1 (**************************************************************************)
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 include "decidable_kit/eqtype.ma".
16 include "decidable_kit/list_aux.ma".
18 record finType : Type ≝ {
21 enum_uniq : ∀x:fsort. count fsort (cmp fsort x) enum = (S O)
24 definition segment : nat → eqType ≝
25 λn.sub_eqType nat_eqType (λx:nat_eqType.ltb x n).
27 definition is_some : ∀d:eqType. option d → bool ≝
28 λd:eqType.λo:option d.notb (cmp (option_eqType d) (None ?) o).
31 λA,B:Type.λp:A→option B.λl:list A.
33 (λx,acc. match (p x) with [None ⇒ acc | (Some y) ⇒ cons B y acc]) (nil B) l.
35 definition segment_enum ≝
36 λbound.filter ? ? (if_p nat_eqType (λx.ltb x bound)) (iota O bound).
38 lemma iota_ltb : ∀x,p:nat. mem nat_eqType x (iota O p) = ltb x p.
39 intros (x p); elim p; simplify;[reflexivity]
40 apply (cmpP nat_eqType x n); intros (E); rewrite > H; clear H; simplify;
41 [1: symmetry; apply (p2bT ? ? (lebP ? ?)); rewrite > E; apply le_n;
42 |2: rewrite < (leb_eqb x n); rewrite > E; reflexivity;]
46 ∀d1,d2:eqType.∀x:d2.∀l:list d1.∀p:d1 → option d2.
47 (∀y.mem d1 y l = true →
48 match (p y) with [None ⇒ false | (Some q) ⇒ cmp d2 x q] = false) →
49 mem d2 x (filter d1 d2 p l) = false.
50 intros 5 (d1 d2 x l p);
51 elim l; simplify; [reflexivity]
52 generalize in match (refl_eq ? (p a));
53 generalize in match (p a) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b; intros (Hpt);
54 [1: apply H; intros (y Hyl);
56 rewrite > Hyl; rewrite > orbC; reflexivity;
57 |2: simplify; apply (cmpP d2 x s); simplify; intros (E);
58 [1: rewrite < (H1 a); simplify; [rewrite > Hpt; rewrite > E]
59 simplify; rewrite > cmp_refl; reflexivity
60 |2: apply H; intros; apply H1; simplify; rewrite > H2;
61 rewrite > orbC; reflexivity]]
65 ∀d:eqType.∀p:d→bool.∀l:list d.
66 (∀x:d.mem d x l = true → notb (p x) = true) → count d p l = O.
67 intros 3 (d p l); elim l; simplify; [1: reflexivity]
68 generalize in match (refl_eq ? (p a));
69 generalize in match (p a) in ⊢ (? ? ? % → %); intros 1 (b);
71 [2:intros (Hpt); apply H; intros; apply H1; simplify;
72 apply (cmpP d x a); [2: rewrite > H2;]; intros; reflexivity;
73 |1:intros (H2); lapply (H1 a); [2:simplify; rewrite > cmp_refl; simplify; autobatch]
74 rewrite > H2 in Hletin; simplify in Hletin; destruct Hletin]
77 lemma segment_finType : nat → finType.
