--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/decidable_kit/fintype/".
+
+include "decidable_kit/eqtype.ma".
+include "decidable_kit/list_aux.ma".
+
+record finType : Type ≝ {
+ fsort :> eqType;
+ enum : list fsort;
+ enum_uniq : ∀x:fsort. count fsort (cmp fsort x) enum = (S O)
+}.
+
+definition segment : nat → eqType ≝
+ λn.sub_eqType nat_eqType (λx:nat_eqType.ltb x n).
+
+definition is_some : ∀d:eqType. option d → bool ≝
+ λd:eqType.λo:option d.notb (cmp (option_eqType d) (None ?) o).
+
+definition filter ≝
+ λA,B:Type.λp:A→option B.λl:list A.
+ foldr A ?
+ (λx,acc. match (p x) with [None ⇒ acc | (Some y) ⇒ cons B y acc]) (nil B) l.
+
+definition segment_enum ≝
+ λbound.filter ? ? (if_p nat_eqType (λx.ltb x bound)) (iota O bound).
+
+lemma iota_ltb : ∀x,p:nat. mem nat_eqType x (iota O p) = ltb x p.
+intros (x p); elim p; simplify;[reflexivity]
+apply (cmpP nat_eqType x n); intros (E); rewrite > H; clear H; simplify;
+[1: symmetry; apply (p2bT ? ? (lebP ? ?)); rewrite > E; apply le_n;
+|2: rewrite < (leb_eqb x n); rewrite > E; reflexivity;]
+qed.
+
+lemma mem_filter :
+ ∀d1,d2:eqType.∀x:d2.∀l:list d1.∀p:d1 → option d2.
+ (∀y.mem d1 y l = true →
+ match (p y) with [None ⇒ false | (Some q) ⇒ cmp d2 x q] = false) →
+ mem d2 x (filter d1 d2 p l) = false.
+intros 5 (d1 d2 x l p);
+elim l; simplify; [reflexivity]
+generalize in match (refl_eq ? (p t));
+generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b; intros (Hpt);
+[1: apply H; intros (y Hyl);
+ apply H1; simplify;
+ rewrite > Hyl; rewrite > orbC; reflexivity;
+|2: simplify; apply (cmpP d2 x s); simplify; intros (E);
+ [1: rewrite < (H1 t); simplify; [rewrite > Hpt; rewrite > E]
+ simplify; rewrite > cmp_refl; reflexivity
+ |2: apply H; intros; apply H1; simplify; rewrite > H2;
+ rewrite > orbC; reflexivity]]
+qed.
+
+lemma count_O :
+ ∀d:eqType.∀p:d→bool.∀l:list d.
+ (∀x:d.mem d x l = true → notb (p x) = true) → count d p l = O.
+intros 3 (d p l); elim l; simplify; [1: reflexivity]
+generalize in match (refl_eq ? (p t));
+generalize in match (p t) in ⊢ (? ? ? % → %); intros 1 (b);
+cases b; simplify;
+[2:intros (Hpt); apply H; intros; apply H1; simplify;
+ apply (cmpP d x t); [2: rewrite > H2;]; intros; reflexivity;
+|1:intros (H2); lapply (H1 t); [2:simplify; rewrite > cmp_refl; simplify; autobatch]
+ rewrite > H2 in Hletin; simplify in Hletin; destruct Hletin]
+qed.
+
+lemma segment_finType : nat → finType.
