Updating.

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
index 189f2f1524598296296b945f3a4841685f05acdb..a1e6fbb230c6f7cd93e444aa96a4fc681bfbacd1 100644 (file)
--- a/helm/software/matita/nlibrary/arithmetics/nat.ma
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -13,6 +13,7 @@
 (**************************************************************************)
 
 (* include "higher_order_defs/functions.ma". *)
+include "hints_declaration.ma".
 include "basics/functions.ma".
 include "basics/eq.ma". 
 
@@ -22,9 +23,17 @@ ninductive nat : Type[0] ≝
   
 interpretation "Natural numbers" 'N = nat.
 
-default "natural numbers" cic:/matita/ng/arithmetics/nat/nat.ind.
+alias num (instance 0) = "nnatural number".
 
-alias num (instance 0) = "natural number".
+(*
+nrecord pos : Type ≝
+ {n:>nat; is_pos: n ≠ 0}.
+
+ncoercion nat_to_pos: ∀n:nat. n ≠0 →pos ≝ mk_pos on 
+*)
+
+(* default "natural numbers" cic:/matita/ng/arithmetics/nat/nat.ind.
+*)
 
 ndefinition pred ≝
  λn. match n with
@@ -35,8 +44,7 @@ ntheorem pred_Sn : ∀n. n = pred (S n).
 //. nqed.
 
 ntheorem injective_S : injective nat nat S.
-(* nwhd; #a; #b;#H;napplyS pred_Sn. *)
-//. nqed.
+//; nqed.
 
 (*
 ntheorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
@@ -88,32 +96,46 @@ nlet rec plus n m ≝
 
 interpretation "natural plus" 'plus x y = (plus x y).
 
-ntheorem plus_n_O: ∀n:nat. n = n+O.
-#n; nelim n; /2/; nqed.
+ntheorem plus_O_n: ∀n:nat. n = 0+n.
+//; nqed.
 
-ntheorem plus_n_Sm : ∀n,m:nat. S (n+m) = n+(S m).
+(*
+ntheorem plus_Sn_m: ∀n,m:nat. S (n + m) = S n + m.
+//; nqed.
+*)
+
+ntheorem plus_n_O: ∀n:nat. n = n+0.
 #n; nelim n; nnormalize; //; nqed.
 
-ntheorem plus_n_SO : ∀n:nat. S n = n+(S O).
-//; nqed.
+ntheorem plus_n_Sm : ∀n,m:nat. S (n+m) = n + S m.
+#n; nelim n; nnormalize; //; nqed.
 
-ntheorem sym_plus: ∀n,m:nat. n+m = m+n.
+(*
+ntheorem plus_Sn_m1: ∀n,m:nat. S m + n = n + S m.
 #n; nelim n; nnormalize; //; nqed.
+*)
+
+(*
+ntheorem plus_n_SO : ∀n:nat. S n = n+S O.
+//; nqed. *)
+
+ntheorem symmetric_plus: symmetric ? plus.
+#n; nelim n; nnormalize; //; nqed. 
 
 ntheorem associative_plus : associative nat plus.
 #n; nelim n; nnormalize; //; nqed.
 
-(* ntheorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
-\def associative_plus. *)
+ntheorem assoc_plus1: ∀a,b,c. b + (a + c) = a + (b + c).
+//; nqed.
 
 ntheorem injective_plus_r: ∀n:nat.injective nat nat (λm.n+m).
 #n; nelim n; nnormalize; /3/; nqed.
 
 (* ntheorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
-\def injective_plus_r. *)
+\def injective_plus_r. 
 
 ntheorem injective_plus_l: ∀m:nat.injective nat nat (λn.n+m).
-/2/; nqed.
+/2/; nqed. *)
 
 (* ntheorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
 \def injective_plus_l. *)
@@ -130,103 +152,59 @@ interpretation "natural times" 'times x y = (times x y).
 ntheorem times_Sn_m: ∀n,m:nat. m+n*m = S n*m.
 //; nqed.
 
