X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fgcd.ma;h=dcdbc7b7ad302405a445bef9cc1bcf9bdb1f4235;hb=68dbcd02022874a025a9444aa1125b0458816fbb;hp=0568536dcb0f6d4cbaef1b94c3feb53d79b17824;hpb=06a19bec47845ecffe3bf9d9a95d3d4dadf76861;p=helm.git

diff --git a/helm/software/matita/library/nat/gcd.ma b/helm/software/matita/library/nat/gcd.ma
index 0568536dc..dcdbc7b7a 100644
--- a/helm/software/matita/library/nat/gcd.ma
+++ b/helm/software/matita/library/nat/gcd.ma
@@ -12,9 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/gcd".
-
 include "nat/primes.ma".
+include "nat/lt_arith.ma".
 
 let rec gcd_aux p m n: nat \def
 match divides_b n m with
@@ -56,7 +55,7 @@ qed.
 theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
 p \divides (m \mod n) \to p \divides n \to p \divides m. 
 intros.elim H1.elim H2.
-apply (witness p m ((n1*(m / n))+n2)).
+apply (witness p m ((n2*(m / n))+n1)).
 rewrite > distr_times_plus.
 rewrite < H3.
 rewrite < assoc_times.
@@ -100,7 +99,6 @@ qed.
 theorem divides_gcd_nm: \forall n,m.
 gcd n m \divides m \land gcd n m \divides n.
 intros.
-(*CSC: simplify simplifies too much because of a redex in gcd *)
 change with
 (match leb n m with
   [ true \Rightarrow 
@@ -163,42 +161,109 @@ intros.
 exact (proj1 ? ? (divides_gcd_nm n m)).
 qed.
 
+
+theorem divides_times_gcd_aux: \forall p,m,n,d,c. 
+O \lt c \to O < n \to n \le m \to n \le p \to
+d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd_aux p m n. 
+intro.
+elim p
+[ absurd (O < n)
+  [ assumption
+  | apply le_to_not_lt.
+    assumption
+  ]
+| simplify.
+  cut (n1 \divides m \lor n1 \ndivides m)
+  [ elim Hcut
+    [ rewrite > divides_to_divides_b_true
+      [ simplify.
+        assumption
+      | assumption
+      | assumption
+      ]
+    | rewrite > not_divides_to_divides_b_false
+      [ simplify.
+        apply H
+        [ assumption
+        | cut (O \lt m \mod n1 \lor O = m \mod n1)
+          [ elim Hcut1
+            [ assumption
+            | absurd (n1 \divides m)
+              [ apply mod_O_to_divides
+                [ assumption
+                | apply sym_eq.
+                  assumption
+                ]
+              | assumption
+              ]
+            ]
+          | apply le_to_or_lt_eq.
+            apply le_O_n
+          ]
+        | apply lt_to_le.
+          apply lt_mod_m_m.
+          assumption
+        | apply le_S_S_to_le.
+          apply (trans_le ? n1)
+          [ change with (m \mod n1 < n1).
+            apply lt_mod_m_m.
+            assumption
+          | assumption
+          ]
+        | assumption
+        | rewrite < times_mod
+          [ rewrite < (sym_times c m).
+            rewrite < (sym_times c n1).
+            apply divides_mod
+            [ rewrite > (S_pred c)
+              [ rewrite > (S_pred n1)
+                [ apply (lt_O_times_S_S)
+                | assumption
+                ]
+              | assumption
+              ]
+            | assumption
+            | assumption
+            ]
+          | assumption
+          | assumption
+          ]
+        ]
+      | assumption
+      | assumption
+      ]
+    ]
+  | apply (decidable_divides n1 m).
+    assumption
+  ]
+]
+qed.
+
+(*a particular case of the previous theorem (setting c=1)*)
 theorem divides_gcd_aux: \forall p,m,n,d. O < n \to n \le m \to n \le p \to
 d \divides m \to d \divides n \to d \divides gcd_aux p m n. 
-intro.elim p.
-absurd (O < n).assumption.apply le_to_not_lt.assumption.
-simplify.
-cut (n1 \divides m \lor n1 \ndivides m).
-elim Hcut.
-rewrite > divides_to_divides_b_true.
-simplify.assumption.
-assumption.assumption.
-rewrite > not_divides_to_divides_b_false.
-simplify.
-apply H.
-cut (O \lt m \mod n1 \lor O = m \mod n1).
-elim Hcut1.assumption.
-absurd (n1 \divides m).apply mod_O_to_divides.
-assumption.apply sym_eq.assumption.assumption.
-apply le_to_or_lt_eq.apply le_O_n.
-apply lt_to_le.
-apply lt_mod_m_m.assumption.
-apply le_S_S_to_le.
-apply (trans_le ? n1).
-change with (m \mod n1 < n1).
-apply lt_mod_m_m.assumption.assumption.
-assumption.
-apply divides_mod.assumption.assumption.assumption.
-assumption.assumption.
-apply (decidable_divides n1 m).assumption.
+intros.
+rewrite > (times_n_SO (gcd_aux p m n)).
+rewrite < (sym_times (S O)).
+apply (divides_times_gcd_aux)
+[ apply (lt_O_S O)
+| assumption
+| assumption
+| assumption
+| rewrite > (sym_times (S O)).
+  rewrite < (times_n_SO m).
+  assumption
+| rewrite > (sym_times (S O)).
+  rewrite < (times_n_SO n).
+  assumption
+]
 qed.
 
