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_s11_necessary_cause_L5 : MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\ Possible p) -> exists c, Gives c A /\ Necessary c /\ ~ MemberOf c A /\ c <> A.
Proof.
intros H1 H2 H3 H8 H10 H11 H12 A [HA HnA] HEx [p [Hp Hpp]].
assert (Hmem: MemberOf p A) by (apply (proj2 (HA p Hp)); exact Hpp).
assert (HnE: ~Ess A).
{ apply H10. exists p; exact Hmem. intros q Hq. assert (Hq2: q <> A) by (intro E; subst; exact (HnA Hq)). pose proof (proj1 (HA q Hq2) Hq) as Hq3. destruct Hq3 as [_ H]. exact H. }
assert (HT: ThroughAnother A) by (apply NNPP; intro HN; apply HnE; apply H8; auto).
destruct HT as [c Hc].
assert (Hth: forall x y, clos_trans Thing Gives x y -> Through y x).
{ intros x y h. induction h as [x y Hg | x y z h1 IH1 h2 IH2]. exact (H2 y x Hg). exact (H3 z y x IH2 IH1). }
assert (Hcl: forall x y, clos_trans Thing Gives x y -> exists w, Gives w y /\ (w = x \/ clos_trans Thing Gives x w)).
{ intros x y h. induction h as [x y Hg | x y z h1 IH1 h2 IH2].
  - exists x; auto.
  - destruct IH2 as [w [Hw [E|C]]]; exists w; split; auto.
    + right; subst; exact h1.
    + right; eapply t_trans; eauto. }
assert (Hnm: ~MemberOf c A).
{ intro Hm. pose proof (H11 c A c Hc Hm) as Hcc. destruct (Hcl c c Hcc) as [w [Hw [E|C]]].
  - subst. exact (H1 _ _ Hw (H2 _ _ Hw)).
  - exact (H1 c w Hw (Hth c w C)). }
assert (Hne: c <> A).
{ intro E. subst. exact (H1 _ _ Hc (H2 _ _ Hc)). }
refine (ex_intro _ c (conj Hc (conj _ (conj Hnm Hne)))).
split.
- exact (H12 c A Hc).
- apply NNPP; intro Hn. apply Hnm. apply (proj2 (HA c Hne)). split; [exact (H12 c A Hc) | exact Hn].
Qed.

End Pilot.

Print Assumptions imkan_s11_necessary_cause_L5.