Section finite_t_weak_dec_powerset.
(* We built a list containing all weakly decidable predicates, ie the weakly decidable powerset build as from atoms x = _ and x <> _  *)
Variable (X : Type).
Let wdec (R : X -> Prop) := forall x, R x \/ ~ R x.
Let pset_fin_t (l : list X) : { ll | forall R (_ : wdec R),
exists T, In T ll
/\ forall x, In x l -> R x <-> T x }.
End finite_t_weak_dec_powerset.
(* We show that there is a finite_t bound over weakly (hence also strongly) decidable binary relations upto equivalence. Notice that it is not guaranteed that the list l below contains only decidable relations *)

Theorem finite_t_weak_dec_rels X :
finite_t X -> { l | forall R : X -> X -> Prop,
(forall x y, R x y \/ ~ R x y)
-> exists T, In T l /\ forall x y, R x y <-> T x y }.