+ntheorem times_O_n: ∀n:nat. O = O*n.
+//; nqed.
+
 ntheorem times_n_O: ∀n:nat. O = n*O.
 #n; nelim n; //; nqed.
 
 ntheorem times_n_Sm : ∀n,m:nat. n+(n*m) = n*(S m).
-#n; nelim n; nnormalize; /2/; nqed.
+#n; nelim n; nnormalize; //; nqed.
+
+ntheorem symmetric_times : symmetric nat times. 
+#n; nelim n; nnormalize; //; nqed.
+
+(* variant sym_times : \forall n,m:nat. n*m = m*n \def
+symmetric_times. *)
+
+ntheorem distributive_times_plus : distributive nat times plus.
+#n; nelim n; nnormalize; //; nqed.
+
+ntheorem distributive_times_plus_r:  
+\forall a,b,c:nat. (b+c)*a = b*a + c*a.
+//; nqed.
+
+ntheorem associative_times: associative nat times.
+#n; nelim n; nnormalize; //; nqed.
 
+nlemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
+//; nqed. 
+
+(* ci servono questi risultati? 
 ntheorem times_O_to_O: ∀n,m:nat.n*m=O → n=O ∨ m=O.
-napply nat_elim2; (* /2/ slow! *)
-  ##[#n; #H; @1; //;
-  ##|#n; #H; @2; //; (* ?? *)
-  ##|#n; #m; #H; #H1; napply False_ind;napply not_eq_O_S;
-  (* why two steps? *) /2/;
-  ##]
-nqed.
+napply nat_elim2; /2/; 
+#n; #m; #H; nnormalize; #H1; napply False_ind;napply not_eq_O_S;
+//; nqed.
   
 ntheorem times_n_SO : ∀n:nat. n = n * S O.
-//; nqed.
-
-(*
+#n; //; nqed.
 
-ntheorem times_SSO_n : ∀n:nat. n + n = 2 * n.
-intros.
-simplify.
-rewrite < plus_n_O.
-reflexivity.
-qed.
+ntheorem times_SSO_n : ∀n:nat. n + n = (S(S O)) * n.
+nnormalize; //; nqed.
 
-alias num (instance 0) = "natural number".
-lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
-intro.
-simplify.rewrite < plus_n_Sm.reflexivity.
-qed.
+nlemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
+//; nqed.
 
-theorem or_eq_eq_S: \forall n.\exists m. 
+ntheorem or_eq_eq_S: \forall n.\exists m. 
 n = (S(S O))*m \lor n = S ((S(S O))*m).
-intro.elim n
-  [apply (ex_intro ? ? O).
-   left.reflexivity
-  |elim H. elim H1
-    [apply (ex_intro ? ? a).
-     right.apply eq_f.assumption
-    |apply (ex_intro ? ? (S a)).
-     left. rewrite > H2.
-     apply sym_eq.
-     apply times_SSO
-    ]
-  ]
-qed.
-
-theorem symmetric_times : symmetric nat times. 
-unfold symmetric.
-intros.elim x
-  [simplify.apply times_n_O. 
-  | simplify.rewrite > H.apply times_n_Sm.]
-qed.
-
-variant sym_times : \forall n,m:nat. n*m = m*n \def
-symmetric_times.
-
-theorem distributive_times_plus : distributive nat times plus.
-unfold distributive.
-intros.elim x.
-simplify.reflexivity.
-simplify.
-demodulate all.
-(*
-rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
-apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
-rewrite > assoc_plus.reflexivity. *)
-qed.
-
-variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
-\def distributive_times_plus.
-
-theorem associative_times: associative nat times.
-unfold associative.
-intros.
-elim x. simplify.apply refl_eq. 
-simplify.
-demodulate all.
-(*
-rewrite < sym_times.
->>>>>>> .r9770
-rewrite > distr_times_plus.
-rewrite < sym_times.
-rewrite < (sym_times (times n y) z).
-rewrite < H.apply refl_eq.
+#n; nelim n;
+  ##[@; /2/;
+  ##|#a; #H; nelim H; #b;#or;nelim or;#aeq;
+    ##[@ b; @ 2; //;
+    ##|@ (S b); @ 1; /2/;
+    ##]
+nqed.
 *)
-qed.
-
-variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
-associative_times.
 
-lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
-intros. 
-demodulate. reflexivity.
-(* autobatch paramodulation. *)
-qed.
+(************************** compare ****************************)
 
-*)
\ No newline at end of file