2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/deqlist.ma".
14 (****** DeqSet: a set with a decidable equality ******)
16 record FinSet : Type[1] ≝
17 { FinSetcarr:> DeqSet;
18 enum: list FinSetcarr;
19 enum_unique: uniqueb FinSetcarr enum = true;
20 enum_complete : ∀x:FinSetcarr. memb ? x enum = true
23 notation < "𝐅" non associative with precedence 90
25 interpretation "FinSet" 'bigF = (mk_FinSet ???).
28 lemma bool_enum_unique: uniqueb ? [true;false] = true.
31 lemma bool_enum_complete: ∀x:bool. memb ? x [true;false] = true.
35 mk_FinSet DeqBool [true;false] bool_enum_unique bool_enum_complete.
37 unification hint 0 ≔ ;
39 (* ---------------------------------------- *) ⊢
44 lemma eqbnat_true : ∀n,m. eqb n m = true ↔ n = m.
45 #n #m % [@eqb_true_to_eq | @eq_to_eqb_true]
48 definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
50 lemma lt_to_le : ∀n,m. n < m → n ≤ m.
53 let rec enumnaux n m ≝
54 match n return (λn.n ≤ m → list (Σx.x < m)) with
55 [ O ⇒ λh.[ ] | S p ⇒ λh:p < m.(mk_Sig ?? p h)::enumnaux p m (lt_to_le p m h)].
57 definition enumn ≝ λn.enumnaux n n (le_n n).
59 definition Nat_to ≝ λn. DeqSig DeqNat (λi.i<n).
61 (* lemma prova : ∀n. carr (Nat_to n) = (Σx.x<n). // *)
63 lemma memb_enumn: ∀m,n,i:DeqNat. ∀h:n ≤ m. ∀k: i < m. n ≤ i →
64 (¬ (memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h))) = true.
65 #m #n elim n -n // #n #Hind #i #ltm #k #ltni @sym_eq @noteq_to_eqnot @sym_not_eq
66 % #H cases (orb_true_l … H)
67 [whd in ⊢ (??%?→?); #H1 @(absurd … ltni) @le_to_not_lt
68 >(eqb_true_to_eq … H1) @le_n
69 |<(notb_notb (memb …)) >Hind normalize /2 by lt_to_le, absurd/
73 lemma enumn_unique_aux: ∀n,m. ∀h:n ≤ m. uniqueb (Nat_to m) (enumnaux n m h) = true.
74 #n elim n -n // #n #Hind #m #h @true_to_andb_true // @memb_enumn //
77 lemma enumn_unique: ∀n.uniqueb (Nat_to n) (enumn n) = true.
81 (* definition ltb ≝ λn,m.leb (S n) m. *)
82 lemma enumn_complete_aux: ∀n,m,i.∀h:n ≤m.∀k:i<m.i<n →
83 memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h) = true.
85 [normalize #n #i #_ #_ #Hfalse @False_ind /2/
86 |#n #Hind #m #i #h #k #lein whd in ⊢ (??%?);
87 cases (le_to_or_lt_eq … (le_S_S_to_le … lein))
88 [#ltin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = false)
89 [normalize @not_eq_to_eqb_false @lt_to_not_eq @ltin]
91 |#eqin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = true)
92 [normalize @eq_to_eqb_true //
98 lemma enumn_complete: ∀n.∀i:Nat_to n. memb ? i (enumn n) = true.
99 #n whd in ⊢ (%→?); * #i #ltin @enumn_complete_aux //
102 definition initN ≝ λn.
103 mk_FinSet (Nat_to n) (enumn n) (enumn_unique n) (enumn_complete n).
105 example tipa: ∀n.∃x: initN (S n). pi1 … x = n.
106 #n @ex_intro [whd @mk_Sig [@n | @le_n] | //] qed.
109 definition enum_option ≝ λA:DeqSet.λl.
110 None A::(map ?? (Some A) l).
112 lemma enum_option_def : ∀A:FinSet.∀l.
