(* Progetto FreeScale *)
(* *)
(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
-(* Ultima modifica: 05/08/2009 *)
+(* Sviluppo: 2008-2010 *)
(* *)
(* ********************************************************************** *)
(* OTTALI *)
(* ****** *)
+(*
ndefinition oct_destruct_aux ≝
Πn1,n2:oct.ΠP:Prop.n1 = n2 →
match eq_oct n1 n2 with [ true ⇒ P → P | false ⇒ P ].
nnormalize;
napply (λx.x).
nqed.
+*)
-nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
- #n1; #n2;
- nelim n1;
+nlemma eq_to_eqoct : ∀n1,n2.n1 = n2 → eq_oct n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
nelim n2;
nnormalize;
napply refl_eq.
nqed.
-nlemma eqoct_to_eq : ∀n1,n2.eq_oct n1 n2 = true → n1 = n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+nlemma neqoct_to_neq : ∀n1,n2.eq_oct n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_oct n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqoct n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
##]
nqed.
-nlemma eq_to_eqoct : ∀n1,n2.n1 = n2 → eq_oct n1 n2 = true.
+nlemma eqoct_to_eq : ∀n1,n2.eq_oct n1 n2 = true → n1 = n2.
#n1; #n2;
ncases n1;
ncases n2;
nnormalize;
##[ ##1,10,19,28,37,46,55,64: #H; napply refl_eq
- ##| ##*: #H; napply (oct_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
##]
nqed.
-nlemma decidable_oct : ∀x,y:oct.decidable (x = y).
- #x; #y;
- nnormalize;
- nelim x;
- nelim y;
- ##[ ##1,10,19,28,37,46,55,64: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …);
- nnormalize; #H;
- napply False_ind;
- napply (oct_destruct … H)
- ##]
+nlemma neq_to_neqoct : ∀n1,n2.n1 ≠ n2 → eq_oct n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_oct n1 n2));
+ napply (not_to_not (eq_oct n1 n2 = true) (n1 = n2) ? H);
+ napply (eqoct_to_eq n1 n2).
nqed.
-nlemma neqoct_to_neq : ∀n1,n2.eq_oct n1 n2 = false → n1 ≠ n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,10,19,28,37,46,55,64: #H; napply (bool_destruct … H)
- ##| ##*: #H; #H1; napply (oct_destruct … H1)
+nlemma decidable_oct : ∀x,y:oct.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_oct x y = true) (eq_oct x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqoct_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqoct_to_neq … H))
##]
nqed.
-nlemma neq_to_neqoct : ∀n1,n2.n1 ≠ n2 → eq_oct n1 n2 = false.
+nlemma symmetric_eqoct : symmetricT oct bool eq_oct.
#n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,10,19,28,37,46,55,64: #H; nelim (H (refl_eq …))
- ##| ##*: #H; napply refl_eq
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_oct n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqoct n1 n2 H);
+ napply (symmetric_eq ? (eq_oct n2 n1) false);
+ napply (neq_to_neqoct n2 n1 (symmetric_neq ? n1 n2 H))
##]
nqed.