X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fng_assembly%2Fcommon%2Fnat_lemmas.ma;h=06ace91b2581313105126405c4ea51603ffbccaf;hb=bc389dd4724959688aafc1ede450794f47b8d0b5;hp=bbce188e521db6730749d2fb03bdeda18cb94175;hpb=29cfb9e2961e62c836cb50217905c0594a074e81;p=helm.git

diff --git a/helm/software/matita/contribs/ng_assembly/common/nat_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/nat_lemmas.ma
index bbce188e5..06ace91b2 100644
--- a/helm/software/matita/contribs/ng_assembly/common/nat_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/common/nat_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 nat_destruct_S_S : ∀n1,n2:nat.S n1 = S n2 → n1 = n2.
  napply refl_eq.
 nqed.
 
+(* !!! da togliere *)
 nlemma nat_destruct_0_S : ∀n:nat.O = S n → False.
  #n; #H;
  nchange with (match S n with [ O ⇒ True | S a ⇒ False ]);
@@ -43,6 +44,7 @@ nlemma nat_destruct_0_S : ∀n:nat.O = S n → False.
  napply I.
 nqed.
 
+(* !!! da togliere *)
 nlemma nat_destruct_S_0 : ∀n:nat.S n = O → False.
  #n; #H;
  nchange with (match S n with [ O ⇒ True | S a ⇒ False ]);
@@ -51,26 +53,6 @@ nlemma nat_destruct_S_0 : ∀n:nat.S n = O → False.
  napply I.
 nqed.
 
-nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
- nnormalize;
- #n1;
- nelim n1;
- ##[ ##1: #n2;
-          nelim n2;
-          nnormalize;
-          ##[ ##1: napply refl_eq
-          ##| ##2: #n3; #H; napply refl_eq
-          ##]
- ##| ##2: #n2; #H; #n3;
-          nnormalize;
-          ncases n3;
-          nnormalize;
-          ##[ ##1: napply refl_eq
-          ##| ##2: #n4; napply (H n4)
-          ##]
- ##]
-nqed.
-
 nlemma eq_to_eqnat : ∀n1,n2:nat.n1 = n2 → eq_nat n1 n2 = true.
  #n1;
  nelim n1;
@@ -78,18 +60,26 @@ nlemma eq_to_eqnat : ∀n1,n2:nat.n1 = n2 → eq_nat n1 n2 = true.
           nelim n2;
           nnormalize;
           ##[ ##1: #H; napply refl_eq
-          ##| ##2: #n3; #H; #H1; nelim (nat_destruct_0_S ? H1)
+          ##| ##2: #n3; #H; #H1; ndestruct (*nelim (nat_destruct_0_S ? H1)*)
           ##]
  ##| ##2: #n2; #H; #n3; #H1;
           ncases n3 in H1:(%) ⊢ %;
           nnormalize;
-          ##[ ##1: #H1; nelim (nat_destruct_S_0 ? H1)
+          ##[ ##1: #H1; ndestruct (*nelim (nat_destruct_S_0 ? H1)*)
           ##| ##2: #n4; #H1;
                    napply (H n4 (nat_destruct_S_S … H1))
           ##]
  ##]
 nqed.
 
+nlemma neqnat_to_neq : ∀n1,n2:nat.(eq_nat n1 n2 = false → n1 ≠ n2).
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_nat n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqnat n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
 nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
  #n1;
  nelim n1;
@@ -97,12 +87,12 @@ nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
           nelim n2;
           nnormalize;
           ##[ ##1: #H; napply refl_eq
-          ##| ##2: #n3; #H; #H1; napply (bool_destruct … (O = S n3) H1) 
+          ##| ##2: #n3; #H; #H1; ndestruct (*napply (bool_destruct … (O = S n3) H1)*)
           ##]
  ##| ##2: #n2; #H; #n3; #H1;
           ncases n3 in H1:(%) ⊢ %;
           nnormalize;
-          ##[ ##1: #H1; napply (bool_destruct … (S n2 = O) H1)
+          ##[ ##1: #H1; ndestruct (*napply (bool_destruct … (S n2 = O) H1)*)
           ##| ##2: #n4; #H1;
                    nrewrite > (H n4 H1);
                    napply refl_eq
@@ -110,53 +100,28 @@ nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
  ##]
 nqed.
 
-nlemma decidable_nat : ∀x,y:nat.decidable (x = y).
- #x; nelim x; nnormalize;
- ##[ ##1: #y; ncases y;
-          ##[ ##1: napply (or2_intro1 (O = O) (O ≠ O) (refl_eq …))
-          ##| ##2: #n2; napply (or2_intro2 (O = (S n2)) (O ≠ (S n2)) ?);
-                   nnormalize; #H; napply (nat_destruct_0_S n2 H)
-          ##]
- ##| ##2: #n1; #H; #y; ncases y;
-          ##[ ##1: napply (or2_intro2 ((S n1) = O) ((S n1) ≠ O) ?);
-                   nnormalize; #H1; napply (nat_destruct_S_0 n1 H1)
-          ##| ##2: #n2; napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (H n2) …);
-                   ##[ ##1: #H1; napply (or2_intro1 (? = ?) (? ≠ ?) …);
-                            nrewrite > H1; napply refl_eq
-                   ##| ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) …);
-                            nnormalize; #H2; napply (H1 (nat_destruct_S_S … H2))
-                   ##]
-          ##]
- ##]
+nlemma neq_to_neqnat : ∀n1,n2:nat.n1 ≠ n2 → eq_nat n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_nat n1 n2));
+ napply (not_to_not (eq_nat n1 n2 = true) (n1 = n2) ? H);
+ napply (eqnat_to_eq n1 n2).
 nqed.
 
-nlemma neq_to_neqnat : ∀n1,n2:nat.n1 ≠ n2 → eq_nat n1 n2 = false.
- #n1; nelim n1;
- ##[ ##1: #n2; ncases n2;
-          ##[ ##1: nnormalize; #H; nelim (H (refl_eq …))
-          ##| ##2: #nn2; #H; nnormalize; napply refl_eq
-          ##]
- ##| ##2: #nn1; #H; #n2; ncases n2;
-          ##[ ##1: #H1; nnormalize; napply refl_eq
-          ##| ##2: #nn2; nnormalize; #H1;
-                   napply (H nn2 ?); nnormalize; #H2;
-                   nrewrite > H2 in H1:(%); #H1;
-                   napply (H1 (refl_eq …))
-          ##]
+nlemma decidable_nat : ∀x,y:nat.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_nat x y = true) (eq_nat x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqnat_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqnat_to_neq … H))
  ##]
 nqed.
 
-nlemma neqnat_to_neq : ∀n1,n2:nat.(eq_nat n1 n2 = false → n1 ≠ n2).
- #n1; nelim n1;
- ##[ ##1: #n2; ncases n2;
-          ##[ ##1: nnormalize; #H; napply (bool_destruct … H)
-          ##| ##2: #nn2; nnormalize; #H; #H1; nelim (nat_destruct_0_S … H1)  
-          ##]
- ##| ##2: #nn1; #H; #n2; ncases n2;
-          ##[ ##1: nnormalize; #H1; #H2; nelim (nat_destruct_S_0 … H2)
-          ##| ##2: #nn2; nnormalize; #H1; #H2; napply (H nn2 H1 ?);
-                   napply (nat_destruct_S_S … H2)
-          ##]
+nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_nat n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqnat n1 n2 H);
+          napply (symmetric_eq ? (eq_nat n2 n1) false);
+          napply (neq_to_neqnat n2 n1 (symmetric_neq ? n1 n2 H))
  ##]
 nqed.