X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Ffermat_little_theorem.ma;h=cc18a8bb9e4249005d96e772d7e41f4d01409720;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=aafdc2f19e95a18967eb2b34cea2dc4bce3129e3;hpb=7273c698dd60c1a8a0f35b44376acb548c6a4a33;p=helm.git

diff --git a/helm/matita/library/nat/fermat_little_theorem.ma b/helm/matita/library/nat/fermat_little_theorem.ma
index aafdc2f19..cc18a8bb9 100644
--- a/helm/matita/library/nat/fermat_little_theorem.ma
+++ b/helm/matita/library/nat/fermat_little_theorem.ma
@@ -14,77 +14,237 @@
 
 set "baseuri" "cic:/matita/nat/fermat_little_theorem".
 
+include "nat/exp.ma".
 include "nat/gcd.ma".
 include "nat/permutation.ma".
+include "nat/congruence.ma".
+
+theorem permut_S_mod: \forall n:nat. permut (S_mod (S n)) n.
+intro.unfold permut.split.intros.
+unfold S_mod.
+apply le_S_S_to_le.
+change with ((S i) \mod (S n) < S n).
+apply lt_mod_m_m.
+unfold lt.apply le_S_S.apply le_O_n.
+unfold injn.intros.
+apply inj_S.
+rewrite < (lt_to_eq_mod i (S n)).
+rewrite < (lt_to_eq_mod j (S n)).
+cut (i < n \lor i = n).
+cut (j < n \lor j = n).
+elim Hcut.
+elim Hcut1.
+(* i < n, j< n *)
+rewrite < mod_S.
+rewrite < mod_S.
+apply H2.unfold lt.apply le_S_S.apply le_O_n.
+rewrite > lt_to_eq_mod.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+rewrite > lt_to_eq_mod.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.assumption.
+(* i < n, j=n *)
+unfold S_mod in H2.
+simplify.
+apply False_ind.
+apply (not_eq_O_S (i \mod (S n))).
+apply sym_eq.
+rewrite < (mod_n_n (S n)).
+rewrite < H4 in \vdash (? ? ? (? %?)).
+rewrite < mod_S.assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+rewrite > lt_to_eq_mod.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+(* i = n, j < n *)
+elim Hcut1.
+apply False_ind.
+apply (not_eq_O_S (j \mod (S n))).
+rewrite < (mod_n_n (S n)).
+rewrite < H3 in \vdash (? ? (? %?) ?).
+rewrite < mod_S.assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+rewrite > lt_to_eq_mod.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.apply le_O_n.
+(* i = n, j= n*)
+rewrite > H3.
+rewrite > H4.
+reflexivity.
+apply le_to_or_lt_eq.assumption.
+apply le_to_or_lt_eq.assumption.
+unfold lt.apply le_S_S.assumption.
+unfold lt.apply le_S_S.assumption.
+qed.
+
+(*
+theorem eq_fact_pi: \forall n,m:nat. n < m \to n! = pi n (S_mod m).
+intro.elim n.
+simplify.reflexivity.
+change with (S n1)*n1!=(S_mod m n1)*(pi n1 (S_mod m)).
+unfold S_mod in \vdash (? ? ? (? % ?)). 
+rewrite > lt_to_eq_mod.
+apply eq_f.apply H.apply (trans_lt ? (S n1)).
+simplify. apply le_n.assumption.assumption.
+qed.
+*)
+
+theorem prime_to_not_divides_fact: \forall p:nat. prime p \to \forall n:nat.
+n \lt p \to \not divides p n!.
+intros 3.elim n.unfold Not.intros.
+apply (lt_to_not_le (S O) p).
+unfold prime in H.elim H.
+assumption.apply divides_to_le.unfold lt.apply le_n.
+assumption.
+change with (divides p ((S n1)*n1!) \to False).
+intro.
+cut (divides p (S n1) \lor divides p n1!).
+elim Hcut.apply (lt_to_not_le (S n1) p).
+assumption.
+apply divides_to_le.unfold lt.apply le_S_S.apply le_O_n.
+assumption.apply H1.
+apply (trans_lt ? (S n1)).unfold lt. apply le_n.
+assumption.assumption.
+apply divides_times_to_divides.
+assumption.assumption.
+qed.
 