79 letin fsort ≝ (segment bound);
80 letin enum ≝ (segment_enum bound);
81 cut (∀x:fsort. count fsort (cmp fsort x) enum = (S O));
82 [ apply (mk_finType fsort enum Hcut)
83 | intros (x); cases x (n Hn); simplify in Hn; clear x;
84 generalize in match Hn; generalize in match Hn; clear Hn;
87 generalize in match bound in ⊢ (% → ? → ? ? (? ? ? (? ? ? ? %)) ?);
88 intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ simplify in Hm; destruct Hm ]
89 simplify; cases (eqP bool_eqType (ltb p bound) true); simplify;
91 unfold segment in ⊢ (? ? match ? % ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
92 unfold nat_eqType in ⊢ (? ? match % with [_ ⇒ ?|_ ⇒ ?] ?);
93 simplify; apply (cmpP nat_eqType n p); intros (Enp); simplify;
94 [2:rewrite > IH; [1,3: autobatch]
95 rewrite < ltb_n_Sm in Hm; rewrite > Enp in Hm;
96 rewrite > orbC in Hm; assumption;
97 |1:clear IH; rewrite > (count_O fsort); [reflexivity]
98 intros 1 (x); rewrite < Enp; cases x (y Hy);
99 intros (ABS); clear x; unfold segment; unfold notb; simplify;
100 apply (cmpP ? n y); intros (Eny); simplify; [2:reflexivity]
101 rewrite < ABS; symmetry; clear ABS;
102 generalize in match Hy; clear Hy;rewrite < Eny;
103 simplify; intros (Hn); apply (mem_filter nat_eqType fsort); intros (w Hw);
104 fold simplify (sort nat_eqType); (* CANONICAL?! *)
105 cases (in_sub_eq nat_eqType (λx:nat_eqType.ltb x bound) w);
106 simplify; [2: reflexivity]
107 generalize in match H1; clear H1; cases s (r Pr); clear s; intros (H1);
108 unfold fsort; unfold segment; simplify; simplify in H1; rewrite > H1;
109 rewrite > iota_ltb in Hw; apply (p2bF ? ? (eqP nat_eqType ? ?));
110 unfold Not; intros (Enw); rewrite > Enw in Hw;
111 rewrite > ltb_refl in Hw; destruct Hw]
112 |2:rewrite > IH; [1:reflexivity|3:assumption]
113 rewrite < ltb_n_Sm in Hm;
114 cases (b2pT ? ?(orbP ? ?) Hm);[1: assumption]
115 rewrite > (b2pT ? ? (eqbP ? ?) H1) in Hn;
116 rewrite > Hn in H; cases (H ?); reflexivity]]
119 let rec uniq (d:eqType) (l:list d) on l : bool ≝
122 | (cons x tl) ⇒ andb (notb (mem d x tl)) (uniq d tl)].
124 lemma uniq_mem : ∀d:eqType.∀x:d.∀l:list d.uniq d (x::l) = true → mem d x l = false.
125 intros (d x l H); simplify in H; lapply (b2pT ? ? (andbP ? ?) H) as H1; clear H;
126 cases H1 (H2 H3); lapply (b2pT ? ?(negbP ?) H2); assumption;
129 lemma andbA : ∀a,b,c.andb a (andb b c) = andb (andb a b) c.
130 intros; cases a; cases b; cases c; reflexivity; qed.
132 lemma andbC : ∀a,b. andb a b = andb b a.
133 intros; cases a; cases b; reflexivity; qed.
136 ∀d:eqType.∀x:d.∀l:list d. uniq d (x::l) = andb (negb (mem d x l)) (uniq d l).
137 intros (d x l); elim l; simplify; [reflexivity]
138 apply (cmpP d x a); intros (E); simplify ; try rewrite > E; [reflexivity]
139 rewrite > andbA; rewrite > andbC in ⊢ (? ? (? % ?) ?); rewrite < andbA;
140 rewrite < H; rewrite > andbC in ⊢ (? ? ? (? % ?)); rewrite < andbA; reflexivity;
143 lemma count_O_mem : ∀d:eqType.∀x:d.∀l:list d.ltb O (count d (cmp d x) l) = mem d x l.
144 intros 3 (d x l); elim l [reflexivity] simplify; rewrite < H; cases (cmp d x a);
147 lemma uniqP : ∀d:eqType.∀l:list d.
148 reflect (∀x:d.mem d x l = true → count d (cmp d x) l = (S O)) (uniq d l).
149 intros (d l); apply prove_reflect; elim l; [1: simplify in H1; destruct H1 | 3: simplify in H; destruct H]
150 [1: generalize in match H2; simplify in H2;
151 lapply (b2pT ? ? (orbP ? ?) H2) as H3; clear H2;
152 cases H3; clear H3; intros;
153 [2: lapply (uniq_mem ? ? ? H1) as H4; simplify; apply (cmpP d x a);
154 intros (H5); simplify;
155 [1: rewrite > count_O; [reflexivity]
156 intros (y Hy); rewrite > H5 in H2 H3 ⊢ %; clear H5; clear x;
157 rewrite > H2 in H4; destruct H4;
158 |2: simplify; apply H;
159 rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1);
161 |1: simplify; rewrite > H2; simplify; rewrite > count_O; [reflexivity]
162 intros (y Hy); rewrite > (b2pT ? ? (eqP d ? ?) H2) in H3 ⊢ %;
163 clear H2; clear x; lapply (uniq_mem ? ? ? H1) as H4;
164 apply (cmpP d a y); intros (E); [2: reflexivity].
