Towards chebyshev.

[helm.git] / matita / library / nat / primes.ma

diff --git a/matita/library/nat/primes.ma b/matita/library/nat/primes.ma
index a95b2e88fddda2344f0fafc448593de57404d470..ec7118980e1a82a53609a14bd76c0371e6a83585 100644 (file)
--- a/matita/library/nat/primes.ma
+++ b/matita/library/nat/primes.ma
@@ -190,6 +190,103 @@ rewrite > H2.rewrite < H3.
 simplify.exact (not_le_Sn_n O).
 qed.
 
+(*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.
+elim (le_to_or_lt_eq O n (le_O_n n))
+  [rewrite > plus_n_O.
+   rewrite < (divides_to_mod_O ? ? H H1).
+   apply sym_eq.
+   apply div_mod.
+   assumption
+  |elim H1.
+   generalize in match H2.
+   rewrite < H.
+   simplify.
+   intro.
+   rewrite > H3.
+   reflexivity
+  ]
+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.
+apply (inj_times_l1 (n/d))
+  [apply (lt_times_n_to_lt d)
+    [apply (divides_to_lt_O ? ? H H1).
+    |rewrite > divides_to_div;assumption
+    ]
+  |rewrite > divides_to_div
+    [rewrite > sym_times.
+     rewrite > divides_to_div
+      [reflexivity
+      |assumption
+      ]
+    |apply (witness ? ? d).
+     apply sym_eq.
+     apply divides_to_div.
+     assumption
+    ]
+  ]
+qed.
+
+theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
+O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
+intros.
+elim H1.
+rewrite > H2.
+rewrite > (sym_times c n2).
+cut(O \lt c)
+[ rewrite > (lt_O_to_div_times n2 c)
+  [ rewrite < assoc_times.
+    rewrite > (lt_O_to_div_times (a *n2) c)
+    [ reflexivity
+    | assumption
+    ]
+  | assumption
+  ]  
+| apply (divides_to_lt_O c b);
+    assumption.
+]
+qed.
+
+
 (* boolean divides *)
 definition divides_b : nat \to nat \to bool \def
 \lambda n,m :nat. (eqb (m \mod n) O).
@@ -223,7 +320,8 @@ 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
@@ -281,6 +379,22 @@ 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.
@@ -293,6 +407,18 @@ 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))
@@ -389,6 +515,11 @@ 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.