New entry: fermat's little theorem (almost complete).

[helm.git] / helm / matita / library / nat / fermat_little_theorem.ma

diff --git a/helm/matita/library/nat/fermat_little_theorem.ma b/helm/matita/library/nat/fermat_little_theorem.ma
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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/nat/fermat_little_theorem".
+
+include "nat/gcd.ma".
+include "nat/permutation.ma".
+
+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 lt_mod_m_m.
+simplify in H.elim H.
+simplify.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.
+unfold injn.intros.
+apply nat_compare_elim i j.
+(* i < j *)
+intro.
+absurd j-i \lt p.
+simplify.
+rewrite > S_pred p.
+apply le_S_S.
+apply le_plus_to_minus.
+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 le_SO_minus.assumption.
+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 sym_eq.
+apply H4.
+(* i = j *)
+intro. assumption.
+(* j < i *)
+intro.
+absurd i-j \lt p.
+simplify.
+rewrite > S_pred p.
+apply le_S_S.
+apply le_plus_to_minus.
+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 le_SO_minus.assumption.
+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 H4.
+qed.
+