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19 (*#***********************************************************************)
21 (* v * The Coq Proof Assistant / The Coq Development Team *)
23 (* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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27 (* // * This file is distributed under the terms of the *)
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33 (*i $Id: Rprod.v,v 1.10.2.1 2004/07/16 19:31:13 herbelin Exp $ i*)
35 include "Arith/Compare.ma".
37 include "Reals/Rbase.ma".
39 include "Reals/Rfunctions.ma".
41 include "Reals/Rseries.ma".
43 include "Reals/PartSum.ma".
45 include "Reals/Binomial.ma".
48 Open Local Scope R_scope.
53 inline procedural "cic:/Coq/Reals/Rprod/prod_f_SO.con" as definition.
57 inline procedural "cic:/Coq/Reals/Rprod/prod_SO_split.con" as lemma.
61 inline procedural "cic:/Coq/Reals/Rprod/prod_SO_pos.con" as lemma.
65 inline procedural "cic:/Coq/Reals/Rprod/prod_SO_Rle.con" as lemma.
67 (* Application to factorial *)
69 inline procedural "cic:/Coq/Reals/Rprod/fact_prodSO.con" as lemma.
71 inline procedural "cic:/Coq/Reals/Rprod/le_n_2n.con" as lemma.
73 (* We prove that (N!)\178\<=(2N-k)!*k! forall k in [|O;2N|] *)
75 inline procedural "cic:/Coq/Reals/Rprod/RfactN_fact2N_factk.con" as lemma.
79 inline procedural "cic:/Coq/Reals/Rprod/INR_fact_lt_0.con" as lemma.
81 (* We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *)
83 inline procedural "cic:/Coq/Reals/Rprod/C_maj.con" as lemma.