); @Hi @Pl'
742  ]
743]
744qed. 
745
746
747
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