113 enum_option A l = None A :: (map ?? (Some A) l).
116 lemma enum_option_unique: ∀A:DeqSet.∀l.
118 uniqueb ? (enum_option A l) = true.
119 #A #l #uA @true_to_andb_true
120 [generalize in match uA; -uA #_ elim l [%]
121 #a #tl #Hind @sym_eq @noteq_to_eqnot % #H
122 cases (orb_true_l … (sym_eq … H))
123 [#H1 @(absurd (None A = Some A a)) [@(\P H1) | % #H2 destruct]
124 |-H #H >H in Hind; normalize /2/
126 |@unique_map_inj // #a #a1 #H destruct %
130 lemma enum_option_complete: ∀A:DeqSet.∀l.
131 (∀x:A. memb A x l = true) →
132 ∀x:DeqOption A. memb ? x (enum_option A l) = true.
133 #A #l #Hl * // #a @memb_cons @memb_map @Hl
136 definition FinOption ≝ λA:FinSet.
137 mk_FinSet (DeqOption A)
138 (enum_option A (enum A))
139 (enum_option_unique … (enum_unique A))
140 (enum_option_complete … (enum_complete A)).
142 unification hint 0 ≔ C;
145 (* ---------------------------------------- *) ⊢
146 option T ≡ FinSetcarr X.
149 definition enum_sum ≝ λA,B:DeqSet.λl1.λl2.
150 (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
152 lemma enum_sum_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 l2 =
153 (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
156 lemma enum_sum_unique: ∀A,B:DeqSet.∀l1,l2.
157 uniqueb A l1 = true → uniqueb B l2 = true →
158 uniqueb ? (enum_sum A B l1 l2) = true.
159 #A #B #l1 #l2 elim l1
160 [#_ #ul2 @unique_map_inj // #b1 #b2 #Hinr destruct //
161 |#a #tl #Hind #uA #uB @true_to_andb_true
162 [@sym_eq @noteq_to_eqnot % #H
163 cases (memb_append … (sym_eq … H))
164 [#H1 @(absurd (memb ? a tl = true))
165 [@(memb_map_inj …H1) #a1 #a2 #Hinl destruct //
166 |<(andb_true_l … uA) @eqnot_to_noteq //
169 [normalize #H destruct
170 |#b #tlB #Hind #membH cases (orb_true_l … membH)
171 [#H lapply (\P H) #H1 destruct |@Hind]
174 |@Hind // @(andb_true_r … uA)
179 lemma enum_sum_complete: ∀A,B:DeqSet.∀l1,l2.
180 (∀x:A. memb A x l1 = true) →
181 (∀x:B. memb B x l2 = true) →
182 ∀x:DeqSum A B. memb ? x (enum_sum A B l1 l2) = true.
183 #A #B #l1 #l2 #Hl1 #Hl2 *
184 [#a @memb_append_l1 @memb_map @Hl1
185 |#b @memb_append_l2 @memb_map @Hl2
189 definition FinSum ≝ λA,B:FinSet.
190 mk_FinSet (DeqSum A B)
191 (enum_sum A B (enum A) (enum B))
192 (enum_sum_unique … (enum_unique A) (enum_unique B))
193 (enum_sum_complete … (enum_complete A) (enum_complete B)).
195 include alias "basics/types.ma".
197 unification hint 0 ≔ C1,C2;
201 (* ---------------------------------------- *) ⊢
202 T1+T2 ≡ FinSetcarr X.
206 definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
207 compose ??? (mk_Prod A B) l1 l2.
209 lemma enum_prod_unique: ∀A,B,l1,l2.
210 uniqueb A l1 = true → uniqueb B l2 = true →
211 uniqueb ? (enum_prod A B l1 l2) = true.
