X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Ffactorization.ma;h=8c50d1d7db1e5612e160776b39a59e55ab21fece;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=37b5ea1ddb8e5d82d19ff44acc362d07f0fbb8ce;hpb=55b82bd235d82ff7f0a40d980effe1efde1f5073;p=helm.git

diff --git a/helm/software/matita/library/nat/factorization.ma b/helm/software/matita/library/nat/factorization.ma
index 37b5ea1dd..8c50d1d7d 100644
--- a/helm/software/matita/library/nat/factorization.ma
+++ b/helm/software/matita/library/nat/factorization.ma
@@ -15,22 +15,40 @@
 set "baseuri" "cic:/matita/nat/factorization".
 
 include "nat/ord.ma".
-include "nat/gcd.ma".
-include "nat/nth_prime.ma".
 
 (* the following factorization algorithm looks for the largest prime
    factor. *)
 definition max_prime_factor \def \lambda n:nat.
 (max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)).
 
+theorem lt_SO_max_prime: \forall m. S O <  m \to 
+S O < max m (λi:nat.primeb i∧divides_b i m).
+intros.
+apply (lt_to_le_to_lt ? (smallest_factor m))
+  [apply lt_SO_smallest_factor.assumption
+  |apply f_m_to_le_max
+    [apply le_smallest_factor_n
+    |apply true_to_true_to_andb_true
+      [apply prime_to_primeb_true.
+       apply prime_smallest_factor_n.
+       assumption
+      |apply divides_to_divides_b_true
+        [apply lt_O_smallest_factor.apply lt_to_le.assumption
+        |apply divides_smallest_factor_n.
+         apply lt_to_le.assumption
+        ]
+      ]
+    ]
+  ]
+qed.
 (* max_prime_factor is indeed a factor *)
 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;
@@ -45,20 +63,22 @@ 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.autobatch new.
+         (*       
+       apply divides_smallest_factor_n;
         apply (trans_lt ? (S O));
         [ unfold lt; apply le_n;
-        | assumption; ] ] ]
-  | apply prime_to_nth_prime;
+        | assumption; ] *) ] ]
+  | autobatch. 
+    (* 
+    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 
 max_prime_factor n \le max_prime_factor m.
-intros.change with
-((max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)) \le
-(max m (\lambda p:nat.eqb (m \mod (nth_prime p)) O))).
+intros.unfold max_prime_factor.
 apply f_m_to_le_max.
 apply (trans_le ? n).
 apply le_max_n.apply divides_to_le.assumption.assumption.
@@ -67,15 +87,48 @@ 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.
+autobatch.
+autobatch.
+(*
+  [ 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 max_prime_factor_to_not_p_ord_O : \forall n,p,r.
+  (S O) < n \to
+  p = max_prime_factor n \to
+  p_ord n (nth_prime p) \neq pair nat nat O r.
+intros.unfold Not.intro.
+apply (p_ord_O_to_not_divides ? ? ? ? H2)
+  [apply (trans_lt ? (S O))[apply lt_O_S|assumption]
+  |rewrite > H1.
+   apply divides_max_prime_factor_n.
+   assumption
+  ]
 qed.
 
 theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
@@ -90,16 +143,30 @@ elim Hcut.assumption.
 absurd (nth_prime (max_prime_factor n) \divides r).
 rewrite < H4.
 apply divides_max_prime_factor_n.
-assumption.
-change with (nth_prime (max_prime_factor n) \divides r \to False).
+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.autobatch 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.autobatch 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.
@@ -127,6 +194,33 @@ assumption.apply sym_eq.assumption.assumption.assumption.
 apply (le_to_or_lt_eq ? p H1).
 qed.
 
