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 }.

Theorem finite_t_weak_dec_powerset :
finite_t X
-> { l | forall R, wdec R -> exists T, In T l
/\ forall x, R x <-> T x }.