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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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "nat/chinese_reminder.ma".
+include "nat/iteration2.ma".
+
+(*a new definition of totient, which uses sigma_p instead of sigma *)
+(*there's a little difference between this definition and the classic one:
+  the classic definition of totient is:
+   
+    phi (n) is the number of naturals i (less or equal than n) so then gcd (i,n) = 1.
+   (so this definition considers the values i=1,2,...,n)
+  
+  sigma_p doesn't work on the value n (but the first value it works on is (pred n))
+  but works also on 0. That's not a problem, in fact
+   - if n <> 1, gcd (n,0) <>1 and gcd (n,n) = n <> 1. 
+   - if n = 1, then Phi(n) = 1, and (totient n), as defined below, returns 1. 
+   
+ *)
+definition totient : nat \to nat \def
+\lambda n.sigma_p n (\lambda m. eqb (gcd m n) (S O)) (\lambda m.S O).
+
+lemma totient1: totient (S(S(S(S(S(S O)))))) = ?.
+[|simplify.
+                                
+theorem totient_times: \forall n,m:nat. (gcd m n) = (S O) \to
+totient (n*m) = (totient n)*(totient m).
+intros.
+unfold totient.
+apply (nat_case1 n)
+[ apply (nat_case1 m)
+  [ intros.
+    simplify.
+    reflexivity
+  | intros.
+    simplify.
+    reflexivity
+  ]
+| apply (nat_case1 m)
+  [ intros.
+    change in \vdash (? ? ? (? ? %)) with (O).
+    rewrite > (sym_times (S m1) O).
+    rewrite > sym_times in \vdash (? ? ? %).
+    simplify.
+    reflexivity  
+  | intros.
+    rewrite > H2 in H.
+    rewrite > H1 in H.    
+    apply (sigma_p_times m2 m1 ? ? ? 
+            (\lambda b,a. cr_pair (S m2) (S m1) a b) 
+            (\lambda x. x \mod (S m2)) (\lambda x. x \mod (S m1)))
+   [intros.unfold cr_pair.
+        apply (le_to_lt_to_lt ? (pred ((S m2)*(S m1))))
+          [unfold min.
+           apply transitive_le;
+            [2: apply le_min_aux_r | skip | apply le_n]
+          |unfold lt.
+           apply (nat_case ((S m2)*(S m1)))
+            [apply le_n|intro.apply le_n]
+          ]
+       |intros.
+        generalize in match (mod_cr_pair (S m2) (S m1) a b H3 H4 H).
+        intro.elim H5.
+        apply H6
+       |intros.
+        generalize in match (mod_cr_pair (S m2) (S m1) a b H3 H4 H).
+        intro.elim H5.
+        apply H7
+       |intros.
+        generalize in match (mod_cr_pair (S m2) (S m1) a b H3 H4 H).
+        intro.elim H5.
+        apply eqb_elim
+          [intro.
+           rewrite > eq_to_eqb_true
+             [rewrite > eq_to_eqb_true
+               [reflexivity
+               |rewrite < H6.
+                rewrite > sym_gcd.
+                rewrite > gcd_mod
+                  [apply (gcd_times_SO_to_gcd_SO ? ? (S m1))
+                    [unfold lt.apply le_S_S.apply le_O_n
+                    |unfold lt.apply le_S_S.apply le_O_n
+                    |assumption
+                    ]
+                  |unfold lt.apply le_S_S.apply le_O_n
+                  ]
+               ]           
+            |rewrite < H7.
+             rewrite > sym_gcd.
+             rewrite > gcd_mod
+               [apply (gcd_times_SO_to_gcd_SO ? ? (S m2))
+                  [unfold lt.apply le_S_S.apply le_O_n
+                  |unfold lt.apply le_S_S.apply le_O_n
+                  |rewrite > sym_times.assumption
+                  ]
+               |unfold lt.apply le_S_S.apply le_O_n
+               ]
+            ]
+          |intro.
+           apply eqb_elim
+           [intro.apply eqb_elim
+              [intro.apply False_ind.
+               apply H8.
+               apply eq_gcd_times_SO
+                 [unfold lt.apply le_S_S.apply le_O_n.
+                 |unfold lt.apply le_S_S.apply le_O_n.
+                 |rewrite < gcd_mod
+                    [rewrite > H6.
+                     rewrite > sym_gcd.assumption
+                    |unfold lt.apply le_S_S.apply le_O_n
+                    ]
+                 |rewrite < gcd_mod
+                    [rewrite > H7.
+                     rewrite > sym_gcd.assumption
+                    |unfold lt.apply le_S_S.apply le_O_n
+                    ]
+                 ]
+              |intro.reflexivity
+              ]
+           |intro.reflexivity
+           ]
+         ]
+       ]
+     ]
+   ]
+qed.