Fact fractran_step_no_zero_den l x y : fractran_regular l -> l // x → y -> x = 0 -> y = 0.Fact fractran_step_no_zero_num l : Forall (fun c => fst c <> 0) l -> forall x y, l // x → y -> y = 0 -> x = 0.Fact fractran_rt_no_zero_den l n y : fractran_regular l -> fractran_steps l n 0 y -> y = 0.Fact fractran_rt_no_zero_num l : Forall (fun c => fst c <> 0) l -> forall n x, fractran_steps l n x 0 -> x = 0.Fact fractran_zero_num l x : Exists (fun c => fst c = 0) l -> exists y, l // x → y.Corollary FRACTRAN_HALTING_0_num l x : Exists (fun c => fst c = 0) l -> FRACTRAN_HALTING (l,x) <-> False.

Lemma fractran_step_zero l : Forall (fun c => fst c <> 0) l -> forall x y, x <> 0 -> l // x → y <-> remove_zero_den l // x → y.