reorganization

diff --git a/matita/library/nat/div_and_mod.ma b/matita/library/nat/div_and_mod.ma
index 2f186dd31ae1ff0cf31eb5971c59c388d14ba349..a9f40cc896cd6c5edc0fa8f9433af97080602b9a 100644 (file)
--- a/matita/library/nat/div_and_mod.ma
+++ b/matita/library/nat/div_and_mod.ma
@@ -296,3 +296,17 @@ qed.
 
 variant inj_times_l1:\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q
 \def lt_O_to_injective_times_l.
+
+(* n_divides computes the pair (div,mod) *)
+
+(* p is just an upper bound, acc is an accumulator *)
+let rec n_divides_aux p n m acc \def
+  match n \mod m with
+  [ O \Rightarrow 
+    match p with
+      [ O \Rightarrow pair nat nat acc n
+      | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)]
+  | (S a) \Rightarrow pair nat nat acc n].
+
+(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
+definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.

diff --git a/matita/library/nat/primes1.ma b/matita/library/nat/primes1.ma
deleted file mode 100644 (file)
index 3ec61ee..0000000
--- a/matita/library/nat/primes1.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
-(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
-(*      \   /                                                             *)
-(*       \ /        Matita is distributed under the terms of the          *)
-(*        v         GNU Lesser General Public License Version 2.1         *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/primes1".
-
-include "datatypes/constructors.ma".
-include "nat/primes.ma".
-
-(* p is just an upper bound, acc is an accumulator *)
-let rec n_divides_aux p n m acc \def
-  match n \mod m with
-  [ O \Rightarrow 
-    match p with
-      [ O \Rightarrow pair nat nat acc n
-      | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)]
-  | (S a) \Rightarrow pair nat nat acc n].
-
-(* n_divides n m = <q,r> if m divides n q times, with remainder r *)
-definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O.
-
-(*
-theorem n_divides_to_Prop: \forall n,m,p,a. 
-  match n_divides_aux p n m a with
-  [ (pair q r) \Rightarrow n = m \sup a *r].
-intros.
-apply nat_case (n \mod m). *)
-

diff --git a/matita/library/nat/sigma_and_pi.ma b/matita/library/nat/sigma_and_pi.ma
index 4f5f6cba008299370b94fb0e95f7b0b6248407a6..fe3a2d8b5d417502ac99e5572a1a759a7e448bba 100644 (file)
--- a/matita/library/nat/sigma_and_pi.ma
+++ b/matita/library/nat/sigma_and_pi.ma
@@ -15,7 +15,6 @@
 set "baseuri" "cic:/matita/nat/sigma_and_pi".
 
 include "nat/factorial.ma".
-include "nat/lt_arith.ma".
 include "nat/exp.ma".
 
 let rec sigma n f m \def