(* ********************************************************************** *)
(* 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 *)
(* *)
(* ********************************************************************** *)
napply refl_eq.
nqed.
+(* !!! da togliere *)
nlemma option_destruct_some_none : ∀T.∀x:T.Some T x = None T → False.
#T; #x; #H;
nchange with (match Some T x with [ None ⇒ True | Some a ⇒ False ]);
napply I.
nqed.
+(* !!! da togliere *)
nlemma option_destruct_none_some : ∀T.∀x:T.None T = Some T x → False.
#T; #x; #H;
nchange with (match Some T x with [ None ⇒ True | Some a ⇒ False ]);
nqed.
nlemma symmetric_eqoption :
-∀T:Type.∀op1,op2:option T.∀f:T → T → bool.
+∀T:Type.∀f:T → T → bool.
(symmetricT T bool f) →
- (eq_option T op1 op2 f = eq_option T op2 op1 f).
- #T; #op1; #op2; #f; #H;
- nelim op1;
- nelim op2;
+ (∀op1,op2:option T.
+ (eq_option T f op1 op2 = eq_option T f op2 op1)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
nnormalize;
##[ ##1: napply refl_eq
##| ##2,3: #H; napply refl_eq
nqed.
nlemma eq_to_eqoption :
-∀T.∀op1,op2:option T.∀f:T → T → bool.
+∀T.∀f:T → T → bool.
(∀x1,x2:T.x1 = x2 → f x1 x2 = true) →
- (op1 = op2 → eq_option T op1 op2 f = true).
- #T; #op1; #op2; #f; #H;
- nelim op1;
- nelim op2;
+ (∀op1,op2:option T.
+ (op1 = op2 → eq_option T f op1 op2 = true)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
nnormalize;
##[ ##1: #H1; napply refl_eq
- ##| ##2: #a; #H1; nelim (option_destruct_none_some ?? H1)
- ##| ##3: #a; #H1; nelim (option_destruct_some_none ?? H1)
+ ##| ##2: #a; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (option_destruct_none_some ?? H1)
+ ##| ##3: #a; #H1;
+ (* !!! ndestruct: assert false *)
+ nelim (option_destruct_some_none ?? H1)
##| ##4: #a; #a0; #H1;
nrewrite > (H … (option_destruct_some_some … H1));
napply refl_eq
nqed.
nlemma eqoption_to_eq :
-∀T.∀op1,op2:option T.∀f:T → T → bool.
+∀T.∀f:T → T → bool.
(∀x1,x2:T.f x1 x2 = true → x1 = x2) →
- (eq_option T op1 op2 f = true → op1 = op2).
- #T; #op1; #op2; #f; #H;
- nelim op1;
- nelim op2;
+ (∀op1,op2:option T.
+ (eq_option T f op1 op2 = true → op1 = op2)).
+ #T; #f; #H;
+ #op1; #op2; nelim op1; nelim op2;
nnormalize;
##[ ##1: #H1; napply refl_eq
- ##| ##2,3: #a; #H1; napply (bool_destruct … H1)
+ ##| ##2,3: #a; #H1; ndestruct (*napply (bool_destruct … H1)*)
##| ##4: #a; #a0; #H1;
nrewrite > (H … H1);
napply refl_eq
##]
nqed.
+nlemma decidable_option :
+∀T.(Πx,y:T.decidable (x = y)) →
+ (∀x,y:option T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+ ##| ##2: #yy; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H1;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_none_some T … H1)
+ ##]
+ ##| ##2: #xx; #y; ncases y;
+ ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_some_none T … H2)
+ ##| ##2: #yy; nnormalize; napply (or2_elim (xx = yy) (xx ≠ yy) ? (H …));
+ ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H2;
+ napply (H1 (option_destruct_some_some T … H2))
+ ##| ##1: #H1; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H1; napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
nlemma neq_to_neqoption :
-∀T.∀op1,op2:option T.∀f:T → T → bool.
+∀T.∀f:T → T → bool.
(∀x1,x2:T.x1 ≠ x2 → f x1 x2 = false) →
- (op1 ≠ op2 → eq_option T op1 op2 f = false).
- #T; #op1; #op2; #f; #H;
- nelim op1;
- nelim op2;
- nnormalize;
- ##[ ##1: #H1; napply False_ind; napply (H1 (refl_eq …))
- ##| ##2,3: #a; #H1; napply refl_eq
- ##| ##4: #a; #a0; #H1;
- napply H;
- napply (neqf_to_neq … a0 a (λx.Some ? x) H1)
+ (∀op1,op2:option T.
+ (op1 ≠ op2 → eq_option T f op1 op2 = false)).
+ #T; #f; #H; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; nelim (H1 (refl_eq …))
+ ##| ##2: #yy; nnormalize; #H1; napply refl_eq
+ ##]
+ ##| ##2: #xx; #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; napply refl_eq
+ ##| ##2: #yy; nnormalize; #H1; napply (H xx yy …);
+ nnormalize; #H2; nrewrite > H2 in H1:(%); #H1;
+ napply (H1 (refl_eq …))
+ ##]
##]
nqed.
nlemma neqoption_to_neq :
-∀T.∀op1,op2:option T.∀f:T → T → bool.
+∀T.∀f:T → T → bool.
(∀x1,x2:T.f x1 x2 = false → x1 ≠ x2) →
- (eq_option T op1 op2 f = false → op1 ≠ op2).
- #T; #op1; #op2; #f; #H;
- nelim op1;
- nelim op2;
- nnormalize;
- ##[ ##1: #H1; napply (bool_destruct … H1)
- ##| ##2: #a; #H1; #H2; napply (option_destruct_none_some ? a H2)
- ##| ##3: #a; #H1; #H2; napply (option_destruct_some_none ? a H2)
- ##| ##4: #a; #a0; #H1; #H2;
- napply (H a0 a H1);
- napply (option_destruct_some_some ? a0 a H2)
+ (∀op1,op2:option T.
+ (eq_option T f op1 op2 = false → op1 ≠ op2)).
+ #T; #f; #H; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1;
+ ndestruct (*napply (bool_destruct … H1)*)
+ ##| ##2: #yy; nnormalize; #H1; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_none_some T … H2)
+ ##]
+ ##| ##2: #xx; #op2; ncases op2;
+ ##[ ##1: nnormalize; #H1; #H2;
+ (* !!! ndestruct: assert false *)
+ napply (option_destruct_some_none T … H2)
+ ##| ##2: #yy; nnormalize; #H1; #H2; napply (H xx yy H1 ?);
+ napply (option_destruct_some_some T … H2)
+ ##]
##]
nqed.