213 #a #tl #Hind #l2 #H1 #H2 @uniqueb_append
214 [@unique_map_inj [#x #y #Heq @(eq_f … \snd … Heq) | //]
215 |@Hind // @(andb_true_r … H1)
216 |#p #H3 cases (memb_map_to_exists … H3) #b *
217 #Hmemb #eqp <eqp @(not_to_not ? (memb ? a tl = true))
218 [2: @sym_not_eq @eqnot_to_noteq @sym_eq @(andb_true_l … H1)
221 |#a1 #tl1 #Hind2 #H4 cases (memb_append … H4) -H4 #H4
222 [cases (memb_map_to_exists … H4) #b1 * #memb1 #H destruct (H)
223 normalize >(\b (refl ? a)) //
224 |@orb_true_r2 @Hind2 @H4
231 lemma enum_prod_complete:∀A,B:DeqSet.∀l1,l2.
232 (∀a. memb A a l1 = true) → (∀b.memb B b l2 = true) →
233 ∀p. memb ? p (enum_prod A B l1 l2) = true.
234 #A #B #l1 #l2 #Hl1 #Hl2 * #a #b @memb_compose //
238 λA,B:FinSet.mk_FinSet (DeqProd A B)
239 (enum_prod A B (enum A) (enum B))
240 (enum_prod_unique A B … (enum_unique A) (enum_unique B))
241 (enum_prod_complete A B … (enum_complete A) (enum_complete B)).
243 unification hint 0 ≔ C1,C2;
247 (* ---------------------------------------- *) ⊢
248 T1×T2 ≡ FinSetcarr X.
250 (* graph of a function *)
252 definition graph_of ≝ λA,B.λf:A→B.
253 Σp:A×B.f (\fst p) = \snd p.
255 definition graph_enum ≝ λA,B:FinSet.λf:A→B.
256 filter ? (λp.f (\fst p) == \snd p) (enum (FinProd A B)).
258 lemma graph_enum_unique : ∀A,B,f.
259 uniqueb ? (graph_enum A B f) = true.
260 #A #B #f @uniqueb_filter @(enum_unique (FinProd A B))
263 lemma graph_enum_correct: ∀A,B:FinSet.∀f:A→B. ∀a,b.
264 memb ? 〈a,b〉 (graph_enum A B f) = true → f a = b.
265 #A #B #f #a #b #membp @(\P ?) @(filter_true … membp)
268 lemma graph_enum_complete: ∀A,B:FinSet.∀f:A→B. ∀a,b.
269 f a = b → memb ? 〈a,b〉 (graph_enum A B f) = true.
270 #A #B #f #a #b #eqf @memb_filter_l [@(\b eqf)]
271 @enum_prod_complete //
276 definition enum_fun_raw: ∀A,B:DeqSet.list A → list B → list (list (DeqProd A B)) ≝
277 λA,B,lA,lB.foldr A (list (list (DeqProd A B)))
278 (λa.compose ??? (λb. cons ? 〈a,b〉) lB) [[]] lA.
280 lemma enum_fun_raw_cons: ∀A,B,a,lA,lB.
281 enum_fun_raw A B (a::lA) lB =
282 compose ??? (λb. cons ? 〈a,b〉) lB (enum_fun_raw A B lA lB).
286 definition is_functional ≝ λA,B:DeqSet.λlA:list A.λl: list (DeqProd A B).
287 map ?? (fst A B) l = lA.
289 definition carr_fun ≝ λA,B:FinSet.
290 DeqSig (DeqList (DeqProd A B)) (is_functional A B (enum A)).
292 definition carr_fun_l ≝ λA,B:DeqSet.λl.
293 DeqSig (DeqList (DeqProd A B)) (is_functional A B l).
295 lemma compose_spec1 : ∀A,B,C:DeqSet.∀f:A→B→C.∀a:A.∀b:B.∀lA:list A.∀lB:list B.
296 a ∈ lA = true → b ∈ lB = true → ((f a b) ∈ (compose A B C f lA lB)) = true.
297 #A #B #C #f #a #b #lA elim lA
298 [normalize #lB #H destruct
299 |#a1 #tl #Hind #lB #Ha #Hb cases (orb_true_l ?? Ha) #Hcase
300 [>(\P Hcase) normalize @memb_append_l1 @memb_map //
301 |@memb_append_l2 @Hind //
306 lemma compose_cons: ∀A,B,C.∀f:A→B→C.∀l1,l2,a.
