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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/decidable_kit/fintype/".
17 include "decidable_kit/eqtype.ma".
18 include "decidable_kit/list_aux.ma".
20 record finType : Type ≝ {
23 enum_uniq : ∀x:fsort. count fsort (cmp fsort x) enum = (S O)
26 definition segment : nat → eqType ≝
27 λn.sub_eqType nat_eqType (λx:nat_eqType.ltb x n).
29 definition is_some : ∀d:eqType. option d → bool ≝
30 λd:eqType.λo:option d.notb (cmp (option_eqType d) (None ?) o).
33 λA,B:Type.λp:A→option B.λl:list A.
35 (λx,acc. match (p x) with [None ⇒ acc | (Some y) ⇒ cons B y acc]) (nil B) l.
37 definition segment_enum ≝
38 λbound.filter ? ? (if_p nat_eqType (λx.ltb x bound)) (iota O bound).
40 lemma iota_ltb : ∀x,p:nat. mem nat_eqType x (iota O p) = ltb x p.
41 intros (x p); elim p; simplify;[reflexivity]
42 apply (cmpP nat_eqType x n); intros (E); rewrite > H; clear H; simplify;
43 [1: symmetry; apply (p2bT ? ? (lebP ? ?)); rewrite > E; apply le_n;
44 |2: rewrite < (leb_eqb x n); rewrite > E; reflexivity;]
48 ∀d1,d2:eqType.∀x:d2.∀l:list d1.∀p:d1 → option d2.
49 (∀y.mem d1 y l = true →
50 match (p y) with [None ⇒ false | (Some q) ⇒ cmp d2 x q] = false) →
51 mem d2 x (filter d1 d2 p l) = false.
52 intros 5 (d1 d2 x l p);
53 elim l; simplify; [reflexivity]
54 generalize in match (refl_eq ? (p t));
55 generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b; intros (Hpt);
56 [1: apply H; intros (y Hyl);
58 rewrite > Hyl; rewrite > orbC; reflexivity;
59 |2: simplify; apply (cmpP d2 x s); simplify; intros (E);
60 [1: rewrite < (H1 t); simplify; [rewrite > Hpt; rewrite > E]
61 simplify; rewrite > cmp_refl; reflexivity
62 |2: apply H; intros; apply H1; simplify; rewrite > H2;
63 rewrite > orbC; reflexivity]]
67 ∀d:eqType.∀p:d→bool.∀l:list d.
68 (∀x:d.mem d x l = true → notb (p x) = true) → count d p l = O.
69 intros 3 (d p l); elim l; simplify; [1: reflexivity]
70 generalize in match (refl_eq ? (p t));
71 generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b);
73 [2:intros (Hpt); apply H; intros; apply H1; simplify;
74 apply (cmpP d x t); [2: rewrite > H2;]; intros; reflexivity;
75 |1:intros (H2); lapply (H1 t); [2:simplify; rewrite > cmp_refl; simplify; autobatch]
76 rewrite > H2 in Hletin; simplify in Hletin; destruct Hletin]
79 lemma segment_finType : nat → finType.
