freescale porting

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

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 f4525c463aafdb8b9538b0af606790bbfe00e5df..06ace91b2581313105126405c4ea51603ffbccaf 100644 (file)
--- a/helm/software/matita/contribs/ng_assembly/common/nat_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/common/nat_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                                                  *)
 (*                                                                        *)
 (* ********************************************************************** *)
 
@@ -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;
@@ -78,17 +60,25 @@ 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. 
+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;
@@ -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
@@ -110,6 +100,31 @@ nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
  ##]
 nqed.
 
+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 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 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.
+
 nlemma Sn_p_n_to_S_npn : ∀n1,n2.(S n1) + n2 = S (n1 + n2).
  #n1;
  nelim n1;