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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(*i        $Id: DiscrR.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $       i*)

Require RIneq.
Require Omega.
V7only [Import R_scope.]. Open Local Scope R_scope.

Lemma Rlt_R0_R2 : ``0<2``.
Replace ``2`` with (INR (2)); [Apply lt_INR_0; Apply lt_O_Sn | Reflexivity].
Qed.

Lemma Rplus_lt_pos : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``.
Intros.
Apply Rlt_trans with x.
Assumption. 
Pattern 1 x; Rewrite <- Rplus_Or.
Apply Rlt_compatibility.
Assumption.
Qed.

Lemma IZR_eq : (z1,z2:Z) z1=z2 -> (IZR z1)==(IZR z2).
Intros; Rewrite H; Reflexivity.
Qed.

Lemma IZR_neq : (z1,z2:Z) `z1<>z2` -> ``(IZR z1)<>(IZR z2)``.
Intros; Red; Intro; Elim H; Apply eq_IZR; Assumption.
Qed.

Tactic Definition DiscrR :=
  Try Match Context With
  | [ |- ~(?1==?2) ] -> Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_neq; Try Discriminate | Reflexivity] | Reflexivity] | Reflexivity].

Recursive Tactic Definition Sup0 :=
  Match Context With
  | [ |- ``0<1`` ] -> Apply Rlt_R0_R1
  | [ |- ``0<?1`` ] -> Repeat (Apply Rmult_lt_pos Orelse Apply Rplus_lt_pos; Try Apply Rlt_R0_R1 Orelse Apply Rlt_R0_R2)
  | [ |- ``?1>0`` ] -> Change ``0<?1``; Sup0.

Tactic Definition SupOmega := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_lt; Omega | Reflexivity] | Reflexivity] | Reflexivity].
  
Recursive Tactic Definition Sup :=
  Match Context With
  | [ |- (Rgt ?1 ?2) ] -> Change ``?2<?1``; Sup
  | [ |- ``0<?1`` ] -> Sup0
  | [ |- (Rlt (Ropp ?1) R0) ] -> Rewrite <- Ropp_O; Sup
  | [ |- (Rlt (Ropp ?1) (Ropp ?2)) ] -> Apply Rlt_Ropp; Sup
  | [ |- (Rlt (Ropp ?1) ?2) ] -> Apply Rlt_trans with ``0``; Sup
  | [ |- (Rlt ?1 ?2) ] -> SupOmega
  | _ -> Idtac.

Tactic Definition RCompute := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_eq; Try Reflexivity | Reflexivity] | Reflexivity] | Reflexivity].