81 letin fsort ≝ (segment bound);
82 letin enum ≝ (segment_enum bound);
83 cut (∀x:fsort. count fsort (cmp fsort x) enum = (S O));
84 [ apply (mk_finType fsort enum Hcut)
85 | intros (x); cases x (n Hn); simplify in Hn; clear x;
86 generalize in match Hn; generalize in match Hn; clear Hn;
89 generalize in match bound in ⊢ (% → ? → ? ? (? ? ? (? ? ? ? %)) ?);
90 intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ simplify in Hm; destruct Hm ]
91 simplify; cases (eqP bool_eqType (ltb p bound) true); simplify;
93 unfold segment in ⊢ (? ? match ? % ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
94 unfold nat_eqType in ⊢ (? ? match % with [_ ⇒ ?|_ ⇒ ?] ?);
95 simplify; apply (cmpP nat_eqType n p); intros (Enp); simplify;
96 [2:rewrite > IH; [1,3: autobatch]
97 rewrite < ltb_n_Sm in Hm; rewrite > Enp in Hm;
98 rewrite > orbC in Hm; assumption;
99 |1:clear IH; rewrite > (count_O fsort); [reflexivity]
100 intros 1 (x); rewrite < Enp; cases x (y Hy);
101 intros (ABS); clear x; unfold segment; unfold notb; simplify;
102 apply (cmpP ? n y); intros (Eny); simplify; [2:reflexivity]
103 rewrite < ABS; symmetry; clear ABS;
104 generalize in match Hy; clear Hy;rewrite < Eny;
105 simplify; intros (Hn); apply (mem_filter nat_eqType fsort); intros (w Hw);
106 fold simplify (sort nat_eqType); (* CANONICAL?! *)
107 cases (in_sub_eq nat_eqType (λx:nat_eqType.ltb x bound) w);
108 simplify; [2: reflexivity]
109 generalize in match H1; clear H1; cases s (r Pr); clear s; intros (H1);
110 unfold fsort; unfold segment; simplify; simplify in H1; rewrite > H1;
111 rewrite > iota_ltb in Hw; apply (p2bF ? ? (eqP nat_eqType ? ?));
112 unfold Not; intros (Enw); rewrite > Enw in Hw;
113 rewrite > ltb_refl in Hw; destruct Hw]
114 |2:rewrite > IH; [1:reflexivity|3:assumption]
115 rewrite < ltb_n_Sm in Hm;
116 cases (b2pT ? ?(orbP ? ?) Hm);[1: assumption]
117 rewrite > (b2pT ? ? (eqbP ? ?) H1) in Hn;
118 rewrite > Hn in H; cases (H ?); reflexivity]]
121 let rec uniq (d:eqType) (l:list d) on l : bool ≝
124 | (cons x tl) ⇒ andb (notb (mem d x tl)) (uniq d tl)].
126 lemma uniq_mem : ∀d:eqType.∀x:d.∀l:list d.uniq d (x::l) = true → mem d x l = false.
127 intros (d x l H); simplify in H; lapply (b2pT ? ? (andbP ? ?) H) as H1; clear H;
128 cases H1 (H2 H3); lapply (b2pT ? ?(negbP ?) H2); assumption;
131 lemma andbA : ∀a,b,c.andb a (andb b c) = andb (andb a b) c.
132 intros; cases a; cases b; cases c; reflexivity; qed.
134 lemma andbC : ∀a,b. andb a b = andb b a.
135 intros; cases a; cases b; reflexivity; qed.
138 ∀d:eqType.∀x:d.∀l:list d. uniq d (x::l) = andb (negb (mem d x l)) (uniq d l).
139 intros (d x l); elim l; simplify; [reflexivity]
140 apply (cmpP d x t); intros (E); simplify ; try rewrite > E; [reflexivity]
141 rewrite > andbA; rewrite > andbC in ⊢ (? ? (? % ?) ?); rewrite < andbA;
142 rewrite < H; rewrite > andbC in ⊢ (? ? ? (? % ?)); rewrite < andbA; reflexivity;
145 lemma count_O_mem : ∀d:eqType.∀x:d.∀l:list d.ltb O (count d (cmp d x) l) = mem d x l.
146 intros 3 (d x l); elim l [reflexivity] simplify; rewrite < H; cases (cmp d x t);
149 lemma uniqP : ∀d:eqType.∀l:list d.
150 reflect (∀x:d.mem d x l = true → count d (cmp d x) l = (S O)) (uniq d l).
151 intros (d l); apply prove_reflect; elim l; [1: simplify in H1; destruct H1 | 3: simplify in H; destruct H]
152 [1: generalize in match H2; simplify in H2;
153 lapply (b2pT ? ? (orbP ? ?) H2) as H3; clear H2;
154 cases H3; clear H3; intros;
155 [2: lapply (uniq_mem ? ? ? H1) as H4; simplify; apply (cmpP d x t);
156 intros (H5); simplify;
157 [1: rewrite > count_O; [reflexivity]
158 intros (y Hy); rewrite > H5 in H2 H3 ⊢ %; clear H5; clear x;
159 rewrite > H2 in H4; destruct H4;
160 |2: simplify; apply H;
161 rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1);
163 |1: simplify; rewrite > H2; simplify; rewrite > count_O; [reflexivity]
164 intros (y Hy); rewrite > (b2pT ? ? (eqP d ? ?) H2) in H3 ⊢ %;
165 clear H2; clear x; lapply (uniq_mem ? ? ? H1) as H4;
166 apply (cmpP d t y); intros (E); [2: reflexivity].
