(* We call this morphism slow to remind that it should only be used
   for proofs, not for computations. *)
Definition SlowMorph {R1 R2 : ConstructiveReals}
  : CRcarrier R1 -> CRcarrier R2
  := fun x => let (y,_) := CR_complete R2 _ (CRmorph_rat_cv _ (Rcv_cauchy_mod _ x (CR_Q_limit_cv x)))
              in y.

(* Specifications: *)
Lemma CauchyMorph_rat : forall {R1 R2 : ConstructiveReals} (q : Q), SlowMorph (CR_of_Q R1 q) == CR_of_Q R2 q.
Lemma SlowMorph_increasing_Qr : forall {R1 R2 : ConstructiveReals} (x : CRcarrier R1) (q : Q), CR_of_Q R1 q < x -> CR_of_Q R2 q < SlowMorph x.
Lemma SlowMorph_increasing_Ql : forall {R1 R2 : ConstructiveReals} (x : CRcarrier R1) (q : Q), x < CR_of_Q R1 q -> SlowMorph x < CR_of_Q R2 q.
Lemma SlowMorph_increasing : forall {R1 R2 : ConstructiveReals} (x y : CRcarrier R1), x < y -> @SlowMorph R1 R2 x < SlowMorph y.