auto rewritten with only one tail recursive function.

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

diff --git a/matita/library/nat/factorization.ma b/matita/library/nat/factorization.ma
index 715a9795f1dc8d8115049931277a7022a166ed60..f5b147005f84ff8b9490bff1f66ec44a04524192 100644 (file)
--- a/matita/library/nat/factorization.ma
+++ b/matita/library/nat/factorization.ma
@@ -27,10 +27,10 @@ definition max_prime_factor \def \lambda n:nat.
 theorem divides_max_prime_factor_n:
   \forall n:nat. (S O) < n
   \to nth_prime (max_prime_factor n) \divides n.
-intros; apply divides_b_true_to_divides;
-[ apply lt_O_nth_prime_n;
-| apply (f_max_true  (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n);
-  cut (\exists i. nth_prime i = smallest_factor n);
+intros.
+apply divides_b_true_to_divides.
+apply (f_max_true  (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n);
+cut (\exists i. nth_prime i = smallest_factor n);
   [ elim Hcut.
     apply (ex_intro nat ? a);
     split;
@@ -46,13 +46,17 @@ intros; apply divides_b_true_to_divides;
       [ apply (trans_lt ? (S O));
         [ unfold lt; apply le_n;
         | apply lt_SO_smallest_factor; assumption; ]
-      | apply divides_smallest_factor_n;
+      | letin x \def le.auto new.
+         (*       
+       apply divides_smallest_factor_n;
         apply (trans_lt ? (S O));
         [ unfold lt; apply le_n;
-        | assumption; ] ] ]
-  | apply prime_to_nth_prime;
+        | assumption; ] *) ] ]
+  | auto. 
+    (* 
+    apply prime_to_nth_prime;
     apply prime_smallest_factor_n;
-    assumption; ] ]
+    assumption; *) ] 
 qed.
 
 theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to 
@@ -66,15 +70,35 @@ apply divides_to_divides_b_true.
 cut (prime (nth_prime (max_prime_factor n))).
 apply lt_O_nth_prime_n.apply prime_nth_prime.
 cut (nth_prime (max_prime_factor n) \divides n).
-apply (transitive_divides ? n).
-apply divides_max_prime_factor_n.
-assumption.assumption.
-apply divides_b_true_to_divides.
-apply lt_O_nth_prime_n.
-apply divides_to_divides_b_true.
-apply lt_O_nth_prime_n.
-apply divides_max_prime_factor_n.
-assumption.
+auto.
+auto.
+(*
+  [ apply (transitive_divides ? n);
+    [ apply divides_max_prime_factor_n.
+      assumption.
+    | assumption. 
+    ]
+  | apply divides_b_true_to_divides;
+    [ apply lt_O_nth_prime_n.
+    | apply divides_to_divides_b_true;
+      [ apply lt_O_nth_prime_n.
+      | apply divides_max_prime_factor_n.
+        assumption.
+      ]
+    ]
+  ]
+*)  
+qed.
+
+theorem divides_to_max_prime_factor1 : \forall n,m. O < n \to O < m \to n \divides m \to 
+max_prime_factor n \le max_prime_factor m.
+intros 3.
+elim (le_to_or_lt_eq ? ? H)
+  [apply divides_to_max_prime_factor
+    [assumption|assumption|assumption]
+  |rewrite < H1.
+   simplify.apply le_O_n.
+  ]
 qed.
 
 theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
@@ -91,13 +115,28 @@ rewrite < H4.
 apply divides_max_prime_factor_n.
 assumption.unfold Not.
 intro.
-cut (r \mod (nth_prime (max_prime_factor n)) \neq O).
-apply Hcut1.apply divides_to_mod_O.
-apply lt_O_nth_prime_n.assumption.
-apply (p_ord_aux_to_not_mod_O n n ? q r).
-apply lt_SO_nth_prime_n.assumption.
-apply le_n.
-rewrite < H1.assumption.
+cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
+  [unfold Not in Hcut1.auto new.
+    (*
+    apply Hcut1.apply divides_to_mod_O;
+    [ apply lt_O_nth_prime_n.
+    | assumption.
+    ]
+    *)
+  |letin z \def le.
+   cut(pair nat nat q r=p_ord_aux n n (nth_prime (max_prime_factor n)));
+   [2: rewrite < H1.assumption.|letin x \def le.auto width = 4 depth = 2]
+   (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
+  ].
+(*
+    apply (p_ord_aux_to_not_mod_O n n ? q r);
+    [ apply lt_SO_nth_prime_n.
+    | assumption.
+    | apply le_n.
+    | rewrite < H1.assumption.
+    ]
+  ].
+*)  
 apply (le_to_or_lt_eq (max_prime_factor r)  (max_prime_factor n)).
 apply divides_to_max_prime_factor.
 assumption.assumption.
@@ -169,24 +208,23 @@ match f with
 | nfa_one \Rightarrow (S O)
 | (nfa_proper g) \Rightarrow defactorize_aux g O]. 
 
-theorem lt_O_defactorize_aux: \forall f:nat_fact.\forall i:nat.
-O < defactorize_aux f i.
-intro.elim f.simplify.unfold lt. 
-rewrite > times_n_SO.
-apply le_times.
-change with (O < nth_prime i).
-apply lt_O_nth_prime_n.
-change with (O < exp (nth_prime i) n).
-apply lt_O_exp.
-apply lt_O_nth_prime_n.
-simplify.unfold lt.
-rewrite > times_n_SO.
-apply le_times.
-change with (O < exp (nth_prime i) n).
-apply lt_O_exp.
-apply lt_O_nth_prime_n.
-change with (O < defactorize_aux n1 (S i)).
-apply H.
+theorem lt_O_defactorize_aux:
+ \forall f:nat_fact.
+ \forall i:nat.
+ O < defactorize_aux f i.
+intro; elim f;
+[1,2:
+  simplify; unfold lt;
+  rewrite > times_n_SO;
+  apply le_times;
+  [ change with (O < nth_prime i);
+    apply lt_O_nth_prime_n;
+  |2,3:
+    change with (O < exp (nth_prime i) n);
+    apply lt_O_exp;
+    apply lt_O_nth_prime_n;
+  | change with (O < defactorize_aux n1 (S i));
+    apply H; ] ]
 qed.
 
 theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
@@ -256,13 +294,13 @@ apply (nat_case n).
 left.split.assumption.reflexivity.
 intro.right.rewrite > Hcut2.
 simplify.unfold lt.apply le_S_S.apply le_O_n.
-cut (r \lt (S O) \or r=(S O)).
+cut (r < (S O) ∨ r=(S O)).
 elim Hcut2.absurd (O=r).
 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
 unfold Not.intro.
 cut (O=n1).
 apply (not_le_Sn_O O).
-rewrite > Hcut3 in \vdash (? ? %).
+rewrite > Hcut3 in ⊢ (? ? %).
 assumption.rewrite > Hcut. 
 rewrite < H6.reflexivity.
 assumption.
@@ -361,7 +399,6 @@ apply (not_eq_O_S (S m1)).
 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
 apply le_to_or_lt_eq.apply le_O_n.
 (* prova del cut *)
-goal 20.
 apply (p_ord_aux_to_exp (S(S m1))).
 apply lt_O_nth_prime_n.
 assumption.
@@ -608,4 +645,3 @@ intros.
 apply injective_defactorize.
 apply defactorize_factorize.
 qed.
-