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[helm.git] / matita / contribs / library_auto / auto / nat / factorial.ma

diff --git a/matita/contribs/library_auto/auto/nat/factorial.ma b/matita/contribs/library_auto/auto/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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/factorial".
+
+include "auto/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/library_autobatch/nat/factorial/fact.con n).
+
+theorem le_SO_fact : \forall n. (S O) \le n!.
+intro.
+elim n
+[ autobatch
+  (*simplify.
+  apply le_n*)
+| change with ((S O) \le (S n1)*n1!).
+  autobatch
+  (*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.
+  autobatch
+  (*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)));autobatch
+  (*[ 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)));autobatch 
+  (*[ 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.
+  autobatch
+  (*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.
+    autobatch
+    (*rewrite < plus_n_O.
+    apply le_plus_n*)
+  | apply le_times_r.
+    autobatch
+    (*apply le_SSO_fact.
+    simplify.
+    unfold lt.
+    apply le_S_S_to_le.
+    exact H*)
+  ]
+]
+qed.
+