(no commit message)

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

diff --git a/matita/library/nat/primes.ma b/matita/library/nat/primes.ma
index 50b7d1221bfdca8211184a664add8c73aa392bd5..d2e89b8f1b56de4e2de72fc7673f008d909e01cf 100644 (file)
--- a/matita/library/nat/primes.ma
+++ b/matita/library/nat/primes.ma
@@ -193,23 +193,19 @@ qed.
 (* boolean divides *)
 definition divides_b : nat \to nat \to bool \def
 \lambda n,m :nat. (eqb (m \mod n) O).
-  
+
 theorem divides_b_to_Prop :
 \forall n,m:nat. O < n \to
 match divides_b n m with
 [ true \Rightarrow n \divides m
 | false \Rightarrow n \ndivides m].
-intros.
-change with 
-match eqb (m \mod n) O with
-[ true \Rightarrow n \divides m
-| false \Rightarrow n \ndivides m].
+intros.unfold divides_b.
 apply eqb_elim.
 intro.simplify.apply mod_O_to_divides.assumption.assumption.
 intro.simplify.unfold Not.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
 qed.
 
-theorem divides_b_true_to_divides :
+theorem divides_b_true_to_divides1:
 \forall n,m:nat. O < n \to
 (divides_b n m = true ) \to n \divides m.
 intros.
@@ -221,7 +217,21 @@ rewrite < H1.apply divides_b_to_Prop.
 assumption.
 qed.
 
-theorem divides_b_false_to_not_divides :
+theorem divides_b_true_to_divides:
+\forall n,m:nat. divides_b n m = true \to n \divides m.
+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
+    ]
+  |intros.
+   apply divides_b_true_to_divides1
+    [apply lt_O_S|assumption]
+  ]
+qed.
+
+theorem divides_b_false_to_not_divides1:
 \forall n,m:nat. O < n \to
 (divides_b n m = false ) \to n \ndivides m.
 intros.
@@ -233,9 +243,25 @@ rewrite < H1.apply divides_b_to_Prop.
 assumption.
 qed.
 
+theorem divides_b_false_to_not_divides:
+\forall n,m:nat. divides_b n m = false \to n \ndivides m.
+intros 2.apply (nat_case n)
+  [apply (nat_case m)
+    [simplify.unfold Not.intros.
+     apply not_eq_true_false.assumption
+    |unfold Not.intros.elim H1.
+     apply (not_eq_O_S m1).apply sym_eq.
+     assumption
+    ]
+  |intros.
+   apply divides_b_false_to_not_divides1
+    [apply lt_O_S|assumption]
+  ]
+qed.
+
 theorem decidable_divides: \forall n,m:nat.O < n \to
 decidable (n \divides m).
-intros.change with ((n \divides m) \lor n \ndivides m).
+intros.unfold decidable.
 cut 
 (match divides_b n m with
 [ true \Rightarrow n \divides m
@@ -267,6 +293,21 @@ absurd (n \divides m).assumption.assumption.
 reflexivity.
 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))
+  [assumption
+  |apply False_ind.
+   elim H1.
+   rewrite < H2 in H1.
+   simplify in H1.
+   apply (lt_to_not_eq O n H).
+   apply sym_eq.
+   apply eqb_true_to_eq.
+   assumption
+  ]
+qed.
+
 (* divides and pi *)
 theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,m,i:nat. 
 m \le i \to i \le n+m \to f i \divides pi n f m.
@@ -327,10 +368,8 @@ theorem not_divides_S_fact: \forall n,i:nat.
 (S O) < i \to i \le n \to i \ndivides S n!.
 intros.
 apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
-change with ((eqb ((S n!) \mod i) O) = false).
-rewrite > mod_S_fact.simplify.reflexivity.
-assumption.assumption.
+unfold divides_b.
+rewrite > mod_S_fact[simplify.reflexivity|assumption|assumption].
 qed.
 
 (* prime *)
@@ -346,6 +385,10 @@ theorem not_prime_SO: \lnot (prime (S O)).
 unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
 qed.
 
+theorem prime_to_lt_O: \forall p. prime p \to O < p.
+intros.elim H.apply lt_to_le.assumption.
+qed.
+
 (* smallest factor *)
 definition smallest_factor : nat \to nat \def
 \lambda n:nat. 
@@ -405,7 +448,6 @@ intro.apply (nat_case m).intro. simplify.
 apply (witness ? ? (S O)). simplify.reflexivity.
 intros.
 apply divides_b_true_to_divides.
-apply (lt_O_smallest_factor ? H).
 change with 
 (eqb ((S(S m1)) \mod (min_aux m1 (S(S m1)) 
   (\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true).
@@ -420,8 +462,8 @@ qed.
   
 theorem le_smallest_factor_n : 
 \forall n:nat. smallest_factor n \le n.
-intro.apply (nat_case n).simplify.reflexivity.
-intro.apply (nat_case m).simplify.reflexivity.
+intro.apply (nat_case n).simplify.apply le_n.
+intro.apply (nat_case m).simplify.apply le_n.
 intro.apply divides_to_le.
 unfold lt.apply le_S_S.apply le_O_n.
 apply divides_smallest_factor_n.
@@ -435,8 +477,6 @@ apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
 intros.
 apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
-change with ((eqb ((S(S m1)) \mod i) O) = false).
 apply (lt_min_aux_to_false 
 (\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S(S m1)) m1 i).
 cut ((S(S O)) = (S(S m1)-m1)).
@@ -448,7 +488,7 @@ qed.
 
 theorem prime_smallest_factor_n : 
 \forall n:nat. (S O) < n \to prime (smallest_factor n).
-intro. change with ((S(S O)) \le n \to (S O) < (smallest_factor n) \land 
+intro.change with ((S(S O)) \le n \to (S O) < (smallest_factor n) \land 
 (\forall m:nat. m \divides smallest_factor n \to (S O) < m \to  m = (smallest_factor n))).
 intro.split.
 apply lt_SO_smallest_factor.assumption.
@@ -525,11 +565,11 @@ match eqb (smallest_factor (S(S m1))) (S(S m1)) with
 [ true \Rightarrow prime (S(S m1))
 | false \Rightarrow \lnot (prime (S(S m1)))].
 apply (eqb_elim (smallest_factor (S(S m1))) (S(S m1))).
-intro.change with (prime (S(S m1))).
+intro.simplify.
 rewrite < H.
 apply prime_smallest_factor_n.
 unfold lt.apply le_S_S.apply le_S_S.apply le_O_n.
-intro.change with (\lnot (prime (S(S m1)))).
+intro.simplify.
 change with (prime (S(S m1)) \to False).
 intro.apply H.
 apply prime_to_smallest_factor.
@@ -557,7 +597,7 @@ apply primeb_to_Prop.
 qed.
 
 theorem decidable_prime : \forall n:nat.decidable (prime n).
-intro.change with ((prime n) \lor \lnot (prime n)).
+intro.unfold decidable.
 cut 
 (match primeb n with
 [ true \Rightarrow prime n