Section decidable_upto.
(* decidability of existential quantification for P : X -> Prop when X is finite upto some binary relation R which is a morphism for P *)
Variable (X : Type) (R : X -> X -> Prop)
(P : X -> Prop) (HP : forall x, decidable (P x))
(HR : forall x y, R x y -> P x <-> P y).

Theorem decidable_list_upto_ex l :
(forall x, exists y, In y l /\ R x y)
-> decidable (exists x, P x).