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/lists/listb.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 [#H1 @(absurd … (\P H1)) % #Hfalse
68 cut (∀A,P,a,a1,h,h1.mk_Sig A P a h = mk_Sig A P a1 h1 → a = a1)
69 [#A #P #a #a1 #h #h1 #H destruct (H) %] #Hcut
70 lapply (Hcut nat (λi.i<m) i n ? ? Hfalse) #Hfalse @(absurd … ltni)
71 @le_to_not_lt >Hfalse @le_n
72 |<(notb_notb (memb …)) >Hind normalize /2/
77 lemma enumn_unique_aux: ∀n,m. ∀h:n ≤ m. uniqueb (Nat_to m) (enumnaux n m h) = true.
78 #n elim n -n // #n #Hind #m #h @true_to_andb_true // @memb_enumn //
81 lemma enumn_unique: ∀n.uniqueb (Nat_to n) (enumn n) = true.
85 (* definition ltb ≝ λn,m.leb (S n) m. *)
86 lemma enumn_complete_aux: ∀n,m,i.∀h:n ≤m.∀k:i<m.i<n →
87 memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h) = true.
89 [normalize #n #i #_ #_ #Hfalse @False_ind /2/
90 |#n #Hind #m #i #h #k #lein whd in ⊢ (??%?);
91 cases (le_to_or_lt_eq … (le_S_S_to_le … lein))
92 [#ltin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = false)
93 [normalize @not_eq_to_eqb_false @lt_to_not_eq @ltin]
95 |#eqin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = true)
96 [normalize @eq_to_eqb_true //
102 lemma enumn_complete: ∀n.∀i:Nat_to n. memb ? i (enumn n) = true.
103 #n whd in ⊢ (%→?); * #i #ltin @enumn_complete_aux //
106 definition initN ≝ λn.
107 mk_FinSet (Nat_to n) (enumn n) (enumn_unique n) (enumn_complete n).
109 example tipa: ∀n.∃x: initN (S n). pi1 … x = n.
110 #n @ex_intro [whd @mk_Sig [@n | @le_n] | //] qed.
113 definition enum_sum ≝ λA,B:DeqSet.λl1.λl2.
114 (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
116 lemma enum_sum_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 l2 =
117 (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
120 lemma enum_sum_unique: ∀A,B:DeqSet.∀l1,l2.
121 uniqueb A l1 = true → uniqueb B l2 = true →
122 uniqueb ? (enum_sum A B l1 l2) = true.
123 #A #B #l1 #l2 elim l1
124 [#_ #ul2 @unique_map_inj // #b1 #b2 #Hinr destruct //
125 |#a #tl #Hind #uA #uB @true_to_andb_true
126 [@sym_eq @noteq_to_eqnot % #H
127 cases (memb_append … (sym_eq … H))
128 [#H1 @(absurd (memb ? a tl = true))
129 [@(memb_map_inj …H1) #a1 #a2 #Hinl destruct //
130 |<(andb_true_l … uA) @eqnot_to_noteq //
133 [normalize #H destruct
134 |#b #tlB #Hind #membH cases (orb_true_l … membH)
135 [#H lapply (\P H) #H1 destruct |@Hind]
138 |@Hind // @(andb_true_r … uA)
143 lemma enum_sum_complete: ∀A,B:DeqSet.∀l1,l2.
144 (∀x:A. memb A x l1 = true) →
145 (∀x:B. memb B x l2 = true) →
146 ∀x:DeqSum A B. memb ? x (enum_sum A B l1 l2) = true.
147 #A #B #l1 #l2 #Hl1 #Hl2 *
148 [#a @memb_append_l1 @memb_map @Hl1
149 |#b @memb_append_l2 @memb_map @Hl2
153 definition FinSum ≝ λA,B:FinSet.
154 mk_FinSet (DeqSum A B)
155 (enum_sum A B (enum A) (enum B))
156 (enum_sum_unique … (enum_unique A) (enum_unique B))
157 (enum_sum_complete … (enum_complete A) (enum_complete B)).
159 include alias "basics/types.ma".
161 unification hint 0 ≔ C1,C2;
165 (* ---------------------------------------- *) ⊢
166 T1+T2 ≡ FinSetcarr X.
170 definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
171 compose ??? (mk_Prod A B) l1 l2.
173 axiom enum_prod_unique: ∀A,B,l1,l2.
174 uniqueb A l1 = true → uniqueb B l2 = true →
175 uniqueb ? (enum_prod A B l1 l2) = true.
177 lemma enum_prod_complete:∀A,B:DeqSet.∀l1,l2.
178 (∀a. memb A a l1 = true) → (∀b.memb B b l2 = true) →
179 ∀p. memb ? p (enum_prod A B l1 l2) = true.
180 #A #B #l1 #l2 #Hl1 #Hl2 * #a #b @memb_compose //
184 λA,B:FinSet.mk_FinSet (DeqProd A B)
185 (enum_prod A B (enum A) (enum B))
186 (enum_prod_unique A B … (enum_unique A) (enum_unique B))
187 (enum_prod_complete A B … (enum_complete A) (enum_complete B)).
189 unification hint 0 ≔ C1,C2;
193 (* ---------------------------------------- *) ⊢
194 T1×T2 ≡ FinSetcarr X.