Towards chebyshev.

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

diff --git a/matita/library/nat/gcd.ma b/matita/library/nat/gcd.ma
index ded9d4843ac6cf3cb4da2b4f564568db2df2e02b..9bbce8144ff8885a7c7346bdb9b7937983bd5f7e 100644 (file)
--- a/matita/library/nat/gcd.ma
+++ b/matita/library/nat/gcd.ma
@@ -749,6 +749,7 @@ apply gcd_O_to_eq_O.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 qed.
 
+(* primes and divides *)
 theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
 n \divides p \lor n \divides q.
 intros.
@@ -802,6 +803,30 @@ cut (n \divides p \lor n \ndivides p)
   ]
 qed.
 
+theorem divides_exp_to_divides: 
+\forall p,n,m:nat. prime p \to 
+p \divides n \sup m \to p \divides n.
+intros 3.elim m.simplify in H1.
+apply (transitive_divides p (S O)).assumption.
+apply divides_SO_n.
+cut (p \divides n \lor p \divides n \sup n1).
+elim Hcut.assumption.
+apply H.assumption.assumption.
+apply divides_times_to_divides.assumption.
+exact H2.
+qed.
+
+theorem divides_exp_to_eq: 
+\forall p,q,m:nat. prime p \to prime q \to
+p \divides q \sup m \to p = q.
+intros.
+unfold prime in H1.
+elim H1.apply H4.
+apply (divides_exp_to_divides p q m).
+assumption.assumption.
+unfold prime in H.elim H.assumption.
+qed.
+
 theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
 gcd m n = (S O) \to gcd m p = (S O) \to gcd m (n*p) = (S O).
 intros.
@@ -877,4 +902,5 @@ cut (n \divides p \lor n \ndivides p)
  |apply (decidable_divides n p).
   assumption.
  ]
-qed.
\ No newline at end of file
+qed.
+