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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
Require RIneq.
Require Omega.
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].
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