(* Maximum *)

Definition CRmax {R : ConstructiveReals} (x y : CRcarrier R) : CRcarrier R
  := (x + y + CRabs _ (y - x)) * CR_of_Q _ (1#2).Add Parametric Morphism {R : ConstructiveReals} : CRmax
    with signature (CReq R) ==> (CReq R) ==> (CReq R)
      as CRmax_morph.#[global]
Instance CRmax_morphT
  : forall {R : ConstructiveReals},
    CMorphisms.Proper
      (CMorphisms.respectful (CReq R) (CMorphisms.respectful (CReq R) (CReq R))) (@CRmax R).

Lemma CRlt_max : forall {R : ConstructiveReals} (x y z : CRcarrier R), CRmax x y < z -> prod (x < z) (y < z).