307 compose A B C f (a::l1) l2 =
308 (map ?? (f a) l2)@(compose A B C f l1 l2).
311 lemma compose_spec2 : ∀A,B,C:DeqSet.∀f:A→B→C.∀c:C.∀lA:list A.∀lB:list B.
312 c ∈ (compose A B C f lA lB) = true →
313 ∃a,b.a ∈ lA = true ∧ b ∈ lB = true ∧ c = f a b.
314 #A #B #C #f #c #lA elim lA
315 [normalize #lB #H destruct
316 |#a1 #tl #Hind #lB >compose_cons #Hc cases (memb_append … Hc) #Hcase
317 [lapply(memb_map_to_exists … Hcase) * #b * #Hb #Hf
319 |lapply(Hind ? Hcase) * #a2 * #b * * #Ha #Hb #Hf %{a2} %{b} % // % //
325 definition compose2 ≝
326 λA,B:DeqSet.λa:A.λl. compose B (carr_fun_l A B l) (carr_fun_l A B (a::l))
327 (λb,tl. mk_Sig ?? (〈a,b〉::(pi1 … tl)) ?).
328 normalize @eq_f @(pi2 … tl)
331 let rec Dfoldr (A:Type[0]) (B:list A → Type[0])
332 (f:∀a:A.∀l.B l → B (a::l)) (b:B [ ]) (l:list A) on l : B l ≝
333 match l with [ nil ⇒ b | cons a l ⇒ f a l (Dfoldr A B f b l)].
335 definition empty_graph: ∀A,B:DeqSet. carr_fun_l A B [].
338 definition enum_fun: ∀A,B:DeqSet.∀lA:list A.list B → list (carr_fun_l A B lA) ≝
339 λA,B,lA,lB.Dfoldr A (λl.list (carr_fun_l A B l))
340 (λa,l.compose2 ?? a l lB) [empty_graph A B] lA.
342 lemma mem_enum_fun: ∀A,B:DeqSet.∀lA,lB.∀x:carr_fun_l A B lA.
343 pi1 … x ∈ map ?? (pi1 … ) (enum_fun A B lA lB) = true →
344 x ∈ enum_fun A B lA lB = true .
345 #A #B #lA #lB #x @(memb_map_inj
346 (DeqSig (DeqList (DeqProd A B))
347 (λx0:DeqList (DeqProd A B).is_functional A B lA x0))
348 (DeqList (DeqProd A B)) (pi1 …))
349 * #l1 #H1 * #l2 #H2 #Heq lapply H1 lapply H2 >Heq //
352 lemma enum_fun_cons: ∀A,B,a,lA,lB.
353 enum_fun A B (a::lA) lB =
354 compose ??? (λb,tl. mk_Sig ?? (〈a,b〉::(pi1 … tl)) ?) lB (enum_fun A B lA lB).
358 lemma map_map: ∀A,B,C.∀f:A→B.∀g:B→C.∀l.
359 map ?? g (map ?? f l) = map ?? (g ∘ f) l.
360 #A #B #C #f #g #l elim l [//]
361 #a #tl #Hind normalize @eq_f @Hind
364 lemma map_compose: ∀A,B,C,D.∀f:A→B→C.∀g:C→D.∀l1,l2.
365 map ?? g (compose A B C f l1 l2) = compose A B D (λa,b. g (f a b)) l1 l2.
366 #A #B #C #D #f #g #l1 elim l1 [//]
367 #a #tl #Hind #l2 >compose_cons >compose_cons <map_append @eq_f2
371 definition enum_fun_graphs: ∀A,B,lA,lB.
372 map ?? (pi1 … ) (enum_fun A B lA lB) = enum_fun_raw A B lA lB.
