X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fprimes.ma;h=67d8254641379f309669e8dc0fa3ee6e1001caea;hb=4dc47c9675ffd5fa50296ffaa9b5997501518c98;hp=f7d6970063c4cf9b06a1e771c73b50c876bc1894;hpb=db9c252cc8adb9243892203805b203bafe486bfc;p=helm.git

diff --git a/helm/software/matita/library/nat/primes.ma b/helm/software/matita/library/nat/primes.ma
index f7d697006..67d825464 100644
--- a/helm/software/matita/library/nat/primes.ma
+++ b/helm/software/matita/library/nat/primes.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/primes".
-
 include "nat/div_and_mod.ma".
 include "nat/minimization.ma".
 include "nat/sigma_and_pi.ma".
@@ -22,9 +20,8 @@ include "nat/factorial.ma".
 inductive divides (n,m:nat) : Prop \def
 witness : \forall p:nat.m = times n p \to divides n m.
 
-interpretation "divides" 'divides n m = (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m).
-interpretation "not divides" 'ndivides n m =
- (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m)).
+interpretation "divides" 'divides n m = (divides n m).
+interpretation "not divides" 'ndivides n m = (Not (divides n m)).
 
 theorem reflexive_divides : reflexive nat divides.
 unfold reflexive.
@@ -83,36 +80,36 @@ qed.
 theorem divides_plus: \forall n,p,q:nat. 
 n \divides p \to n \divides q \to n \divides p+q.
 intros.
-elim H.elim H1. apply (witness n (p+q) (n2+n1)).
+elim H.elim H1. apply (witness n (p+q) (n1+n2)).
 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
 qed.
 
 theorem divides_minus: \forall n,p,q:nat. 
 divides n p \to divides n q \to divides n (p-q).
 intros.
-elim H.elim H1. apply (witness n (p-q) (n2-n1)).
+elim H.elim H1. apply (witness n (p-q) (n1-n2)).
 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
 qed.
 
 theorem divides_times: \forall n,m,p,q:nat. 
 n \divides p \to m \divides q \to n*m \divides p*q.
 intros.
-elim H.elim H1. apply (witness (n*m) (p*q) (n2*n1)).
+elim H.elim H1. apply (witness (n*m) (p*q) (n1*n2)).
 rewrite > H2.rewrite > H3.
-apply (trans_eq nat ? (n*(m*(n2*n1)))).
-apply (trans_eq nat ? (n*(n2*(m*n1)))).
+apply (trans_eq nat ? (n*(m*(n1*n2)))).
+apply (trans_eq nat ? (n*(n1*(m*n2)))).
 apply assoc_times.
 apply eq_f.
-apply (trans_eq nat ? ((n2*m)*n1)).
+apply (trans_eq nat ? ((n1*m)*n2)).
 apply sym_eq. apply assoc_times.
-rewrite > (sym_times n2 m).apply assoc_times.
+rewrite > (sym_times n1 m).apply assoc_times.
 apply sym_eq. apply assoc_times.
 qed.
 
 theorem transitive_divides: transitive ? divides.
 unfold.
 intros.
-elim H.elim H1. apply (witness x z (n2*n)).
+elim H.elim H1. apply (witness x z (n1*n)).
 rewrite > H3.rewrite > H2.
 apply assoc_times.
 qed.
@@ -152,7 +149,7 @@ qed.
 
 theorem antisymmetric_divides: antisymmetric nat divides.
 unfold antisymmetric.intros.elim H. elim H1.
-apply (nat_case1 n2).intro.
+apply (nat_case1 n1).intro.
 rewrite > H3.rewrite > H2.rewrite > H4.
 rewrite < times_n_O.reflexivity.
 intros.
@@ -169,11 +166,11 @@ qed.
 
 (* divides le *)
 theorem divides_to_le : \forall n,m. O < m \to n \divides m \to n \le m.
-intros. elim H1.rewrite > H2.cut (O < n2).
-apply (lt_O_n_elim n2 Hcut).intro.rewrite < sym_times.
+intros. elim H1.rewrite > H2.cut (O < n1).
+apply (lt_O_n_elim n1 Hcut).intro.rewrite < sym_times.
 simplify.rewrite < sym_plus.
 apply le_plus_n.
-elim (le_to_or_lt_eq O n2).
+elim (le_to_or_lt_eq O n1).
 assumption.
 absurd (O<m).assumption.
 rewrite > H2.rewrite < H3.rewrite < times_n_O.
@@ -190,8 +187,33 @@ rewrite > H2.rewrite < H3.
 simplify.exact (not_le_Sn_n O).
 qed.
 
