(* for fixed n i j, the function v => exp i <v(j),...,v(n-1+j)> has a diophantine representation *)
Let exp_dio n i j y : 𝔻P y -> 𝔻R (fun v => y v = exp i (fun2vec j n v)).
(* for a fixed n, the relation  ν 0 = ps 1 * (qs 1)^(ν 1) * ... * (qs n)^(ν n)  has a diophantine representation *)
Hint Resolve exp_dio.
Fact exp_diophantine n : 𝔻R (fun ν => ν 0 = ps 1 * exp 1 (fun2vec 0 n (fun x => ν (S x)))).
End exp_diophantine.
Hint Resolve exp_diophantine.

Theorem FRACTRAN_HALTING_on_exp_diophantine n l :
𝔻R (fun ν => l /F/ ps 1 * exp 1 (fun2vec 0 n ν) ↓).