(* 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 CRmax_min_mult_neg :
forall {R : ConstructiveReals} (p q r:CRcarrier R),
r <= 0 -> CRmax (r * p) (r * q) == r * CRmin p q.