+intros (bound);
+letin fsort ≝ (segment bound);
+letin enum ≝ (segment_enum bound);
+cut (∀x:fsort. count fsort (cmp fsort x) enum = (S O));
+ [ apply (mk_finType fsort enum Hcut)
+ | intros (x); cases x (n Hn); simplify in Hn; clear x;
+ generalize in match Hn; generalize in match Hn; clear Hn;
+ unfold enum;
+ unfold segment_enum;
+ generalize in match bound in ⊢ (% → ? → ? ? (? ? ? (? ? ? ? %)) ?);
+ intros 1 (m); elim m (Hm Hn p IH Hm Hn); [ simplify in Hm; destruct Hm ]
+ simplify; cases (eqP bool_eqType (ltb p bound) true); simplify;
+ [1:unfold fsort;
+ unfold segment in ⊢ (? ? match ? % ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
+ unfold nat_eqType in ⊢ (? ? match % with [_ ⇒ ?|_ ⇒ ?] ?);
+ simplify; apply (cmpP nat_eqType n p); intros (Enp); simplify;
+ [2:rewrite > IH; [1,3: autobatch]
+ rewrite < ltb_n_Sm in Hm; rewrite > Enp in Hm;
+ rewrite > orbC in Hm; assumption;
+ |1:clear IH; rewrite > (count_O fsort); [reflexivity]
+ intros 1 (x); rewrite < Enp; cases x (y Hy);
+ intros (ABS); clear x; unfold segment; unfold notb; simplify;
+ apply (cmpP ? n y); intros (Eny); simplify; [2:reflexivity]
+ rewrite < ABS; symmetry; clear ABS;
+ generalize in match Hy; clear Hy;rewrite < Eny;
+ simplify; intros (Hn); apply (mem_filter nat_eqType fsort); intros (w Hw);
+ fold simplify (sort nat_eqType); (* CANONICAL?! *)
+ cases (in_sub_eq nat_eqType (λx:nat_eqType.ltb x bound) w);
+ simplify; [2: reflexivity]
+ generalize in match H1; clear H1; cases s (r Pr); clear s; intros (H1);
+ unfold fsort; unfold segment; simplify; simplify in H1; rewrite > H1;
+ rewrite > iota_ltb in Hw; apply (p2bF ? ? (eqP nat_eqType ? ?));
+ unfold Not; intros (Enw); rewrite > Enw in Hw;
+ rewrite > ltb_refl in Hw; destruct Hw]
+ |2:rewrite > IH; [1:reflexivity|3:assumption]
+ rewrite < ltb_n_Sm in Hm;
+ cases (b2pT ? ?(orbP ? ?) Hm);[1: assumption]
+ rewrite > (b2pT ? ? (eqbP ? ?) H1) in Hn;
+ rewrite > Hn in H; cases (H ?); reflexivity]]
+qed.
+
+let rec uniq (d:eqType) (l:list d) on l : bool ≝
+ match l with
+ [ nil ⇒ true
+ | (cons x tl) ⇒ andb (notb (mem d x tl)) (uniq d tl)].
+
+lemma uniq_mem : ∀d:eqType.∀x:d.∀l:list d.uniq d (x::l) = true → mem d x l = false.
+intros (d x l H); simplify in H; lapply (b2pT ? ? (andbP ? ?) H) as H1; clear H;
+cases H1 (H2 H3); lapply (b2pT ? ?(negbP ?) H2); assumption;
+qed.
+
+lemma andbA : ∀a,b,c.andb a (andb b c) = andb (andb a b) c.
+intros; cases a; cases b; cases c; reflexivity; qed.
+
+lemma andbC : ∀a,b. andb a b = andb b a.
+intros; cases a; cases b; reflexivity; qed.
+
+lemma uniq_tail :
+ ∀d:eqType.∀x:d.∀l:list d. uniq d (x::l) = andb (negb (mem d x l)) (uniq d l).
+intros (d x l); elim l; simplify; [reflexivity]
+apply (cmpP d x t); intros (E); simplify ; try rewrite > E; [reflexivity]
+rewrite > andbA; rewrite > andbC in ⊢ (? ? (? % ?) ?); rewrite < andbA;
+rewrite < H; rewrite > andbC in ⊢ (? ? ? (? % ?)); rewrite < andbA; reflexivity;
+qed.
+
+lemma count_O_mem : ∀d:eqType.∀x:d.∀l:list d.ltb O (count d (cmp d x) l) = mem d x l.
+intros 3 (d x l); elim l [reflexivity] simplify; rewrite < H; cases (cmp d x t);
+reflexivity; qed.
+
+lemma uniqP : ∀d:eqType.∀l:list d.
+ reflect (∀x:d.mem d x l = true → count d (cmp d x) l = (S O)) (uniq d l).
+intros (d l); apply prove_reflect; elim l; [1: simplify in H1; destruct H1 | 3: simplify in H; destruct H]
+[1: generalize in match H2; simplify in H2;
+ lapply (b2pT ? ? (orbP ? ?) H2) as H3; clear H2;
+ cases H3; clear H3; intros;
+ [2: lapply (uniq_mem ? ? ? H1) as H4; simplify; apply (cmpP d x t);
+ intros (H5); simplify;
+ [1: rewrite > count_O; [reflexivity]
+ intros (y Hy); rewrite > H5 in H2 H3 ⊢ %; clear H5; clear x;
+ rewrite > H2 in H4; destruct H4;
+ |2: simplify; apply H;
+ rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1);
+ assumption;]
+ |1: simplify; rewrite > H2; simplify; rewrite > count_O; [reflexivity]
+ intros (y Hy); rewrite > (b2pT ? ? (eqP d ? ?) H2) in H3 ⊢ %;
+ clear H2; clear x; lapply (uniq_mem ? ? ? H1) as H4;
+ apply (cmpP d t y); intros (E); [2: reflexivity].
