Section zero_cases.
Fact fractran_zero_num_step l :
Exists (fun c => fst c = 0) l
-> forall x, exists y, l /F/ x → y.

Lemma FRACTRAN_HALTING_zero_num l x :
Exists (fun c => fst c = 0) l
-> FRACTRAN_HALTING (l,x) <-> False.