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

diff --git a/matita/library/nat/nth_prime.ma b/matita/library/nat/nth_prime.ma
index 5330f52adbb923ddd84fc91ac1a876b373751ccc..9d01e99f211e0b3a6e65cc7cb567c732aeff1c6e 100644 (file)
--- a/matita/library/nat/nth_prime.ma
+++ b/matita/library/nat/nth_prime.ma
@@ -41,8 +41,8 @@ qed. *)
 theorem smallest_factor_fact: \forall n:nat.
 n < smallest_factor (S n!).
 intros.
-apply not_le_to_lt.
-change with (smallest_factor (S n!) \le n \to False).intro.
+apply not_le_to_lt.unfold Not.
+intro.
 apply (not_divides_S_fact n (smallest_factor(S n!))).
 apply lt_SO_smallest_factor.
 unfold lt.apply le_S_S.apply le_SO_fact.
@@ -63,8 +63,7 @@ split.split.
 apply smallest_factor_fact.
 apply le_smallest_factor_n.
 (* Andrea: ancora hint non lo trova *)
-apply prime_smallest_factor_n.
-change with ((S(S O)) \le S (S n1)!).
+apply prime_smallest_factor_n.unfold lt.
 apply le_S.apply le_SSO_fact.
 unfold lt.apply le_S_S.assumption.
 qed.
@@ -78,7 +77,7 @@ match n with
     min_aux (upper_bound - (S previous_prime)) upper_bound primeb].
     
 (* it works, but nth_prime 4 takes already a few minutes -
-it must compute factorial of 7 ...
+it must compute factorial of 7 ...*)
 
 theorem example11 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
 normalize.reflexivity.
@@ -90,12 +89,16 @@ qed.
 
 theorem example13 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
 normalize.reflexivity.
+qed.
+
+(*
+theorem example14 : nth_prime (S(S(S(S(S O))))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
+normalize.reflexivity.
 *) 
 
 theorem prime_nth_prime : \forall n:nat.prime (nth_prime n).
 intro.
-apply (nat_case n).
-change with (prime (S(S O))).
+apply (nat_case n).simplify.
 apply (primeb_to_Prop (S(S O))).
 intro.
 change with
@@ -115,14 +118,13 @@ apply le_S_S.
 apply le_n_fact_n.
 apply le_smallest_factor_n.
 apply prime_to_primeb_true.
-apply prime_smallest_factor_n.
-change with ((S(S O)) \le S (nth_prime m)!).
+apply prime_smallest_factor_n.unfold lt.
 apply le_S_S.apply le_SO_fact.
 qed.
 
 (* properties of nth_prime *)
 theorem increasing_nth_prime: increasing nth_prime.
-change with (\forall n:nat. (nth_prime n) < (nth_prime (S n))).
+unfold increasing.
 intros.
 change with
 (let previous_prime \def (nth_prime n) in
@@ -157,6 +159,17 @@ intros.apply (trans_lt O (S O)).
 unfold lt. apply le_n.apply lt_SO_nth_prime_n.
 qed.
 
+theorem lt_n_nth_prime_n : \forall n:nat. n \lt nth_prime n.
+intro.
+elim n
+  [apply lt_O_nth_prime_n
+  |apply (lt_to_le_to_lt ? (S (nth_prime n1)))
+    [unfold.apply le_S_S.assumption
+    |apply lt_nth_prime_n_nth_prime_Sn
+    ]
+  ]
+qed.
+
 theorem ex_m_le_n_nth_prime_m: 
 \forall n: nat. nth_prime O \le n \to 
 \exists m. nth_prime m \le n \land n < nth_prime (S m).