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 6241244f39a2c24e5776a3caea7492ad616ef190..f5b147005f84ff8b9490bff1f66ec44a04524192 100644 (file)
--- a/matita/library/nat/factorization.ma
+++ b/matita/library/nat/factorization.ma
@@ -18,133 +18,6 @@ include "nat/ord.ma".
 include "nat/gcd.ma".
 include "nat/nth_prime.ma".
 
-
-theorem prova :
-  \forall n,m:nat.
-  \forall P:nat \to Prop.
-  \forall H:P (S (S O)).
-  \forall H:P (S (S (S O))).
-  \forall H1: \forall x.P x \to O = x.
-   O = S (S (S (S (S O)))).
-   intros.
-   auto paramodulation.
- qed.
- 
-theorem example2:
-\forall x: nat. (x+S O)*(x-S O) = x*x - S O.
-intro;
-apply (nat_case x);
- [ auto paramodulation.|intro.auto paramodulation.]
-qed.
-
-theorem prova3:
-  \forall A:Set.
-  \forall m:A \to A \to A.
-  \forall divides: A \to A \to Prop.
-  \forall o,a,b,two:A.
-  \forall H1:\forall x.m o x = x.
-  \forall H1:\forall x.m x o = x.
-  \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
-  \forall H1:\forall x.m x o = x.
-  \forall H2:\forall x,y.m x y = m y x.
-  \forall H3:\forall x,y,z. m x y = m x z \to y = z. 
-  (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
-  \forall H4:\forall x,y.(divides x y \to (\exists z.m x z = y)). 
-  \forall H4:\forall x,y,z.m x z = y \to divides x y.
-  \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
-  \forall H5:m a a = m two (m b b).
-  \forall H6:\forall x.divides x a \to divides x b \to x = o.
-  two = o.
-  intros.
-  cut (divides two a);
-    [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
-    |elim (H6 ? ? Hcut).
-     cut (divides two b);
-       [ apply (H10 ? Hcut Hcut1).
-       | elim (H8 b b);[assumption.|assumption|
-         apply (H7 ? ? (m a1 a1));
-         apply (H5 two ? ?);rewrite < H9.
-         rewrite < H11.rewrite < H2.
-         apply eq_f.rewrite > H2.rewrite > H4.reflexivity.
-         ]
-         ]
-         ]
-         qed.
-         
-theorem prova31:
-  \forall A:Set.
-  \forall m,f:A \to A \to A.
-  \forall divides: A \to A \to Prop.
-  \forall o,a,b,two:A.
-  \forall H1:\forall x.m o x = x.
-  \forall H1:\forall x.m x o = x.
-  \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
-  \forall H1:\forall x.m x o = x.
-  \forall H2:\forall x,y.m x y = m y x.
-  \forall H3:\forall x,y,z. m x y = m x z \to y = z. 
-  (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
-  \forall H4:\forall x,y.(divides x y \to m x (f x y) = y). 
-  \forall H4:\forall x,y,z.m x z = y \to divides x y.
-  \forall H4:\forall x,y.divides two (m x y) \to divides two x ∨ divides two y.
-  \forall H5:m a a = m two (m b b).
-  \forall H6:\forall x.divides x a \to divides x b \to x = o.
-  two = o.
-  intros.
-  cut (divides two a);
-    [2:elim (H8 a a);[assumption.|assumption|rewrite > H9.auto.]
-    |(*elim (H6 ? ? Hcut). *)
-     cut (divides two b);
-       [ apply (H10 ? Hcut Hcut1).
-       | elim (H8 b b);[assumption.|assumption|
-       
-         apply (H7 ? ? (m (f two a) (f two a)));
-         apply (H5 two ? ?);
-         rewrite < H9.
-         rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
-         rewrite < H2.apply eq_f.
-         rewrite < H4 in \vdash (? ? ? %).
-         rewrite > H2.reflexivity.
-         ]
-         ]
-         ]
-         qed.  
-           
-theorem prova32:
-  \forall A:Set.
-  \forall m,f:A \to A \to A.
-  \forall divides: A \to A \to Prop.
-  \forall o,a,b,two:A.
-  \forall H1:\forall x.m o x = x.
-  \forall H1:\forall x.m x o = x.
-  \forall H1:\forall x,y,z.m x (m y z) = m (m x y) z.
-  \forall H1:\forall x.m x o = x.
-  \forall H2:\forall x,y.m x y = m y x.
-  \forall H3:\forall x,y,z. m x y = m x z \to y = z. 
-  (* \forall H4:\forall x,y.(\exists z.m x z = y) \to divides x y. *)
-  \forall H4:\forall x,y.(divides x y \to m x (f x y) = y). 
-  \forall H4:\forall x,y,z.m x z = y \to divides x y.
-  \forall H4:\forall x.divides two (m x x) \to divides two x.
-  \forall H5:m a a = m two (m b b).
-  \forall H6:\forall x.divides x a \to divides x b \to x = o.
-  two = o.
-  intros.
-  cut (divides two a);[|apply H8;rewrite > H9.auto].
-  apply H10;
-  [ assumption.
-  | apply (H8 b);       
-         apply (H7 ? ? (m (f two a) (f two a)));
-         apply (H5 two ? ?);
-         auto paramodulation.
-         (*
-         rewrite < H9.
-         rewrite < (H6 two a Hcut) in \vdash (? ? ? %).
-         rewrite < H2.apply eq_f.
-         rewrite < H4 in \vdash (? ? ? %).
-         rewrite > H2.reflexivity.
-         *)
-  ]
-qed.
-             
 (* the following factorization algorithm looks for the largest prime
    factor. *)
 definition max_prime_factor \def \lambda n:nat.
@@ -154,10 +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;
@@ -173,17 +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; ]
-      | letin x \def le.auto.
+      | letin x \def le.auto new.
          (*       
        apply divides_smallest_factor_n;
         apply (trans_lt ? (S O));
         [ unfold lt; apply le_n;
         | assumption; ] *) ] ]
-  | letin x \def prime. auto.
+  | 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 
@@ -217,6 +90,17 @@ auto.
 *)  
 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
 p = max_prime_factor n \to 
 (pair nat nat q r) = p_ord n (nth_prime p) \to
@@ -232,7 +116,7 @@ apply divides_max_prime_factor_n.
 assumption.unfold Not.
 intro.
 cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
-  [unfold Not in Hcut1.auto.
+  [unfold Not in Hcut1.auto new.
     (*
     apply Hcut1.apply divides_to_mod_O;
     [ apply lt_O_nth_prime_n.
@@ -241,7 +125,7 @@ cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
     *)
   |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]
+   [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 *)
   ].
 (*
@@ -253,20 +137,13 @@ cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
     ]
   ].
 *)  
-cut (n=r*(nth_prime p)\sup(q));
-  [letin www \def le.letin www1 \def divides.
-   auto.
-(*
 apply (le_to_or_lt_eq (max_prime_factor r)  (max_prime_factor n)).
 apply divides_to_max_prime_factor.
 assumption.assumption.
 apply (witness r n ((nth_prime p) \sup q)).
-*)
- |
 rewrite < sym_times.
 apply (p_ord_aux_to_exp n n ? q r).
 apply lt_O_nth_prime_n.assumption.
-]
 qed.
 
 theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
@@ -522,7 +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.
@@ -769,4 +645,3 @@ intros.
 apply injective_defactorize.
 apply defactorize_factorize.
 qed.
-