- renamed ocaml/ to components/

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

diff --git a/helm/matita/library/nat/factorial.ma b/helm/matita/library/nat/factorial.ma
deleted file mode 100644 (file)
index 14217bb..0000000
--- a/helm/matita/library/nat/factorial.ma
+++ /dev/null
@@ -1,61 +0,0 @@
-(**************************************************************************)
-(*       ___                                                               *)
-(*      ||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         *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/factorial".
-
-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.
-