X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Ffactorization.ma;h=6226f3b6de81997ac3eefa61006d0c54205deafc;hb=f8f3f0bf31de02f543b9bb5e944ea01fd706d3a0;hp=6241244f39a2c24e5776a3caea7492ad616ef190;hpb=beb4e1e9549d5b43e24907dc86c7ef899e487a3c;p=helm.git

diff --git a/helm/software/matita/library/nat/factorization.ma b/helm/software/matita/library/nat/factorization.ma
index 6241244f3..6226f3b6d 100644
--- a/helm/software/matita/library/nat/factorization.ma
+++ b/helm/software/matita/library/nat/factorization.ma
@@ -12,152 +12,41 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/factorization".
-
 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.
 (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;
@@ -167,23 +56,34 @@ intros; apply divides_b_true_to_divides;
       | rewrite > H1;
         apply le_smallest_factor_n; ]
     | rewrite > H1;
-      (*CSC: simplify here does something nasty! *)
       change with (divides_b (smallest_factor n) n = true);
       apply divides_to_divides_b_true;
       [ 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.autobatch.
          (*       
        apply divides_smallest_factor_n;
         apply (trans_lt ? (S O));
         [ unfold lt; apply le_n;
         | assumption; ] *) ] ]
-  | letin x \def prime. auto.
+  | autobatch. 
     (* 
     apply prime_to_nth_prime;
     apply prime_smallest_factor_n;
-    assumption; *) ] ]
+    assumption; *) ] 
+qed.
+
+lemma divides_to_prime_divides : \forall n,m.1 < m \to m < n \to m \divides n \to
+ \exists p.p \leq m \land prime p \land p \divides n.
+intros;apply (ex_intro ? ? (nth_prime (max_prime_factor m)));split
+  [split
+     [apply divides_to_le
+        [apply lt_to_le;assumption
+        |apply divides_max_prime_factor_n;assumption]
+     |apply prime_nth_prime;]
+  |apply (transitive_divides ? ? ? ? H2);apply divides_max_prime_factor_n;
+   assumption]
 qed.
 
 theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to 
@@ -197,8 +97,8 @@ 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).
-auto.
-auto.
+autobatch.
+autobatch.
 (*
   [ apply (transitive_divides ? n);
     [ apply divides_max_prime_factor_n.
@@ -217,6 +117,30 @@ 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 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
 p = max_prime_factor n \to 
 (pair nat nat q r) = p_ord n (nth_prime p) \to
@@ -232,7 +156,7 @@ 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.autobatch.
     (*
     apply Hcut1.apply divides_to_mod_O;
     [ apply lt_O_nth_prime_n.
@@ -241,7 +165,7 @@ 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.autobatch width = 4 depth = 2]
    (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
   ].
 (*
@@ -253,20 +177,13 @@ 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
@@ -287,6 +204,33 @@ 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
@@ -439,7 +383,7 @@ theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
 intro.
 apply (nat_case n).reflexivity.
 intro.apply (nat_case m).reflexivity.
-intro.(*CSC: simplify here does something really nasty *)
+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
@@ -460,7 +404,6 @@ simplify.
 cut ((S(S m1)) = (nth_prime p) \sup q *r).
 cut (O<r).
 rewrite > defactorize_aux_factorize_aux.
-(*CSC: simplify here does something really nasty *)
 change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
 cut ((S (pred q)) = q).
 rewrite > Hcut2.
@@ -478,7 +421,6 @@ apply (divides_max_prime_factor_n (S (S m1))).
 unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
 cut ((S(S m1)) = r).
 rewrite > Hcut3 in \vdash (? (? ? %)).
-(*CSC: simplify here does something really nasty *)
 change with (nth_prime p \divides r \to False).
 intro.
 apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
@@ -522,7 +464,6 @@ 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.
@@ -556,28 +497,86 @@ 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.
@@ -649,21 +648,19 @@ theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
 defactorize_aux f i = defactorize_aux g i \to f = g.
 intro.
 elim f.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
 apply eq_f.
 apply inj_S. apply (inj_exp_r (nth_prime i)).
 apply lt_SO_nth_prime_n.
 assumption.
 apply False_ind.
-apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
-generalize in match H1.
-elim g.
+apply (not_eq_nf_last_nf_cons n2 n n1 i H1).
+elim g in H1 ⊢ %.
 apply False_ind.
 apply (not_eq_nf_last_nf_cons n1 n2 n i).
 apply sym_eq. assumption.
-simplify in H3.
-generalize in match H3.
+simplify in H2.
+generalize in match H2.
 apply (nat_elim2 (\lambda n,n2.
 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
@@ -695,7 +692,7 @@ change with
 [ (nf_last m) \Rightarrow m
 | (nf_cons m g) \Rightarrow m ] = m).
 rewrite > Hcut.simplify.reflexivity.
-apply H4.simplify in H5.
+apply H3.simplify in H4.
 apply (inj_times_r1 (nth_prime i)).
 apply lt_O_nth_prime_n.
 rewrite < assoc_times.rewrite < assoc_times.assumption.
@@ -713,22 +710,21 @@ injective nat_fact_all nat defactorize.
 unfold injective.
 change with (\forall f,g.defactorize f = defactorize g \to f=g).
 intro.elim f.
-generalize in match H.elim g.
+elim g in H ⊢ %.
 (* zero - zero *)
 reflexivity.
 (* zero - one *)
 simplify in H1.
 apply False_ind.
-apply (not_eq_O_S O H1).
+apply (not_eq_O_S O H).
 (* zero - proper *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n O).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
 change with (O < defactorize_aux n O).
 apply lt_O_defactorize_aux.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
 (* one - zero *)
 simplify in H1.
 apply False_ind.
@@ -739,34 +735,33 @@ reflexivity.
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n (S O)).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
 change with ((S O) < defactorize_aux n O).
 apply lt_SO_defactorize_aux.
-generalize in match H.elim g.
+elim g in H ⊢ %.
 (* proper - zero *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n O).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
 change with (O < defactorize_aux n O).
 apply lt_O_defactorize_aux.
 (* proper - one *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n (S O)).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
 change with ((S O) < defactorize_aux n O).
 apply lt_SO_defactorize_aux.
 (* proper - proper *)
 apply eq_f.
 apply (injective_defactorize_aux O).
-exact H1.
+exact H.
 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.
 apply defactorize_factorize.
 qed.
-