From Stdlib Require Import Classical Relations.

Section Pilot.
(* Vocabulary.  Nothing is assumed about any of these. *)
Variable Thing : Type.
Variable Gives : Thing -> Thing -> Prop.     (* Gives y x : y gives existence to x *)
Variable Through : Thing -> Thing -> Prop.   (* Through x y : x has its existence through y *)
Variable Exists : Thing -> Prop.             (* x exists *)
Variable Ess : Thing -> Prop.                (* x's existence is from its own nature *)
Variable PartOf : Thing -> Thing -> Prop.    (* p is a part of u *)
Variable Universe : Thing.                   (* the universe *)

(* Definitions. *)
Definition DependsOn (x y : Thing) : Prop := Gives y x.            (* x depends on y *)
Definition ThroughAnother (x : Thing) : Prop := exists y, Gives y x.
Definition OfItself (x : Thing) : Prop := Exists x /\ ~ ThroughAnother x.
Definition Necessary (x : Thing) : Prop := Exists x /\ Ess x.
Definition Possible (x : Thing) : Prop := Exists x /\ ~ Ess x.
Definition InUniverse (x : Thing) : Prop := x = Universe \/ PartOf x Universe.

(* Approved postulates, written as definitions: a theorem uses one only when
   its statement lists it as a condition. *)
Definition MP001 : Prop := forall x y, Gives y x -> ~ Through y x.   (* a giver does not have its existence through what it gives existence to *)
Definition MP002 : Prop := forall x y, Gives y x -> Through x y.     (* the receiver has its existence through the giver *)
Definition MP003 : Prop := forall x y z, Through x y -> Through y z -> Through x z.  (* existence-through is inherited *)
Definition MP008 : Prop := forall x, Exists x -> ~ ThroughAnother x -> Ess x.       (* existence comes from one's nature or from a giver *)
Definition OP001 : Prop := forall x y, Gives y x -> OfItself y.      (* only what exists of itself gives existence *)
Definition OP006 : Prop := exists x, InUniverse x /\ Possible x.     (* the universe is not wholly necessary *)
Definition SP001 : Prop := exists x, Exists x.                       (* something exists *)

(* Added for the full test: aggregates, possible worlds, and their postulates. *)
Variable MemberOf : Thing -> Thing -> Prop.  (* p is a member of the aggregate a *)
Variable World : Type.
Variable ExistsIn : World -> Thing -> Prop.  (* x exists in world w *)
Definition Independent (x : Thing) : Prop := forall y, ~ DependsOn x y.
Definition NonExistencePossible (x : Thing) : Prop := exists w, ~ ExistsIn w x.
Definition AggregateOfPossibles (A : Thing) : Prop :=
  (forall p, p <> A -> (MemberOf p A <-> Possible p)) /\ ~ MemberOf A A.
Definition MP007 : Prop := forall x, Ess x -> forall w, ExistsIn w x.        (* existence from one's nature cannot fail *)
Definition MP009 : Prop := (exists p, Possible p) -> exists A, AggregateOfPossibles A /\ Exists A.  (* the aggregate of possibles exists *)
Definition MP010 : Prop := forall A, (exists p, MemberOf p A) ->
  (forall p, MemberOf p A -> ~ Ess p) -> ~ Ess A.                            (* possibles do not add up to a necessary *)
Definition MP011 : Prop := forall c A p, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p.  (* giving to a whole gives to its members *)
Definition MP012 : Prop := forall y x, Gives y x -> Exists y.               (* a giver exists *)

Lemma pr001_relational_exhaustive_L5 : forall x, (exists y, (x = y \/ clos_trans Thing DependsOn x y) /\ DependsOn y y) \/ (exists y, (x = y \/ clos_trans Thing DependsOn x y) /\ clos_trans Thing DependsOn y y /\ ~ DependsOn y y) \/ (exists S : Thing -> Prop, S x /\ forall a, S a -> exists b, S b /\ DependsOn a b) \/ ((forall y, ~ DependsOn x y) \/ exists y, clos_trans Thing DependsOn x y /\ forall z, ~ DependsOn y z).
Proof.
intro x.
destruct (classic (forall a, (a = x \/ clos_trans Thing DependsOn x a) -> exists b, (b = x \/ clos_trans Thing DependsOn x b) /\ DependsOn a b)) as [H|H].
- right; right; left. exists (fun a => a = x \/ clos_trans Thing DependsOn x a). split.
  + left; reflexivity.
  + exact H.
- apply not_all_ex_not in H. destruct H as [a Ha].
  pose proof (not_imply_elim _ _ Ha) as H1.
  pose proof (not_imply_elim2 _ _ Ha) as Hn.
  assert (Hno: forall b, ~ DependsOn a b).
  { intros b Hab. apply Hn. exists b. split.
    - right. destruct H1 as [E|Hc].
      + rewrite E in Hab. apply t_step. exact Hab.
      + eapply t_trans. exact Hc. apply t_step. exact Hab.
    - exact Hab. }
  right; right; right.
  destruct H1 as [E|Hc].
  + left. rewrite <- E. exact Hno.
  + right. exists a. split; assumption.
Qed.

End Pilot.

Print Assumptions pr001_relational_exhaustive_L5.