X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2Flibrary_auto%2Fauto%2Fnat%2Ftotient.ma;fp=matita%2Fcontribs%2Flibrary_auto%2Fauto%2Fnat%2Ftotient.ma;h=98b37467669041a290c72b09878e5022d4c3c3c6;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

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+(**************************************************************************)
+(*       __                                                               *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/totient".
+
+include "auto/nat/count.ma".
+include "auto/nat/chinese_reminder.ma".
+
+definition totient : nat \to nat \def
+\lambda n. count n (\lambda m. eqb (gcd m n) (S O)).
+
+theorem totient3: totient (S(S(S O))) = (S(S O)).
+reflexivity.
+qed.
+
+theorem totient6: totient (S(S(S(S(S(S O)))))) = (S(S O)).
+reflexivity.
+qed.
+
+theorem totient_times: \forall n,m:nat. (gcd m n) = (S O) \to
+totient (n*m) = (totient n)*(totient m).
+intro.
+apply (nat_case n)
+[ intros.
+  autobatch
+  (*simplify.
+  reflexivity*)
+| intros 2.
+  apply (nat_case m1)
+  [ rewrite < sym_times.
+    rewrite < (sym_times (totient O)).
+    simplify.
+    intro.
+    reflexivity
+  | intros.
+    unfold totient.     
+    apply (count_times m m2 ? ? ? 
+      (\lambda b,a. cr_pair (S m) (S m2) a b) 
+      (\lambda x. x \mod (S m)) (\lambda x. x \mod (S m2)))
+    [ intros.
+      unfold cr_pair.
+      apply (le_to_lt_to_lt ? (pred ((S m)*(S m2))))
+      [ unfold min.
+        apply le_min_aux_r
+      | autobatch
+        (*unfold lt.
+        apply (nat_case ((S m)*(S m2)))
+        [ apply le_n
+        | intro.
+          apply le_n
+        ]*)
+      ]
+    | intros.
+      generalize in match (mod_cr_pair (S m) (S m2) a b H1 H2 H).
+      intro.
+      elim H3.
+      apply H4
+    | intros.
+      generalize in match (mod_cr_pair (S m) (S m2) a b H1 H2 H).
+      intro.      
+      elim H3.
+      apply H5
+    | intros.
+      generalize in match (mod_cr_pair (S m) (S m2) a b H1 H2 H).
+      intro.
+      elim H3.
+      apply eqb_elim
+      [ intro.
+        rewrite > eq_to_eqb_true
+        [ rewrite > eq_to_eqb_true
+          [ reflexivity
+          | rewrite < H4.
+            rewrite > sym_gcd.
+            rewrite > gcd_mod
+            [ apply (gcd_times_SO_to_gcd_SO ? ? (S m2))
+              [ autobatch
+                (*unfold lt.
+                apply le_S_S.
+                apply le_O_n*)
+              | autobatch
+                (*unfold lt.
+                apply le_S_S.
+                apply le_O_n*)
+              | assumption
+              ]
+            | autobatch
+              (*unfold lt.
+              apply le_S_S.
+              apply le_O_n*)
+            ]
+          ]           
+        | rewrite < H5.
+          rewrite > sym_gcd.
+          rewrite > gcd_mod
+          [ apply (gcd_times_SO_to_gcd_SO ? ? (S m))
+            [ autobatch
+              (*unfold lt.
+              apply le_S_S.
+              apply le_O_n*)
+            | autobatch
+              (*unfold lt.
+              apply le_S_S.
+              apply le_O_n*)
+            | autobatch
+              (*rewrite > sym_times.
+              assumption*)
+            ]
+          | autobatch
+            (*unfold lt.
+            apply le_S_S.
+            apply le_O_n*)
+          ]
+        ]
+      | intro.
+        apply eqb_elim
+        [ intro.
+          apply eqb_elim
+          [ intro.
+            apply False_ind.
+            apply H6.
+            apply eq_gcd_times_SO
+            [ autobatch
+              (*unfold lt.
+              apply le_S_S.
+              apply le_O_n*)
+            | autobatch
+              (*unfold lt.
+              apply le_S_S.
+              apply le_O_n*)
+            | rewrite < gcd_mod
+              [ autobatch
+                (*rewrite > H4.
+                rewrite > sym_gcd.
+                assumption*)
+              | autobatch
+                (*unfold lt.
+                apply le_S_S.
+                apply le_O_n*)
+              ]
+            | rewrite < gcd_mod
+              [ autobatch
+                (*rewrite > H5.
+                rewrite > sym_gcd.
+                assumption*)
+              | autobatch
+                (*unfold lt.
+                apply le_S_S.
+                apply le_O_n*)
+              ]
+            ]
+          | intro.
+            reflexivity
+          ]
+        | intro.
+          reflexivity
+        ]
+      ]
+    ]
+  ]
+]
+qed.
\ No newline at end of file