-theorem divides_d_gcd: \forall m,n,d. 
-d \divides m \to d \divides n \to d \divides gcd n m. 
+theorem divides_d_times_gcd: \forall m,n,d,c. 
+O \lt c \to d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd n m. 
 intros.
-(*CSC: here simplify simplifies too much because of a redex in gcd *)
 change with
-(d \divides
+(d \divides c *
 match leb n m with
   [ true \Rightarrow 
     match n with 
@@ -208,20 +273,63 @@ match leb n m with
     match m with 
     [ O \Rightarrow n
     | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]).
-apply (leb_elim n m).
-apply (nat_case1 n).simplify.intros.assumption.
-intros.
-change with (d \divides gcd_aux (S m1) m (S m1)).
-apply divides_gcd_aux.
-unfold lt.apply le_S_S.apply le_O_n.assumption.apply le_n.assumption.
-rewrite < H2.assumption.
-apply (nat_case1 m).simplify.intros.assumption.
+apply (leb_elim n m)
+[ apply (nat_case1 n)
+  [ simplify.
+    intros.
+    assumption
+  | intros.
+    change with (d \divides c*gcd_aux (S m1) m (S m1)).
+    apply divides_times_gcd_aux
+    [ assumption
+    | unfold lt.
+      apply le_S_S.
+      apply le_O_n
+    | assumption
+    | apply (le_n (S m1))
+    | assumption
+    | rewrite < H3.
+      assumption
+    ]
+  ]
+| apply (nat_case1 m)
+  [ simplify.
+    intros.
+    assumption
+  | intros.
+    change with (d \divides c * gcd_aux (S m1) n (S m1)).
+    apply divides_times_gcd_aux
+    [ unfold lt.
+      change with (O \lt c).
+      assumption
+    | apply lt_O_S
+    | apply lt_to_le.
+      apply not_le_to_lt.
+      assumption
+    | apply (le_n (S m1)).
+    | assumption
+    | rewrite < H3.
+      assumption
+    ]
+  ]
+]
+qed.
+
+(*a particular case of the previous theorem (setting c=1)*)
+theorem divides_d_gcd: \forall m,n,d. 
+d \divides m \to d \divides n \to d \divides gcd n m. 
 intros.
-change with (d \divides gcd_aux (S m1) n (S m1)).
-apply divides_gcd_aux.
-unfold lt.apply le_S_S.apply le_O_n.
-apply lt_to_le.apply not_le_to_lt.assumption.apply le_n.assumption.
-rewrite < H2.assumption.
+rewrite > (times_n_SO (gcd n m)).
+rewrite < (sym_times (S O)).
+apply (divides_d_times_gcd)
+[ apply (lt_O_S O)
+| rewrite > (sym_times (S O)).
+  rewrite < (times_n_SO m).
+  assumption
+| rewrite > (sym_times (S O)).
+  rewrite < (times_n_SO n).
+  assumption
+]
 qed.
 
 theorem eq_minus_gcd_aux: \forall p,m,n.O < n \to n \le m \to n \le p \to
@@ -483,7 +591,6 @@ intro.apply (nat_case n)
 qed.
 
 theorem symmetric_gcd: symmetric nat gcd.
-(*CSC: bug here: unfold symmetric does not work *)
 change with 
 (\forall n,m:nat. gcd n m = gcd m n).
 intros.
@@ -545,9 +652,9 @@ elim H2.
 generalize in match H1.
 rewrite > H3.
 intro.
-cut (O < n2)
-  [elim (gcd_times_SO_to_gcd_SO n n n2 ? ? H4)
-    [cut (gcd n (n*n2) = n)
+cut (O < n1)
+  [elim (gcd_times_SO_to_gcd_SO n n n1 ? ? H4)
+    [cut (gcd n (n*n1) = n)
       [apply (lt_to_not_eq (S O) n)
         [assumption|rewrite < H4.assumption]
       |apply gcd_n_times_nm.assumption
@@ -555,7 +662,7 @@ cut (O < n2)
     |apply (trans_lt ? (S O))[apply le_n|assumption]
     |assumption
     ]
-  |elim (le_to_or_lt_eq O n2 (le_O_n n2));
+  |elim (le_to_or_lt_eq O n1 (le_O_n n1));
     [assumption
     |apply False_ind.
      apply (le_to_not_lt n (S O))
@@ -563,7 +670,7 @@ cut (O < n2)
        apply divides_to_le
         [rewrite > H4.apply lt_O_S
         |apply divides_d_gcd
-          [apply (witness ? ? n2).reflexivity
+          [apply (witness ? ? n1).reflexivity
           |apply divides_n_n
           ]
         ]
@@ -638,6 +745,7 @@ apply gcd_O_to_eq_O.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 qed.
 
