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

diff --git a/matita/library/nat/factorial.ma b/matita/library/nat/factorial.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                                 *)
+(*      \   /                                                             *)
+(*       \ /        Matita is distributed under the terms of the          *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "nat/le_arith.ma".
+
+let rec fact n \def
+  match n with 
+  [ O \Rightarrow (S O)
+  | (S m) \Rightarrow (S m)*(fact m)].
+
+interpretation "factorial" 'fact n = (cic:/matita/nat/factorial/fact.con n).
+
+theorem le_SO_fact : \forall n. (S O) \le n!.
+intro.elim n.simplify.apply le_n.
+change with ((S O) \le (S n1)*n1!).
+apply (trans_le ? ((S n1)*(S O))).simplify.
+apply le_S_S.apply le_O_n.
+apply le_times_r.assumption.
+qed.
+
+theorem le_SSO_fact : \forall n. (S O) < n \to (S(S O)) \le n!.
+intro.apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
+intros.change with ((S (S O)) \le (S m)*m!).
+apply (trans_le ? ((S(S O))*(S O))).apply le_n.
+apply le_times.exact H.apply le_SO_fact.
+qed.
+
+theorem le_n_fact_n: \forall n. n \le n!.
+intro. elim n.apply le_O_n.
+change with (S n1 \le (S n1)*n1!).
+apply (trans_le ? ((S n1)*(S O))).
+rewrite < times_n_SO.apply le_n.
+apply le_times.apply le_n.
+apply le_SO_fact.
+qed.
+
+theorem lt_n_fact_n: \forall n. (S(S O)) < n \to n < n!.
+intro.apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S(S O)) H).
+intros.change with ((S m) < (S m)*m!).
+apply (lt_to_le_to_lt ? ((S m)*(S (S O)))).
+rewrite < sym_times.
+simplify.unfold lt.
+apply le_S_S.rewrite < plus_n_O.
+apply le_plus_n.
+apply le_times_r.apply le_SSO_fact.
+simplify.unfold lt.apply le_S_S_to_le.exact H.
+qed.
+