mod change (-x)

[helm.git] / helm / software / matita / contribs / ng_assembly / num / quatern_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma b/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma
old mode 100755 (executable)
new mode 100644 (file)
index 66a75cc..a3ff945
--- a/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/num/quatern_lemmas.ma
@@ -15,8 +15,8 @@
 (* ********************************************************************** *)
 (*                          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                                                  *)
 (*                                                                        *)
 (* ********************************************************************** *)
 
@@ -27,89 +27,67 @@ include "num/bool_lemmas.ma".
 (* QUATERNARI *)
 (* ********** *)
 
+(*
 ndefinition quatern_destruct_aux ≝
 Πn1,n2:quatern.ΠP:Prop.n1 = n2 →
- match n1 with
-  [ q0 ⇒ match n2 with [ q0 ⇒ P → P | _ ⇒ P ]
-  | q1 ⇒ match n2 with [ q1 ⇒ P → P | _ ⇒ P ]
-  | q2 ⇒ match n2 with [ q2 ⇒ P → P | _ ⇒ P ]
-  | q3 ⇒ match n2 with [ q3 ⇒ P → P | _ ⇒ P ]
-  ].
+ match eq_qu n1 n2 with [ true ⇒ P → P | false ⇒ P ].
 
 ndefinition quatern_destruct : quatern_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 q0 with [ q0 ⇒ False | _ ⇒ True ]);
-                   nrewrite > H; nnormalize; napply I
-          ##]
- ##| ##2: nelim n2; nnormalize; #H;
-          ##[ ##2: napply (λx:P.x)
-          ##| ##*: napply False_ind;
-                   nchange with (match q1 with [ q1 ⇒ False | _ ⇒ True ]);
-                   nrewrite > H; nnormalize; napply I
-          ##]
- ##| ##3: nelim n2; nnormalize; #H;
-          ##[ ##3: napply (λx:P.x)
-          ##| ##*: napply False_ind;
-                   nchange with (match q2 with [ q2 ⇒ False | _ ⇒ True ]);
-                   nrewrite > H; nnormalize; napply I
-          ##]
- ##| ##4: nelim n2; nnormalize; #H;
-          ##[ ##4: napply (λx:P.x)
-          ##| ##*: napply False_ind;
-                   nchange with (match q3 with [ q3 ⇒ False | _ ⇒ True ]);
-                   nrewrite > H; nnormalize; napply I
-          ##]
- ##]
+ nnormalize;
+ napply (λx.x).
 nqed.
+*)
 
-nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
- #n1; #n2;
- nelim n1;
+nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
  nelim n2;
  nnormalize;
  napply refl_eq.
 nqed.
 
-nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; napply refl_eq
- ##| ##*: #H; napply (bool_destruct … H)
+nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_qu n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqqu n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
  ##]
 nqed.
 
-nlemma eq_to_eqqu : ∀n1,n2.n1 = n2 → eq_qu n1 n2 = true.
+nlemma eqqu_to_eq : ∀n1,n2.eq_qu n1 n2 = true → n1 = n2.
  #n1; #n2;
  ncases n1;
  ncases n2;
  nnormalize;
  ##[ ##1,6,11,16: #H; napply refl_eq
- ##| ##*: #H; napply (quatern_destruct … H)
+ ##| ##*: #H; ndestruct (*napply (bool_destruct … H)*)
  ##]
 nqed.
 
-nlemma neqqu_to_neq : ∀n1,n2.eq_qu n1 n2 = false → n1 ≠ n2.
- #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; napply (bool_destruct … H)
- ##| ##*: #H; #H1; napply (quatern_destruct … H1)
+nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_qu n1 n2));
+ napply (not_to_not (eq_qu n1 n2 = true) (n1 = n2) ? H);
+ napply (eqqu_to_eq n1 n2).
+nqed.
+
+nlemma decidable_qu : ∀x,y:quatern.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_qu x y = true) (eq_qu x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqqu_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqqu_to_neq … H))
  ##]
 nqed.
 
-nlemma neq_to_neqqu : ∀n1,n2.n1 ≠ n2 → eq_qu n1 n2 = false.
+nlemma symmetric_eqqu : symmetricT quatern bool eq_qu.
  #n1; #n2;
- ncases n1;
- ncases n2;
- nnormalize;
- ##[ ##1,6,11,16: #H; napply False_ind; napply (H (refl_eq …))
- ##| ##*: #H; napply refl_eq
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_qu n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqqu n1 n2 H);
+          napply (symmetric_eq ? (eq_qu n2 n1) false);
+          napply (neq_to_neqqu n2 n1 (symmetric_neq ? n1 n2 H))
  ##]
 nqed.