167 rewrite > E in H4; rewrite > H4 in Hy; destruct Hy;]
168 |2: rewrite > uniq_tail in H1;
169 generalize in match (refl_eq ? (uniq d l1));
170 generalize in match (uniq d l1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
171 [1: intros (E); rewrite > E in H1; rewrite > andbC in H1; simplify in H1;
172 unfold Not; intros (A); lapply (A t) as A';
173 [1: simplify in A'; rewrite > cmp_refl in A'; simplify in A';
174 destruct A'; rewrite < count_O_mem in H1;
175 rewrite > Hcut in H1; simplify in H1; destruct H1;
176 |2: simplify; rewrite > cmp_refl; reflexivity;]
177 |2: intros (Ul1); lapply (H Ul1); unfold Not; intros (A); apply Hletin;
178 intros (r Mrl1); lapply (A r);
179 [2: simplify; rewrite > Mrl1; cases (cmp d r t); reflexivity]
180 generalize in match Hletin1; simplify; apply (cmpP d r t);
181 simplify; intros (E Hc); [2: assumption]
182 destruct Hc; rewrite < count_O_mem in Mrl1;
183 rewrite > Hcut in Mrl1; simplify in Mrl1; destruct Mrl1;]]
186 lemma mem_finType : ∀d:finType.∀x:d. mem d x (enum d) = true.
187 intros 1 (d); cases d; simplify; intros; rewrite < count_O_mem;
188 rewrite > H; reflexivity;
191 lemma uniq_fintype_enum : ∀d:finType. uniq d (enum d) = true.
192 intros; cases d; simplify; apply (p2bT ? ? (uniqP ? ?)); intros; apply H;
195 lemma sub_enumP : ∀d:finType.∀p:d→bool.∀x:sub_eqType d p.
196 count (sub_eqType d p) (cmp ? x) (filter ? ? (if_p ? p) (enum d)) = (S O).
197 intros (d p x); cases x (t Ht); clear x;
198 generalize in match (mem_finType d t);
199 generalize in match (uniq_fintype_enum d);
200 elim (enum d); [simplify in H1; destruct H1] simplify;
201 cases (in_sub_eq d p t1); simplify;
202 [1:generalize in match H3; clear H3; cases s (r Hr); clear s;
203 simplify; intros (Ert1); generalize in match Hr; clear Hr;
204 rewrite > Ert1; clear Ert1; clear r; intros (Ht1);
205 unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
206 simplify; apply (cmpP ? t t1); simplify; intros (Ett1);
207 [1: cut (count (sub_eqType d p) (cmp (sub_eqType d p) {t,Ht})
208 (filter d (sigma d p) (if_p d p) l) = O); [1:rewrite > Hcut; reflexivity]
209 lapply (uniq_mem ? ? ? H1);
210 generalize in match Ht;
211 rewrite > Ett1; intros (Ht1'); clear Ht1;
212 generalize in match Hletin; elim l; [ reflexivity]
213 simplify; cases (in_sub_eq d p t2); simplify;
214 [1: generalize in match H5; cases s; simplify; intros; clear H5;
215 unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
216 simplify; rewrite > H7; simplify in H4;
217 generalize in match H4; clear H4; apply (cmpP ? t1 t2);
218 simplify; intros; [destruct H5] apply H3; assumption;
220 generalize in match H4; clear H4; simplify; apply (cmpP ? t1 t2);
221 simplify; intros; [destruct H6] assumption;]
222 |2: apply H; [ rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
223 simplify in H2; rewrite > Ett1 in H2; simplify in H2; assumption]
224 |2:rewrite > H; [1:reflexivity|2: rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
225 simplify in H2; generalize in match H2; apply (cmpP ? t t1);
226 intros (E) [2:assumption] clear H; rewrite > E in Ht; rewrite > H3 in Ht;
230 definition sub_finType : ∀d:finType.∀p:d→bool.finType ≝
231 λd:finType.λp:d→bool. mk_finType (sub_eqType d p) (filter ? ? (if_p ? p) (enum d)) (sub_enumP d p).