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(** * Connection between the guarded, non guarded and omniscient choices *)
(** We show that the guarded formulations of the axiom of choice are equivalent to their "omniscient" variant and comes from the non guarded formulation in presence either of the independence of general premises or subset types (themselves derivable from subtypes thanks to proof- irrelevance) *)
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(** ** AC_rel + PI -> GAC_rel and AC_rel + IGP -> GAC_rel and GAC_rel = OAC_rel *)

(* Specifications: *)
Lemma rel_choice_and_proof_irrel_imp_guarded_rel_choice :
RelationalChoice -> ProofIrrelevance -> GuardedRelationalChoice.
Lemma rel_choice_indep_of_general_premises_imp_guarded_rel_choice :
forall A B : Type, inhabited B -> RelationalChoice_on A B ->
IndependenceOfGeneralPremises -> GuardedRelationalChoice_on A B.
Lemma guarded_rel_choice_imp_rel_choice :
forall A B : Type, GuardedRelationalChoice_on A B -> RelationalChoice_on A B.
Lemma subset_types_imp_guarded_rel_choice_iff_rel_choice :
ProofIrrelevance -> (GuardedRelationalChoice <-> RelationalChoice).