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 imkan_s9_cause_outside_L5 : MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\ c <> A /\ ~ Possible c.
Proof.
intros H1 H2 H3 H11 A c HA HG.
assert (L1: forall a b, clos_trans Thing Gives a b -> Through b a).
{ intros a b H. induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].
  - apply H2; exact Hg.
  - eapply H3; [exact IH2 | exact IH1]. }
assert (L2: forall a b, clos_trans Thing Gives a b -> exists y, Gives y b /\ (y = a \/ clos_trans Thing Gives a y)).
{ intros a b H. induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].
  - exists x. split; [exact Hg | left; reflexivity].
  - destruct IH2 as [w [Hw [Hw1 | Hw1]]].
    + exists w. split; [exact Hw | right]. subst w. exact Hxy.
    + exists w. split; [exact Hw | right]. eapply t_trans; [exact Hxy | exact Hw1]. }
assert (Hne: c <> A).
{ intro e. subst. exact (H1 _ _ HG (H2 _ _ HG)). }
assert (Hnm: ~ MemberOf c A).
{ intro Hm. pose proof (H11 _ _ _ HG Hm) as Hc.
  destruct (L2 _ _ Hc) as [y [Hy [e | Hy2]]].
  - subst y. exact (H1 _ _ Hy (H2 _ _ Hy)).
  - exact (H1 _ _ Hy (L1 _ _ Hy2)). }
repeat split; auto.
intro Hp. apply Hnm. destruct HA as [HA1 _]. apply HA1; auto.
Qed.

End Pilot.

Print Assumptions imkan_s9_cause_outside_L5.