(* We call this morphism slow to remind that it should only be used for proofs, not for computations. *)
Definition SlowConstructiveRealsMorphism {R1 R2 : ConstructiveReals}
: @ConstructiveRealsMorphism R1 R2
:= Build_ConstructiveRealsMorphism
R1 R2 SlowMorph CauchyMorph_rat
SlowMorph_increasing.

Lemma CRmorph_cauchy_reverse : forall {R1 R2 : ConstructiveReals}
(f : @ConstructiveRealsMorphism R1 R2)
(un : nat -> CRcarrier R1),
CR_cauchy R2 (fun n => CRmorph f (un n))
-> CR_cauchy R1 un.