rebuilding the library

diff --git a/helm/software/matita/nlibrary/arithmetics/nat.ma b/helm/software/matita/nlibrary/arithmetics/nat.ma
new file mode 100644 (file)
index 0000000..189f2f1
--- /dev/null
+++ b/helm/software/matita/nlibrary/arithmetics/nat.ma
@@ -0,0 +1,232 @@
+(**************************************************************************)
+(*       ___                                                             *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* include "higher_order_defs/functions.ma". *)
+include "basics/functions.ma".
+include "basics/eq.ma". 
+
+ninductive nat : Type[0] ≝
+  | O : nat
+  | S : nat → nat.
+  
+interpretation "Natural numbers" 'N = nat.
+
+default "natural numbers" cic:/matita/ng/arithmetics/nat/nat.ind.
+
+alias num (instance 0) = "natural number".
+
+ndefinition pred ≝
+ λn. match n with
+ [ O ⇒  O
+ | (S p) ⇒ p].
+
+ntheorem pred_Sn : ∀n. n = pred (S n).
+//. nqed.
+
+ntheorem injective_S : injective nat nat S.
+(* nwhd; #a; #b;#H;napplyS pred_Sn. *)
+//. nqed.
+
+(*
+ntheorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
+//. nqed. *)
+
+ntheorem not_eq_S: ∀n,m:nat. n ≠ m → S n ≠ S m.
+/2/; nqed.
+
+ndefinition not_zero: nat → Prop ≝
+ λn: nat. match n with
+  [ O ⇒ False
+  | (S p) ⇒ True ].
+
+ntheorem not_eq_O_S : ∀n:nat. O ≠ S n.
+#n; #eqOS; nchange with (not_zero O); nrewrite > eqOS; //.
+nqed.
+
+ntheorem not_eq_n_Sn : ∀n:nat. n ≠ S n.
+#n; nelim n; /2/; nqed.
+
+ntheorem nat_case:
+ ∀n:nat.∀P:nat → Prop. 
+  (n=O → P O) → (∀m:nat. (n=(S m) → P (S m))) → P n.
+#n; #P; nelim n; /2/; nqed.
+
+ntheorem nat_elim2 :
+ ∀R:nat → nat → Prop.
+  (∀n:nat. R O n) 
+  → (∀n:nat. R (S n) O)
+  → (∀n,m:nat. R n m → R (S n) (S m))
+  → ∀n,m:nat. R n m.
+#R; #ROn; #RSO; #RSS; #n; nelim n;//;
+#n0; #Rn0m; #m; ncases m;/2/; nqed.
+
+ntheorem decidable_eq_nat : \forall n,m:nat.decidable (n=m).
+napply nat_elim2;#n;
+ ##[ ncases n; /2/;
+ ##| /3/;
+ ##| #m; #Hind; ncases Hind; /3/; (* ??? /2/;#neqnm; /3/; *)
+ ##]
+nqed. 
+
+(*************************** plus ******************************)
+
+nlet rec plus n m ≝ 
+ match n with 
+ [ O ⇒ m
+ | S p ⇒ S (plus p 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_n_Sm : ∀n,m:nat. S (n+m) = n+(S m).
+#n; nelim n; nnormalize; //; nqed.
+
+ntheorem plus_n_SO : ∀n:nat. S n = n+(S O).
+//; nqed.
+
+ntheorem sym_plus: ∀n,m:nat. n+m = m+n.
+#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 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. *)
+
+ntheorem injective_plus_l: ∀m:nat.injective nat nat (λn.n+m).
+/2/; nqed.
+
+(* ntheorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
+\def injective_plus_l. *)
+
+(*************************** times *****************************)
+
+nlet rec times n m ≝ 
+ match n with 
+ [ O ⇒ O
+ | S p ⇒ m+(times p m) ].
+
+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_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.
+
+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.
+  
+ntheorem times_n_SO : ∀n:nat. n = n * S O.
+//; nqed.
+
+(*
+
+ntheorem times_SSO_n : ∀n:nat. n + n = 2 * n.
+intros.
+simplify.
+rewrite < plus_n_O.
+reflexivity.
+qed.
+
+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.
+
+theorem 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.
+*)
+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.
+
+*)
\ No newline at end of file