freescale porting

[helm.git] / helm / software / matita / contribs / ng_assembly / common / option_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
index 303d11ad38dfa8600b813d0ccff86ab3244b60bc..1d25b4bd95bfed16bf577492e00721d60aab517f 100644 (file)
--- a/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
@@ -16,7 +16,7 @@
 (*                          Progetto FreeScale                            *)
 (*                                                                        *)
 (*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
-(*   Ultima modifica: 05/08/2009                                          *)
+(*   Sviluppo: 2008-2010                                                  *)
 (*                                                                        *)
 (* ********************************************************************** *)
 
@@ -35,6 +35,7 @@ nlemma option_destruct_some_some : ∀T.∀x1,x2:T.Some T x1 = Some T x2 → x1
  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 ]);
@@ -43,6 +44,7 @@ nlemma option_destruct_some_none : ∀T.∀x:T.Some T x = None T → 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 ]);
@@ -52,12 +54,12 @@ nlemma option_destruct_none_some : ∀T.∀x:T.None T = Some T x → 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
@@ -68,16 +70,20 @@ nlemma symmetric_eqoption :
 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
@@ -85,17 +91,86 @@ nlemma eq_to_eqoption :
 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.∀f:T → T → bool.
+ (∀x1,x2:T.x1 ≠ x2 → f x1 x2 = false) →
+ (∀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.∀f:T → T → bool.
+ (∀x1,x2:T.f x1 x2 = false → x1 ≠ x2) →
+ (∀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.