Notation "l /F/ x → y" := (fractran_step l x y) (at level 70, no associativity).

(* Specifications: *)
Theorem dio_rel_fractran_rt l x y : 𝔻P x -> 𝔻P y -> 𝔻R (fun ν => fractran_compute l (x ν) (y ν)).
Theorem dio_rel_fractran_stop l x : 𝔻P x -> 𝔻R (fun ν => fractran_stop l (x ν)).
Theorem FRACTRAN_HALTING_on_diophantine ll x : 𝔻P x -> 𝔻R (fun ν => FRACTRAN_HALTING (ll,x ν)).
Theorem FRACTRAN_HALTING_diophantine_0 ll : 𝔻R (fun ν => FRACTRAN_HALTING (ll,ν 0)).
Theorem FRACTRAN_HALTING_diophantine l x : 𝔻R (fun _ => FRACTRAN_HALTING (l,x)).