+lemma lt_max_prime_factor_to_not_divides: \forall n,p:nat.
+O < n \to n=S O \lor max_prime_factor n < p \to 
+(nth_prime p \ndivides n).
+intros.unfold Not.intro.
+elim H1
+  [rewrite > H3 in H2.
+   apply (le_to_not_lt (nth_prime p) (S O))
+    [apply divides_to_le[apply le_n|assumption]
+    |apply lt_SO_nth_prime_n
+    ]
+  |apply (not_le_Sn_n p).
+   change with (p < p).
+   apply (le_to_lt_to_lt ? ? ? ? H3).
+   unfold max_prime_factor.
+   apply  f_m_to_le_max
+    [apply (trans_le ? (nth_prime p))
+      [apply lt_to_le.
+       apply lt_n_nth_prime_n
+      |apply divides_to_le;assumption
+      ]
+    |apply eq_to_eqb_true.
+     apply divides_to_mod_O
+      [apply lt_O_nth_prime_n|assumption]
+    ]
+  ]
+qed.
+
 (* datatypes and functions *)
 
 inductive nat_fact : Set \def
@@ -171,24 +265,23 @@ 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.
@@ -258,13 +351,13 @@ 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.
@@ -280,7 +373,8 @@ theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
 intro.
 apply (nat_case n).reflexivity.
 intro.apply (nat_case m).reflexivity.
-intro.change with  
+intro.
+change with  
 (let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
 defactorize (match p_ord (S(S m1)) (nth_prime p) with
 [ (pair q r) \Rightarrow 
@@ -296,8 +390,7 @@ apply (Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
 apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H.
-change with 
-(defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1))).
+simplify.
 cut ((S(S m1)) = (nth_prime p) \sup q *r).
 cut (O<r).
 rewrite > defactorize_aux_factorize_aux.
@@ -361,7 +454,6 @@ 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.
@@ -395,28 +487,86 @@ apply (witness ? ? (n2* (nth_prime i) \sup n)).
 reflexivity.
 qed.
 
-theorem divides_exp_to_divides: 
-\forall p,n,m:nat. prime p \to 
-p \divides n \sup m \to p \divides n.
-intros 3.elim m.simplify in H1.
-apply (transitive_divides p (S O)).assumption.
-apply divides_SO_n.
-cut (p \divides n \lor p \divides n \sup n1).
-elim Hcut.assumption.
-apply H.assumption.assumption.
-apply divides_times_to_divides.assumption.
-exact H2.
-qed.
-
-theorem divides_exp_to_eq: 
-\forall p,q,m:nat. prime p \to prime q \to
-p \divides q \sup m \to p = q.
+lemma eq_p_max: \forall n,p,r:nat. O < n \to
+O < r \to
+r = (S O) \lor (max r (\lambda p:nat.eqb (r \mod (nth_prime p)) O)) < p \to
+p = max_prime_factor (r*(nth_prime p)\sup n).
 intros.
-unfold prime in H1.
-elim H1.apply H4.
-apply (divides_exp_to_divides p q m).
-assumption.assumption.
-unfold prime in H.elim H.assumption.
+apply sym_eq.
+unfold max_prime_factor.
+apply max_spec_to_max.
+split
+  [split
+    [rewrite > times_n_SO in \vdash (? % ?).
+     rewrite > sym_times.
+     apply le_times
+      [assumption
+      |apply lt_to_le.
+       apply (lt_to_le_to_lt ? (nth_prime p))
+        [apply lt_n_nth_prime_n
+        |rewrite > exp_n_SO in \vdash (? % ?).
+         apply le_exp
+          [apply lt_O_nth_prime_n
+          |assumption
+          ]
+        ]
+      ]
+    |apply eq_to_eqb_true.
+     apply divides_to_mod_O
+      [apply lt_O_nth_prime_n
+      |apply (lt_O_n_elim ? H).
+       intro.
+       apply (witness ? ? (r*(nth_prime p \sup m))).
+       rewrite < assoc_times.
+       rewrite < sym_times in \vdash (? ? ? (? % ?)).
+       rewrite > exp_n_SO in \vdash (? ? ? (? (? ? %) ?)).
+       rewrite > assoc_times.
+       rewrite < exp_plus_times.
+       reflexivity
+      ]
+    ]
+  |intros.  
+   apply not_eq_to_eqb_false.
+   unfold Not.intro.
+   lapply (mod_O_to_divides ? ? ? H5)
+    [apply lt_O_nth_prime_n
+    |cut (Not (divides (nth_prime i) ((nth_prime p)\sup n)))
+      [elim H2
+        [rewrite > H6 in Hletin.
+         simplify in Hletin.
+         rewrite < plus_n_O in Hletin.
+         apply Hcut.assumption
+        |elim (divides_times_to_divides ? ? ? ? Hletin)
+          [apply (lt_to_not_le ? ? H3).
+           apply lt_to_le. 
+           apply (le_to_lt_to_lt ? ? ? ? H6).
+           apply f_m_to_le_max
+            [apply (trans_le ? (nth_prime i))
+              [apply lt_to_le.
+               apply lt_n_nth_prime_n
+              |apply divides_to_le[assumption|assumption]
+              ]
+            |apply eq_to_eqb_true.
+             apply divides_to_mod_O
+              [apply lt_O_nth_prime_n|assumption]
+            ]
+          |apply prime_nth_prime
+          |apply Hcut.assumption
+          ]
+        ]
+      |unfold Not.intro.
+       apply (lt_to_not_eq ? ? H3).
+       apply sym_eq.
+       elim (prime_nth_prime p).
+       apply injective_nth_prime.
+       apply H8
+        [apply (divides_exp_to_divides ? ? ? ? H6).
+         apply prime_nth_prime.
+        |apply lt_SO_nth_prime_n
+        ]
+      ]
+    ]
+  ]
 qed.
 
