(* ********************************************************************** *)
(* Progetto FreeScale *)
(* *)
-(* Sviluppato da: Cosimo Oliboni, oliboni@cs.unibo.it *)
-(* Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
(* *)
(* ********************************************************************** *)
(* OTTALI *)
(* ****** *)
+(*
ndefinition oct_destruct_aux ≝
Πn1,n2:oct.ΠP:Prop.n1 = n2 →
- match n1 with
- [ o0 ⇒ match n2 with [ o0 ⇒ P → P | _ ⇒ P ]
- | o1 ⇒ match n2 with [ o1 ⇒ P → P | _ ⇒ P ]
- | o2 ⇒ match n2 with [ o2 ⇒ P → P | _ ⇒ P ]
- | o3 ⇒ match n2 with [ o3 ⇒ P → P | _ ⇒ P ]
- | o4 ⇒ match n2 with [ o4 ⇒ P → P | _ ⇒ P ]
- | o5 ⇒ match n2 with [ o5 ⇒ P → P | _ ⇒ P ]
- | o6 ⇒ match n2 with [ o6 ⇒ P → P | _ ⇒ P ]
- | o7 ⇒ match n2 with [ o7 ⇒ P → P | _ ⇒ P ]
- ].
+ match eq_oct n1 n2 with [ true ⇒ P → P | false ⇒ P ].
ndefinition oct_destruct : oct_destruct_aux.
- #n1; #n2; #P;
+ #n1; #n2; #P; #H;
+ nrewrite < H;
nelim n1;
- ##[ ##1: nelim n2; nnormalize; #H;
- ##[ ##1: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o0 with [ o0 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##2: nelim n2; nnormalize; #H;
- ##[ ##2: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o1 with [ o1 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##3: nelim n2; nnormalize; #H;
- ##[ ##3: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o2 with [ o2 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##4: nelim n2; nnormalize; #H;
- ##[ ##4: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o3 with [ o3 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##5: nelim n2; nnormalize; #H;
- ##[ ##5: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o4 with [ o4 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##6: nelim n2; nnormalize; #H;
- ##[ ##6: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o5 with [ o5 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##7: nelim n2; nnormalize; #H;
- ##[ ##7: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o6 with [ o6 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##| ##8: nelim n2; nnormalize; #H;
- ##[ ##8: napply (λx:P.x)
- ##| ##*: napply False_ind;
- nchange with (match o7 with [ o7 ⇒ False | _ ⇒ True ]);
- nrewrite > H; nnormalize; napply I
- ##]
- ##]
+ 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 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 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 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; napply False_ind; napply (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.