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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Require Export Ring Field.
Local Open Scope R_scope.
Notation Rset := (Eqsth R).
Notation Rext := (Eq_ext Rplus Rmult Ropp).
unfold Rgt.
induction x; simpl; intros.
apply Rlt_trans with (1 + 0).
rewrite Rplus_comm.
apply Rlt_n_Sn.
apply Rplus_lt_compat_l.
rewrite <- (Rmul_0_l Rset Rext RTheory 2).
rewrite Rmult_comm.
apply Rmult_lt_compat_l.
apply Rlt_0_2.
trivial.
rewrite <- (Rmul_0_l Rset Rext RTheory 2).
rewrite Rmult_comm.
apply Rmult_lt_compat_l.
apply Rlt_0_2.
trivial.
replace 1 with (0 + 1).
apply Rlt_n_Sn.
apply Rplus_0_l.
Qed.
red; intros.
specialize (Rgen_phiPOS x).
rewrite H; intro.
apply (Rlt_asym 0 0); trivial.
Qed.
Proof gen_phiZ_complete Rset Rext Rfield Rgen_phiPOS_not_0.

Lemma R_power_theory : power_theory 1%R Rmult (@eq R) N.to_nat pow.