165 rewrite > E in H4; rewrite > H4 in Hy; destruct Hy;]
166 |2: rewrite > uniq_tail in H1;
167 generalize in match (refl_eq ? (uniq d l1));
168 generalize in match (uniq d l1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
169 [1: intros (E); rewrite > E in H1; rewrite > andbC in H1; simplify in H1;
170 unfold Not; intros (A); lapply (A a) as A';
171 [1: simplify in A'; rewrite > cmp_refl in A'; simplify in A';
172 destruct A'; rewrite < count_O_mem in H1;
173 rewrite > Hcut in H1; simplify in H1; destruct H1;
174 |2: simplify; rewrite > cmp_refl; reflexivity;]
175 |2: intros (Ul1); lapply (H Ul1); unfold Not; intros (A); apply Hletin;
176 intros (r Mrl1); lapply (A r);
177 [2: simplify; rewrite > Mrl1; cases (cmp d r a); reflexivity]
178 generalize in match Hletin1; simplify; apply (cmpP d r a);
179 simplify; intros (E Hc); [2: assumption]
180 destruct Hc; rewrite < count_O_mem in Mrl1;
181 rewrite > Hcut in Mrl1; simplify in Mrl1; destruct Mrl1;]]
184 lemma mem_finType : ∀d:finType.∀x:d. mem d x (enum d) = true.
185 intros 1 (d); cases d; simplify; intros; rewrite < count_O_mem;
186 rewrite > H; reflexivity;
189 lemma uniq_fintype_enum : ∀d:finType. uniq d (enum d) = true.
190 intros; cases d; simplify; apply (p2bT ? ? (uniqP ? ?)); intros; apply H;
193 lemma sub_enumP : ∀d:finType.∀p:d→bool.∀x:sub_eqType d p.
194 count (sub_eqType d p) (cmp ? x) (filter ? ? (if_p ? p) (enum d)) = (S O).
195 intros (d p x); cases x (t Ht); clear x;
196 generalize in match (mem_finType d t);
197 generalize in match (uniq_fintype_enum d);
198 elim (enum d); [simplify in H1; destruct H1] simplify;
199 cases (in_sub_eq d p a); simplify;
200 [1:generalize in match H3; clear H3; cases s (r Hr); clear s;
201 simplify; intros (Ert1); generalize in match Hr; clear Hr;
202 rewrite > Ert1; clear Ert1; clear r; intros (Ht1);
203 unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
204 simplify; apply (cmpP ? t a); simplify; intros (Ett1);
205 [1: cut (count (sub_eqType d p) (cmp (sub_eqType d p) {t,Ht})
206 (filter d (sigma d p) (if_p d p) l) = O); [1:rewrite > Hcut; reflexivity]
207 lapply (uniq_mem ? ? ? H1);
208 generalize in match Ht;
209 rewrite > Ett1; intros (Ht1'); clear Ht1;
210 generalize in match Hletin; elim l; [ reflexivity]
211 simplify; cases (in_sub_eq d p a1); simplify;
212 [1: generalize in match H5; cases s; simplify; intros; clear H5;
213 unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
214 simplify; rewrite > H7; simplify in H4;
215 generalize in match H4; clear H4; apply (cmpP ? a a1);
216 simplify; intros; [destruct H5] apply H3; assumption;
218 generalize in match H4; clear H4; simplify; apply (cmpP ? a a1);
219 simplify; intros; [destruct H6] assumption;]
220 |2: apply H; [ rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
221 simplify in H2; rewrite > Ett1 in H2; simplify in H2; assumption]
222 |2:rewrite > H; [1:reflexivity|2: rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
223 simplify in H2; generalize in match H2; apply (cmpP ? t a);
224 intros (E) [2:assumption] clear H; rewrite > E in Ht; rewrite > H3 in Ht;
228 definition sub_finType : ∀d:finType.∀p:d→bool.finType ≝
229 λd:finType.λp:d→bool. mk_finType (sub_eqType d p) (filter ? ? (if_p ? p) (enum d)) (sub_enumP d p).