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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 *)
(************************************************************************)
(*i $Id: DiscrR.v,v 1.21.2.1 2004/07/16 19:31:10 herbelin Exp $ i*)
Require Import RIneq.
Require Import Omega. 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 : forall x y:R, 0 < x -> 0 < y -> 0 < x + y.
intros.
apply Rlt_trans with x.
assumption.
pattern x at 1 in |- *; rewrite <- Rplus_0_r.
apply Rplus_lt_compat_l.
assumption.
Qed.
Lemma IZR_eq : forall z1 z2:Z, z1 = z2 -> IZR z1 = IZR z2.
intros; rewrite H; reflexivity.
Qed.
Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2.
intros; red in |- *; intro; elim H; apply eq_IZR; assumption.
Qed.
Ltac discrR :=
try
match goal with
| |- (?X1 <> ?X2) =>
replace 2 with (IZR 2);
[ replace 1 with (IZR 1);
[ replace 0 with (IZR 0);
[ repeat
rewrite <- plus_IZR ||
rewrite <- mult_IZR ||
rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
apply IZR_neq; try discriminate
| reflexivity ]
| reflexivity ]
| reflexivity ]
end.
Ltac prove_sup0 :=
match goal with
| |- (0 < 1) => apply Rlt_0_1
| |- (0 < ?X1) =>
repeat
(apply Rmult_lt_0_compat || apply Rplus_lt_pos;
try apply Rlt_0_1 || apply Rlt_R0_R2)
| |- (?X1 > 0) => change (0 < X1) in |- *; prove_sup0
end.
Ltac omega_sup :=
replace 2 with (IZR 2);
[ replace 1 with (IZR 1);
[ replace 0 with (IZR 0);
[ repeat
rewrite <- plus_IZR ||
rewrite <- mult_IZR ||
rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
apply IZR_lt; omega
| reflexivity ]
| reflexivity ]
| reflexivity ].
Ltac prove_sup :=
match goal with
| |- (?X1 > ?X2) => change (X2 < X1) in |- *; prove_sup
| |- (0 < ?X1) => prove_sup0
| |- (- ?X1 < 0) => rewrite <- Ropp_0; prove_sup
| |- (- ?X1 < - ?X2) => apply Ropp_lt_gt_contravar; prove_sup
| |- (- ?X1 < ?X2) => apply Rlt_trans with 0; prove_sup
| |- (?X1 < ?X2) => omega_sup
| _ => idtac
end.
Ltac Rcompute :=
replace 2 with (IZR 2);
[ replace 1 with (IZR 1);
[ replace 0 with (IZR 0);
[ repeat
rewrite <- plus_IZR ||
rewrite <- mult_IZR ||
rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus;
apply IZR_eq; try reflexivity
| reflexivity ]
| reflexivity ]
| reflexivity ].
|