+ rewrite > E in H4; rewrite > H4 in Hy; destruct Hy;]
+|2: rewrite > uniq_tail in H1;
+ generalize in match (refl_eq ? (uniq d l1));
+ generalize in match (uniq d l1) in ⊢ (? ? ? % → %); intros 1 (b); cases b; clear b;
+ [1: intros (E); rewrite > E in H1; rewrite > andbC in H1; simplify in H1;
+ unfold Not; intros (A); lapply (A t) as A';
+ [1: simplify in A'; rewrite > cmp_refl in A'; simplify in A';
+ destruct A'; rewrite < count_O_mem in H1;
+ rewrite > Hcut in H1; simplify in H1; destruct H1;
+ |2: simplify; rewrite > cmp_refl; reflexivity;]
+ |2: intros (Ul1); lapply (H Ul1); unfold Not; intros (A); apply Hletin;
+ intros (r Mrl1); lapply (A r);
+ [2: simplify; rewrite > Mrl1; cases (cmp d r t); reflexivity]
+ generalize in match Hletin1; simplify; apply (cmpP d r t);
+ simplify; intros (E Hc); [2: assumption]
+ destruct Hc; rewrite < count_O_mem in Mrl1;
+ rewrite > Hcut in Mrl1; simplify in Mrl1; destruct Mrl1;]]
+qed.
+
+lemma mem_finType : ∀d:finType.∀x:d. mem d x (enum d) = true.
+intros 1 (d); cases d; simplify; intros; rewrite < count_O_mem;
+rewrite > H; reflexivity;
+qed.
+
+lemma uniq_fintype_enum : ∀d:finType. uniq d (enum d) = true.
+intros; cases d; simplify; apply (p2bT ? ? (uniqP ? ?)); intros; apply H;
+qed.
+
+lemma sub_enumP : ∀d:finType.∀p:d→bool.∀x:sub_eqType d p.
+ count (sub_eqType d p) (cmp ? x) (filter ? ? (if_p ? p) (enum d)) = (S O).
+intros (d p x); cases x (t Ht); clear x;
+generalize in match (mem_finType d t);
+generalize in match (uniq_fintype_enum d);
+elim (enum d); [simplify in H1; destruct H1] simplify;
+cases (in_sub_eq d p t1); simplify;
+[1:generalize in match H3; clear H3; cases s (r Hr); clear s;
+ simplify; intros (Ert1); generalize in match Hr; clear Hr;
+ rewrite > Ert1; clear Ert1; clear r; intros (Ht1);
+ unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
+ simplify; apply (cmpP ? t t1); simplify; intros (Ett1);
+ [1: cut (count (sub_eqType d p) (cmp (sub_eqType d p) {t,Ht})
+ (filter d (sigma d p) (if_p d p) l) = O); [1:rewrite > Hcut; reflexivity]
+ lapply (uniq_mem ? ? ? H1);
+ generalize in match Ht;
+ rewrite > Ett1; intros (Ht1'); clear Ht1;
+ generalize in match Hletin; elim l; [ reflexivity]
+ simplify; cases (in_sub_eq d p t2); simplify;
+ [1: generalize in match H5; cases s; simplify; intros; clear H5;
+ unfold sub_eqType in ⊢ (? ? match ? (% ? ?) ? ? with [_ ⇒ ?|_ ⇒ ?] ?);
+ simplify; rewrite > H7; simplify in H4;
+ generalize in match H4; clear H4; apply (cmpP ? t1 t2);
+ simplify; intros; [destruct H5] apply H3; assumption;
+ |2: apply H3;
+ generalize in match H4; clear H4; simplify; apply (cmpP ? t1 t2);
+ simplify; intros; [destruct H6] assumption;]
+ |2: apply H; [ rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
+ simplify in H2; rewrite > Ett1 in H2; simplify in H2; assumption]
+|2:rewrite > H; [1:reflexivity|2: rewrite > uniq_tail in H1; cases (b2pT ? ? (andbP ? ?) H1); assumption]
+ simplify in H2; generalize in match H2; apply (cmpP ? t t1);
+ intros (E) [2:assumption] clear H; rewrite > E in Ht; rewrite > H3 in Ht;
+ destruct Ht;]
+qed.
+
+definition sub_finType : ∀d:finType.∀p:d→bool.finType ≝
+ λd:finType.λp:d→bool. mk_finType (sub_eqType d p) (filter ? ? (if_p ? p) (enum d)) (sub_enumP d p).
+
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