+(* primes and divides *)
 theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
 n \divides p \lor n \divides q.
 intros.
@@ -647,18 +755,14 @@ cut (n \divides p \lor n \ndivides p)
     |right.
      cut (\exists a,b. a*n - b*p = (S O) \lor b*p - a*n = (S O))
        [elim Hcut1.elim H3.elim H4
-         [(* first case *)
-          rewrite > (times_n_SO q).rewrite < H5.
+         [rewrite > (times_n_SO q).rewrite < H5.
           rewrite > distr_times_minus.
+          elim H1.
+          autobatch by witness; 
+         (*
           rewrite > (sym_times q (a1*p)).
           rewrite > (assoc_times a1).
-          elim H1.
-          (*
-             rewrite > H6.
-             applyS (witness n (n*(q*a-a1*n2)) (q*a-a1*n2))
-             reflexivity. *);
-          applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
-          (*
+          applyS (witness n ? ? (refl_eq ? ?)).          
           rewrite < (sym_times n).rewrite < assoc_times.
           rewrite > (sym_times q).rewrite > assoc_times.
           rewrite < (assoc_times a1).rewrite < (sym_times n).
@@ -669,14 +773,18 @@ cut (n \divides p \lor n \ndivides p)
          |(* second case *)
           rewrite > (times_n_SO q).rewrite < H5.
           rewrite > distr_times_minus.
+          elim H1. autobatch by witness; 
+          (*
           rewrite > (sym_times q (a1*p)).
           rewrite > (assoc_times a1).
-          elim H1.rewrite > H6.
+          rewrite > H6.
+          applyS (witness n ? ? (refl_eq ? ?)).
           rewrite < sym_times.rewrite > assoc_times.
           rewrite < (assoc_times q).
           rewrite < (sym_times n).
           rewrite < distr_times_minus.
-          apply (witness ? ? (n2*a1-q*a)).reflexivity
+          apply (witness ? ? (n1*a1-q*a)).reflexivity
+          *)
         ](* end second case *)
      |rewrite < (prime_to_gcd_SO n p)
        [apply eq_minus_gcd|assumption|assumption
@@ -691,6 +799,30 @@ cut (n \divides p \lor n \ndivides p)
   ]
 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.
+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.
+qed.
+
 theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
 gcd m n = (S O) \to gcd m p = (S O) \to gcd m (n*p) = (S O).
 intros.
@@ -746,19 +878,15 @@ cut (n \divides p \lor n \ndivides p)
     |cut (\exists a,b. a*n - b*m = (S O) \lor b*m - a*n = (S O))
       [elim Hcut1.elim H4.elim H5         
         [(* first case *)
+          elim H2. 
           rewrite > (times_n_SO p).rewrite < H6.
           rewrite > distr_times_minus.
-          rewrite > (sym_times p (a1*m)).
-          rewrite > (assoc_times a1).
+          autobatch by witness, divides_minus.
+        |(* second case *)
           elim H2.
-          applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
-         |(* second case *)
           rewrite > (times_n_SO p).rewrite < H6.
           rewrite > distr_times_minus.
-          rewrite > (sym_times p (a1*m)).
-          rewrite > (assoc_times a1).
-          elim H2.
-          applyS (witness n ? ? (refl_eq ? ?)).
+          autobatch by witness, divides_minus.
         ](* end second case *)
      |rewrite < H1.apply eq_minus_gcd.
      ]
@@ -766,4 +894,41 @@ cut (n \divides p \lor n \ndivides p)
  |apply (decidable_divides n p).
   assumption.
  ]
+qed.
+
+(*
+theorem divides_to_divides_times1: \forall p,q,n. prime p \to prime q \to p \neq q \to
+divides p n \to divides q n \to divides (p*q) n.
+intros.elim H3.
+rewrite > H5 in H4.
+elim (divides_times_to_divides ? ? ? H1 H4)
+  [elim H.apply False_ind.
+   apply H2.apply sym_eq.apply H8
+    [assumption
+    |apply prime_to_lt_SO.assumption
+    ]
+  |elim H6.
+   apply (witness ? ? n1).
+   rewrite > assoc_times.
+   rewrite < H7.assumption
+  ]
+qed.
+*)
+
+theorem divides_to_divides_times: \forall p,q,n. prime p  \to p \ndivides q \to
+divides p n \to divides q n \to divides (p*q) n.
+intros.elim H3.
+rewrite > H4 in H2.
+elim (divides_times_to_divides ? ? ? H H2)
+  [apply False_ind.apply H1.assumption
+  |elim H5.
+   autobatch by transitive_divides, H5, reflexive_divides,divides_times.
+   (*
+   apply (witness ? ? n2).
+   rewrite > sym_times in ⊢ (? ? ? (? % ?)).
+   rewrite > assoc_times.
+   autobatch.
+   (*rewrite < H6.assumption*)
+   *)
+  ]
 qed.
\ No newline at end of file