(* This code is copyrighted by its authors; it is distributed under  *)
(* the terms of the LGPL license (see LICENSE and description files) *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                                                                          *)
(*                                                                          *)
(*                   Solange Coupet-Grimal & Line Jakubiec                  *)
(*                                                                          *)
(*                                                                          *)
(*              Laboratoire d'Informatique de Marseille                     *)
(*               CMI-Technopole de Chateau-Gombert                          *)
(*                   39, Rue F. Joliot Curie                                *)
(*                   13453 MARSEILLE Cedex 13                               *)
(*           e-mail:{Solange.Coupet,Line.Jakubiec}@lim.univ-mrs.fr          *)
(*                                                                          *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              May 30th 1996                               *)
(*                                                                          *)
(****************************************************************************)
(*                              Compare_Nat.v                               *)
(****************************************************************************)
Require Export Arith.
Global Set Asymmetric Patterns.
Inductive Or3 (A B C : Prop) : Set :=
| First : A -> Or3 A B C
| Second : B -> Or3 A B C
| Third : C -> Or3 A B C.
Hint Resolve First Second Third.
simple induction n; simple induction m.
apply Second; try trivial with arith.
intros; apply First; try trivial with arith.
apply Third; try trivial with arith.
intros.
elim (H n1); intros Hyp.
apply First; auto with arith.
apply Second; auto with arith.
apply Third; auto with arith.
Defined.
Inductive order : Set :=
| L : order
| E : order
| G : order.
Definition comparison (v1 v2 : nat) :=
match Lt_eq_Gt v1 v2 return order with
| First _ => L
| Second _ => E
| Third _ => G
end.
intros.
unfold Compare_Nat.comparison in |- *.
elim (Lt_eq_Gt v1 v2).
auto with arith.
intros e; absurd (v1 < v2); auto with arith.
rewrite e; auto with arith.
intro; absurd (v1 > v2); auto with arith.
Qed.
Hint Resolve comparisonL.
intros.
unfold Compare_Nat.comparison in |- *.
elim (Lt_eq_Gt v1 v2); auto with arith.
intro; absurd (v1 > v2); auto with arith.
intros e; absurd (v1 > v2); auto with arith.
rewrite e; auto with arith.
Qed.
Hint Resolve comparisonG.
intros.
unfold Compare_Nat.comparison in |- *.
elim (Lt_eq_Gt v1 v2); auto with arith.
intro; absurd (v1 < v2); auto with arith.
rewrite H; auto with arith.
intro; absurd (v1 > v2); auto with arith.
rewrite H; auto with arith.
Qed.
Hint Resolve comparisonE.
intros.
elim (Lt_eq_Gt v1 v2); auto with arith.
intros e.
absurd (Compare_Nat.comparison v1 v2 = L); auto with arith.
rewrite (comparisonE v1 v2 e); discriminate.
intros g.
absurd (Compare_Nat.comparison v1 v2 = L); auto with arith.
rewrite (comparisonG v1 v2 g); discriminate.
Qed.
Hint Resolve inv_comparisonL.
intros v1 v2 c.
elim (Lt_eq_Gt v1 v2); auto with arith.
intros l; absurd (Compare_Nat.comparison v1 v2 = E); auto with arith.
rewrite (comparisonL v1 v2 l); discriminate.
intros g; absurd (Compare_Nat.comparison v1 v2 = E); auto with arith.
rewrite (comparisonG v1 v2 g); discriminate.
Qed.
Hint Resolve inv_comparisonE.
intros v1 v2 c.
elim (Lt_eq_Gt v1 v2); auto with arith.
intros l; absurd (Compare_Nat.comparison v1 v2 = G); auto with arith.
rewrite (comparisonL v1 v2 l); discriminate.
intros e; absurd (Compare_Nat.comparison v1 v2 = G); auto with arith.
rewrite (comparisonE v1 v2 e); discriminate.
Qed.
Hint Resolve inv_comparisonG.
intros v1 v2.
cut (Compare_Nat.comparison v1 v2 = Compare_Nat.comparison v1 v2);
auto with arith.
pattern (Compare_Nat.comparison v1 v2) at 2 3 in |- *.
case (Compare_Nat.comparison v1 v2); intro; auto with arith.
Qed.
intros v1 v2 H.
apply comparisonG.
auto with arith.
Qed.
Hint Resolve comp_sym_LG.
intros v1 v2 H.
apply comparisonL; auto with arith.
Qed.
Hint Resolve comp_sym_GL.

Lemma comp_sym_E :
forall v1 v2 : nat,
Compare_Nat.comparison v1 v2 = E -> Compare_Nat.comparison v2 v1 = E.