--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "num/exadecim.ma".
+include "num/bool_lemmas.ma".
+
+(* *********** *)
+(* ESADECIMALI *)
+(* *********** *)
+
+(*
+ndefinition exadecim_destruct_aux ≝
+Πe1,e2.ΠP:Prop.ΠH:e1 = e2.
+ match eq_ex e1 e2 with [ true ⇒ P → P | false ⇒ P ].
+
+ndefinition exadecim_destruct : exadecim_destruct_aux.
+ #e1; #e2; #P; #H;
+ nrewrite < H;
+ nelim e1;
+ nnormalize;
+ napply (λx.x).
+nqed.
+*)
+
+nlemma eq_to_eqex : ∀n1,n2.n1 = n2 → eq_ex n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqex_to_neq : ∀n1,n2.eq_ex n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_ex n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqex n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqex_to_eq : ∀n1,n2.eq_ex n1 n2 = true → n1 = n2.
+ #n1; #n2;
+ ncases n1;
+ ncases n2;
+ nnormalize;
+ ##[ ##1,18,35,52,69,86,103,120,137,154,171,188,205,222,239,256: #H; napply refl_eq
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma neq_to_neqex : ∀n1,n2.n1 ≠ n2 → eq_ex n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_ex n1 n2));
+ napply (not_to_not (eq_ex n1 n2 = true) (n1 = n2) ? H);
+ napply (eqex_to_eq n1 n2).
+nqed.
+
+nlemma decidable_ex : ∀x,y:exadecim.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_ex x y = true) (eq_ex x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqex_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqex_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqex : symmetricT exadecim bool eq_ex.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_ex n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqex n1 n2 H);
+ napply (symmetric_eq ? (eq_ex n2 n1) false);
+ napply (neq_to_neqex n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.