-
-(*divides and div*)
+(*a variant of or_div_mod *)
+theorem or_div_mod1: \forall n,q. O < q \to
+(divides q (S n)) \land S n = (S (div n q)) * q \lor
+(\lnot (divides q (S n)) \land S n= (div n q) * q + S (n\mod q)).
+intros.elim (or_div_mod n q H);elim H1
+  [left.split
+    [apply (witness ? ? (S (n/q))).
+     rewrite > sym_times.assumption
+    |assumption
+    ]
+  |right.split
+    [intro.
+     apply (not_eq_O_S (n \mod q)).
+     (* come faccio a fare unfold nelleipotesi ? *)
+     cut ((S n) \mod q = O)
+      [rewrite < Hcut.
+       apply (div_mod_spec_to_eq2 (S n) q (div (S n) q) (mod (S n) q) (div n q) (S (mod n q)))
+        [apply div_mod_spec_div_mod.
+         assumption
+        |apply div_mod_spec_intro;assumption
+        ]
+      |apply divides_to_mod_O;assumption
+      ]
+    |assumption
+    ]
+  ]
+qed.
 
 theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
 intro.
@@ -211,6 +233,14 @@ elim (le_to_or_lt_eq O n (le_O_n n))
   ]
 qed.
 
+theorem divides_div: \forall d,n. divides d n \to divides (n/d) n.
+intros.
+apply (witness ? ? d).
+apply sym_eq.
+apply divides_to_div.
+assumption.
+qed.
+
 theorem div_div: \forall n,d:nat. O < n \to divides d n \to 
 n/(n/d) = d.
 intros.
@@ -238,11 +268,11 @@ O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
 intros.
 elim H1.
 rewrite > H2.
-rewrite > (sym_times c n2).
+rewrite > (sym_times c n1).
 cut(O \lt c)
-[ rewrite > (lt_O_to_div_times n2 c)
+[ rewrite > (lt_O_to_div_times n1 c)
   [ rewrite < assoc_times.
-    rewrite > (lt_O_to_div_times (a *n2) c)
+    rewrite > (lt_O_to_div_times (a *n1) c)
     [ reflexivity
     | assumption
     ]
@@ -253,6 +283,28 @@ cut(O \lt c)
 ]
 qed.
 
+theorem eq_div_plus: \forall n,m,d. O < d \to
+divides d n \to divides d m \to
+(n + m ) / d = n/d + m/d.
+intros.
+elim H1.
+elim H2.
+rewrite > H3.rewrite > H4.
+rewrite < distr_times_plus.
+rewrite > sym_times.
+rewrite > sym_times in ⊢ (? ? ? (? (? % ?) ?)).
+rewrite > sym_times in ⊢ (? ? ? (? ? (? % ?))).
+rewrite > lt_O_to_div_times
+  [rewrite > lt_O_to_div_times
+    [rewrite > lt_O_to_div_times
+      [reflexivity
+      |assumption
+      ]
+    |assumption
+    ]
+  |assumption
+  ]
+qed.
 
 (* boolean divides *)
 definition divides_b : nat \to nat \to bool \def
@@ -287,7 +339,8 @@ intros 2.apply (nat_case n)
   [apply (nat_case m)
     [intro.apply divides_n_n
     |simplify.intros.apply False_ind.
-     apply not_eq_true_false.apply sym_eq.assumption
+     apply not_eq_true_false.apply sym_eq.
+     assumption
     ]
   |intros.
    apply divides_b_true_to_divides1
@@ -345,6 +398,22 @@ elim (divides_b n m).reflexivity.
 absurd (n \divides m).assumption.assumption.
 qed.
 
+theorem divides_to_divides_b_true1 : \forall n,m:nat.
+O < m \to n \divides m \to divides_b n m = true.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+  [apply divides_to_divides_b_true
+    [assumption|assumption]
+  |apply False_ind.
+   rewrite < H in H2.
+   elim H2.
+   simplify in H3.
+   apply (not_le_Sn_O O).
+   rewrite > H3 in H1.
+   assumption
+  ]
+qed.
+
 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
 \lnot(n \divides m) \to (divides_b n m) = false.
 intros.
@@ -357,6 +426,18 @@ absurd (n \divides m).assumption.assumption.
 reflexivity.
 qed.
 
+theorem divides_b_div_true: 
+\forall d,n. O < n \to 
+  divides_b d n = true \to divides_b (n/d) n = true.
+intros.
+apply divides_to_divides_b_true1
+  [assumption
+  |apply divides_div.
+   apply divides_b_true_to_divides.
+   assumption
+  ]
+qed.
+
 theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
 intros.
 elim (le_to_or_lt_eq ? ? (le_O_n m))
@@ -453,6 +534,11 @@ theorem prime_to_lt_O: \forall p. prime p \to O < p.
 intros.elim H.apply lt_to_le.assumption.
 qed.
 
+theorem prime_to_lt_SO: \forall p. prime p \to S O < p.
+intros.elim H.
+assumption.
+qed.
+
 (* smallest factor *)
 definition smallest_factor : nat \to nat \def
 \lambda n:nat.