373 #A #B #lA elim lA [normalize //]
374 #a #tl #Hind #lB >(enum_fun_cons A B a tl lB) >enum_fun_raw_cons >map_compose
375 cut (∀lB2. compose B (Σx:DeqList (DeqProd A B).is_functional A B tl x)
376 (DeqList (DeqProd A B))
378 .λb:Σx:DeqList (DeqProd A B).is_functional A B tl x
380 ::pi1 (list (A×B)) (λx:DeqList (DeqProd A B).is_functional A B tl x) b) lB
381 (enum_fun A B tl lB2)
382 =compose B (list (A×B)) (list (A×B)) (λb:B.cons (A×B) 〈a,b〉) lB
383 (enum_fun_raw A B tl lB2))
386 |#b #tlb #Hindb >compose_cons in ⊢ (???%); >compose_cons
387 @eq_f2 [<Hind >map_map // |@Hindb]]]
391 lemma uniqueb_compose: ∀A,B,C:DeqSet.∀f,l1,l2.
392 (∀a1,a2,b1,b2. f a1 b1 = f a2 b2 → a1 = a2 ∧ b1 = b2) →
393 uniqueb ? l1 = true → uniqueb ? l2 = true →
394 uniqueb ? (compose A B C f l1 l2) = true.
395 #A #B #C #f #l1 #l2 #Hinj elim l1 //
396 #a #tl #Hind #HuA #HuB >compose_cons @uniqueb_append
397 [@(unique_map_inj … HuB) #b1 #b2 #Hb1b2 @(proj2 … (Hinj … Hb1b2))
398 |@Hind // @(andb_true_r … HuA)
399 |#c #Hc lapply(memb_map_to_exists … Hc) * #b * #Hb2 #Hfab % #Hc
400 lapply(compose_spec2 … Hc) * #a1 * #b1 * * #Ha1 #Hb1 <Hfab #H
402 [@(proj1 … (Hinj … H))
403 |% #eqa @(absurd … Ha1) % <eqa #H lapply(andb_true_l … HuA) >H
404 normalize #H1 destruct (H1)
409 lemma enum_fun_unique: ∀A,B:DeqSet.∀lA,lB.
410 uniqueb ? lA = true → uniqueb ? lB = true →
411 uniqueb ? (enum_fun A B lA lB) = true.
414 |#a #tlA #Hind #lB #uA #uB lapply (enum_fun_cons A B a tlA lB) #H >H
415 @(uniqueb_compose B (carr_fun_l A B tlA) (carr_fun_l A B (a::tlA)))
416 [#b1 #b2 * #l1 #funl1 * #l2 #funl2 #H1 destruct (H1) /2/
418 |@(Hind … uB) @(andb_true_r … uA)
423 lemma enum_fun_complete: ∀A,B:FinSet.∀l1,l2.
424 (∀x:A. memb A x l1 = true) →
425 (∀x:B. memb B x l2 = true) →
426 ∀x:carr_fun_l A B l1. memb ? x (enum_fun A B l1 l2) = true.
427 #A #B #l1 #l2 #H1 #H2 * #g #H @mem_enum_fun >enum_fun_graphs
428 lapply H -H lapply g -g elim l1
429 [* // #p #tlg normalize #H destruct (H)
430 |#a #tl #Hind #g cases g
431 [normalize in ⊢ (%→?); #H destruct (H)
432 |* #a1 #b #tl1 normalize in ⊢ (%→?); #H
433 cut (is_functional A B tl tl1) [destruct (H) //] #Hfun
434 >(cons_injective_l ????? H)
435 >(enum_fun_raw_cons … ) @(compose_spec1 … (λb. cons ? 〈a,b〉))
442 λA,B:FinSet.mk_FinSet (carr_fun A B)
443 (enum_fun A B (enum A) (enum B))
444 (enum_fun_unique A B … (enum_unique A) (enum_unique B))
445 (enum_fun_complete A B … (enum_complete A) (enum_complete B)).
448 unification hint 0 ≔ C1,C2;
452 (* ---------------------------------------- *) ⊢
453 T1×T2 ≡ FinSetcarr X. *)