 theorem  not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
@@ -427,13 +577,11 @@ change with
 intro.absurd ((nth_prime i) = (nth_prime j)).
 apply (divides_exp_to_eq ? ? (S n)).
 apply prime_nth_prime.apply prime_nth_prime.
-assumption.
-change with ((nth_prime i) = (nth_prime j) \to False).
+assumption.unfold Not.
 intro.cut (i = j).
 apply (not_le_Sn_n i).rewrite > Hcut in \vdash (? ? %).assumption.
 apply (injective_nth_prime ? ? H2).
-change with 
-(nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False).
+unfold Not.simplify.
 intro.
 cut (nth_prime i \divides (nth_prime j) \sup n
 \lor nth_prime i \divides defactorize_aux n1 (S j)).
@@ -441,8 +589,7 @@ elim Hcut.
 absurd ((nth_prime i) = (nth_prime j)).
 apply (divides_exp_to_eq ? ? n).
 apply prime_nth_prime.apply prime_nth_prime.
-assumption.
-change with ((nth_prime i) = (nth_prime j) \to False).
+assumption.unfold Not.
 intro.
 cut (i = j).
 apply (not_le_Sn_n i).rewrite > Hcut1 in \vdash (? ? %).assumption.
@@ -466,8 +613,6 @@ rewrite < Hcut in \vdash (? ? %).
 simplify.apply le_S_S.
 apply le_plus_n.
 apply injective_nth_prime.
-(* uffa, perche' semplifica ? *)
-change with (nth_prime (S(max_p g)+i)= nth_prime i).
 apply (divides_exp_to_eq ? ? (S n)).
 apply prime_nth_prime.apply prime_nth_prime.
 rewrite > H.
@@ -547,16 +692,15 @@ qed.
 
 theorem injective_defactorize_aux: \forall i:nat.
 injective nat_fact nat (\lambda f.defactorize_aux f i).
-change with (\forall i:nat.\forall f,g:nat_fact.
-defactorize_aux f i = defactorize_aux g i \to f = g).
+simplify.
 intros.
-apply (eq_defactorize_aux_to_eq f g i H).
+apply (eq_defactorize_aux_to_eq x y i H).
 qed.
 
 theorem injective_defactorize: 
 injective nat_fact_all nat defactorize.
-change with (\forall f,g:nat_fact_all.
-defactorize f = defactorize g \to f = g).
+unfold injective.
+change with (\forall f,g.defactorize f = defactorize g \to f=g).
 intro.elim f.
 generalize in match H.elim g.
 (* zero - zero *)
@@ -609,11 +753,8 @@ exact H1.
 qed.
 
 theorem factorize_defactorize: 
-\forall f,g: nat_fact_all. factorize (defactorize f) = f.
+\forall f: nat_fact_all. factorize (defactorize f) = f.
 intros.
 apply injective_defactorize.
-(* uffa: perche' semplifica ??? *)
-change with (defactorize(factorize (defactorize f)) = (defactorize f)).
 apply defactorize_factorize.
 qed.
-