--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/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 lt_mod_m_m.
+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)).unfold lt.apply le_n.assumption.
+unfold injn.intros.
+apply (nat_compare_elim i j).
+(* i < j *)
+intro.
+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.
+rewrite > sym_plus.
+apply le_plus_n.
+unfold prime in H.
+elim H.
+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)).
+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)).unfold lt.apply le_n.assumption.
+apply sym_eq.
+apply H4.
+(* i = j *)
+intro. assumption.
+(* j < i *)
+intro.
+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.
+rewrite > sym_plus.
+apply le_plus_n.
+unfold prime in H.
+elim H.
+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)).
+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)).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))
+ [unfold lt.rewrite < (S_pred ? Hcut1).apply le_n.
+ |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) Hcut2).
+ rewrite > eq_fact_pi.
+ 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 ? ? ? ? ? (λm.m))
+ [apply assoc_times
+ |apply sym_times
+ |rewrite < plus_n_Sm.
+ rewrite < plus_n_O.
+ rewrite < (S_pred ? Hcut2).
+ apply permut_mod[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
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ |unfold lt.apply le_S_S_to_le.rewrite < (S_pred ? Hcut1).
+ unfold prime in H.elim H.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.
+
projects / helm.git / blobdiff