(* begin hide *)
Notation RmaxLess2 := Rmax_r (only parsing).
(* end hide *)

(* Specifications: *)
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Lemma Rmax_left : forall x y, y <= x -> Rmax x y = x.
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Lemma Rmax_right : forall x y, x <= y -> Rmax x y = y.
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Lemma Rle_max_compat_r : forall x y z, x <= y -> Rmax x z <= Rmax y z.
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Lemma Rle_max_compat_l : forall x y z, x <= y -> Rmax z x <= Rmax z y.
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Lemma RmaxRmult :
forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q.
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Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0.
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Lemma Rmax_lub : forall x y z:R, x <= z -> y <= z -> Rmax x y <= z.
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Lemma Rmax_lub_lt : forall x y z:R, x < z -> y < z -> Rmax x y < z.
Lemma Rmax_Rlt : forall x y z,
Rmax x y < z <-> x < z /\ y < z.
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Lemma Rmax_neg : forall x y:R, x < 0 -> y < 0 -> Rmax x y < 0.
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(** *        Rabsolu           *)
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Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}.
Definition Rabs r : R :=
match Rcase_abs r with
| left _ => - r
| right _ => r
end.