 theorem permut_mod: \forall p,a:nat. prime p \to
 \lnot divides p a\to permut (\lambda n.(mod (a*n) p)) (pred p).
 unfold permut.intros.
 split.intros.apply le_S_S_to_le.
-apply trans_le ? p.
-change with mod (a*i) p < p.
+apply (trans_le ? p).
+change with (mod (a*i) p < p).
 apply lt_mod_m_m.
-simplify in H.elim H.
-simplify.apply trans_le ? (S (S O)).
+unfold prime in H.elim H.
+unfold lt.apply (trans_le ? (S (S O))).
 apply le_n_Sn.assumption.
 rewrite < S_pred.apply le_n.
 unfold prime in H.
 elim H.
-apply trans_lt ? (S O).simplify.apply le_n.assumption.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.assumption.
 unfold injn.intros.
-apply nat_compare_elim i j.
+apply (nat_compare_elim i j).
 (* i < j *)
 intro.
-absurd j-i \lt p.
-simplify.
-rewrite > S_pred p.
+absurd (j-i \lt p).
+unfold lt.
+rewrite > (S_pred p).
 apply le_S_S.
 apply le_plus_to_minus.
-apply trans_le ? (pred p).assumption.
+apply (trans_le ? (pred p)).assumption.
 rewrite > sym_plus.
 apply le_plus_n.
 unfold prime in H.
 elim H.
-apply trans_lt ? (S O).simplify.apply le_n.assumption.
-apply le_to_not_lt p (j-i).
-apply divides_to_le.simplify.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.assumption.
+apply (le_to_not_lt p (j-i)).
+apply divides_to_le.unfold lt.
 apply le_SO_minus.assumption.
-cut divides p a \lor divides p (j-i).
+cut (divides p a \lor divides p (j-i)).
 elim Hcut.apply False_ind.apply H1.assumption.assumption.
 apply divides_times_to_divides.assumption.
 rewrite > distr_times_minus.
 apply eq_mod_to_divides.
 unfold prime in H.
 elim H.
-apply trans_lt ? (S O).simplify.apply le_n.assumption.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.assumption.
 apply sym_eq.
 apply H4.
 (* i = j *)
 intro. assumption.
 (* j < i *)
 intro.
-absurd i-j \lt p.
-simplify.
-rewrite > S_pred p.
+absurd (i-j \lt p).
+unfold lt.
+rewrite > (S_pred p).
 apply le_S_S.
 apply le_plus_to_minus.
-apply trans_le ? (pred p).assumption.
+apply (trans_le ? (pred p)).assumption.
 rewrite > sym_plus.
 apply le_plus_n.
 unfold prime in H.
 elim H.
-apply trans_lt ? (S O).simplify.apply le_n.assumption.
-apply le_to_not_lt p (i-j).
-apply divides_to_le.simplify.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.assumption.
+apply (le_to_not_lt p (i-j)).
+apply divides_to_le.unfold lt.
 apply le_SO_minus.assumption.
-cut divides p a \lor divides p (i-j).
+cut (divides p a \lor divides p (i-j)).
 elim Hcut.apply False_ind.apply H1.assumption.assumption.
 apply divides_times_to_divides.assumption.
 rewrite > distr_times_minus.
 apply eq_mod_to_divides.
 unfold prime in H.
 elim H.
-apply trans_lt ? (S O).simplify.apply le_n.assumption.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.assumption.
 apply H4.
 qed.
 
+theorem congruent_exp_pred_SO: \forall p,a:nat. prime p \to \lnot divides p a \to
+congruent (exp a (pred p)) (S O) p. 
+intros.
+cut (O < a).
+cut (O < p).
+cut (O < pred p).
+apply divides_to_congruent.
+assumption.
+change with (O < exp a (pred p)).
+apply lt_O_exp.assumption.
+cut (divides p (exp a (pred p)-(S O)) \lor divides p (pred p)!).
+elim Hcut3.
+assumption.
+apply False_ind.
+apply (prime_to_not_divides_fact p H (pred p)).
+change with (S (pred p) \le p).
+rewrite < S_pred.apply le_n.
+assumption.assumption.
+apply divides_times_to_divides. 
+assumption.
+rewrite > times_minus_l.
+rewrite > (sym_times (S O)).
+rewrite < times_n_SO.
+rewrite > (S_pred (pred p)).
+rewrite > eq_fact_pi.
+(* in \vdash (? ? (? % ?)). *)
+rewrite > exp_pi_l.
+apply congruent_to_divides.
+assumption. 
+apply (transitive_congruent p ? 
+(pi (pred (pred p)) (\lambda m. a*m \mod p) (S O))).
+apply (congruent_pi (\lambda m. a*m)).
+assumption.
+cut (pi (pred(pred p)) (\lambda m.m) (S O)
+= pi (pred(pred p)) (\lambda m.a*m \mod p) (S O)).
+rewrite > Hcut3.apply congruent_n_n.
+rewrite < eq_map_iter_i_pi.
+rewrite < eq_map_iter_i_pi.
+apply permut_to_eq_map_iter_i.
+apply assoc_times.
+apply sym_times.
+rewrite < plus_n_Sm.rewrite < plus_n_O.
+rewrite < S_pred.
+apply permut_mod.assumption.
+assumption.assumption.
+intros.cut (m=O).
+rewrite > Hcut3.rewrite < times_n_O.
+apply mod_O_n.apply sym_eq.apply le_n_O_to_eq.
+apply le_S_S_to_le.assumption.
+assumption.
+change with ((S O) \le pred p).
+apply le_S_S_to_le.rewrite < S_pred.
+unfold prime in H.elim H.assumption.assumption.
+unfold prime in H.elim H.apply (trans_lt ? (S O)).
+unfold lt.apply le_n.assumption.
+cut (O < a \lor O = a).
+elim Hcut.assumption.
+apply False_ind.apply H1.
+rewrite < H2.
+apply (witness ? ? O).apply times_n_O.
+apply le_to_or_lt_eq.
+apply le_O_n.
+qed.
+