Euler totient function is multiplicative!

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

diff --git a/helm/matita/library/nat/gcd.ma b/helm/matita/library/nat/gcd.ma
index 792629aaa04c224689edcf2a152a3c2925aa547f..d42a57e8e8db6c1661d480d829bcf1d6289c96de 100644 (file)
--- a/helm/matita/library/nat/gcd.ma
+++ b/helm/matita/library/nat/gcd.ma
@@ -390,6 +390,16 @@ rewrite < H.
 apply divides_gcd_nm.
 qed.
 
+theorem lt_O_gcd:\forall m,n:nat. O < n \to O < gcd m n.
+intros.
+apply (nat_case1 (gcd m n)).
+intros.
+generalize in match (gcd_O_to_eq_O m n H1).
+intros.elim H2.
+rewrite < H4 in \vdash (? ? %).assumption.
+intros.simplify.apply le_S_S.apply le_O_n.
+qed.
+
 theorem symmetric_gcd: symmetric nat gcd.
 change with 
 (\forall n,m:nat. gcd n m = gcd m n).
@@ -418,6 +428,33 @@ qed.
 variant sym_gcd: \forall n,m:nat. gcd n m = gcd m n \def
 symmetric_gcd.
 
+theorem le_gcd_times: \forall m,n,p:nat. O< p \to gcd m n \le gcd m (n*p).
+intros.
+apply (nat_case n).reflexivity.
+intro.
+apply divides_to_le.
+apply lt_O_gcd.
+rewrite > (times_n_O O).
+apply lt_times.simplify.apply le_S_S.apply le_O_n.assumption.
+apply divides_d_gcd.
+apply (transitive_divides ? (S m1)).
+apply divides_gcd_m.
+apply (witness ? ? p).reflexivity.
+apply divides_gcd_n.
+qed.
+
+theorem gcd_times_SO_to_gcd_SO: \forall m,n,p:nat. O < n \to O < p \to 
+gcd m (n*p) = (S O) \to gcd m n = (S O).
+intros.
+apply antisymmetric_le.
+rewrite < H2.
+apply le_gcd_times.assumption.
+change with (O < gcd m n). 
+apply lt_O_gcd.assumption.
+qed.
+
+(* for the "converse" of the previous result see the end  of this development *)
+
 theorem gcd_SO_n: \forall n:nat. gcd (S O) n = (S O).
 intro.
 apply antisym_le.apply divides_to_le.simplify.apply le_n.
@@ -432,7 +469,7 @@ apply gcd_O_to_eq_O.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 qed.
 
-theorem divides_gcd_mod: \forall m,n:nat. O < n \to O < m \to
+theorem divides_gcd_mod: \forall m,n:nat. O < n \to
 divides (gcd m n) (gcd n (m \mod n)).
 intros.
 apply divides_d_gcd.
@@ -442,7 +479,7 @@ apply divides_gcd_m.
 apply divides_gcd_m.
 qed.
 
-theorem divides_mod_gcd: \forall m,n:nat. O < n \to O < m \to
+theorem divides_mod_gcd: \forall m,n:nat. O < n \to
 divides (gcd n (m \mod n)) (gcd m n) .
 intros.
 apply divides_d_gcd.
@@ -453,6 +490,14 @@ apply divides_gcd_m.
 apply divides_gcd_n.
 qed.
 
+theorem gcd_mod: \forall m,n:nat. O < n \to
+(gcd n (m \mod n)) = (gcd m n) .
+intros.
+apply antisymmetric_divides.
+apply divides_mod_gcd.assumption.
+apply divides_gcd_mod.assumption.
+qed.
+
 (* gcd and primes *)
 
 theorem prime_to_gcd_SO: \forall n,m:nat. prime n \to n \ndivides m \to
@@ -515,3 +560,49 @@ apply (decidable_divides n p).
 apply (trans_lt ? (S O)).simplify.apply le_n.
 simplify 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.
+apply antisymmetric_le.
+apply not_lt_to_le.
+unfold Not.intro.
+cut (divides (smallest_factor (gcd m (n*p))) n \lor 
+     divides (smallest_factor (gcd m (n*p))) p).
+elim Hcut.
+apply (not_le_Sn_n (S O)).
+change with ((S O) < (S O)).
+rewrite < H2 in \vdash (? ? %).
+apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
+apply lt_SO_smallest_factor.assumption.
+apply divides_to_le.
+rewrite > H2.simplify.apply le_n.
+apply divides_d_gcd.assumption.
+apply (transitive_divides ? (gcd m (n*p))).
+apply divides_smallest_factor_n.
+apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply divides_gcd_n.
+apply (not_le_Sn_n (S O)).
+change with ((S O) < (S O)).
+rewrite < H3 in \vdash (? ? %).
+apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
+apply lt_SO_smallest_factor.assumption.
+apply divides_to_le.
+rewrite > H3.simplify.apply le_n.
+apply divides_d_gcd.assumption.
+apply (transitive_divides ? (gcd m (n*p))).
+apply divides_smallest_factor_n.
+apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply divides_gcd_n.
+apply divides_times_to_divides.
+apply prime_smallest_factor_n.
+assumption.
+apply (transitive_divides ? (gcd m (n*p))).
+apply divides_smallest_factor_n.
+apply (trans_lt ? (S O)). simplify. apply le_n. assumption.
+apply divides_gcd_m.
+change with (O < gcd m (n*p)).
+apply lt_O_gcd.
+rewrite > (times_n_O O).
+apply lt_times.assumption.assumption.
+qed.