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 HnM] HE [q [Hq Hqp]].
assert (L: forall a b, clos_trans Thing Gives a b -> Through b a).
{ intros a b H. induction H as [a b H|a b c Ha IHa Hb IHb].
  - apply H2; exact H.
  - exact (H3 c b a IHb IHa). }
assert (NC: forall c, ~ clos_trans Thing Gives c c).
{ intros c H. apply clos_trans_t1n in H. inversion H; subst.
  - eapply H1; [eassumption | eapply H2; eassumption].
  - match goal with Hx: clos_trans_1n _ _ _ _ |- _ => apply clos_t1n_trans in Hx end.
    eapply H1; [eassumption | eapply L; eassumption]. }
assert (Hqm: MemberOf q A) by (apply (proj2 (HA q Hq)); exact Hqp).
assert (NEss: ~ Ess A).
{ apply H10.
  - exists q; exact Hqm.
  - intros p Hp. assert (pn: p <> A) by (intro e; subst; exact (HnM Hp)).
    pose proof (proj1 (HA p pn) Hp) as Pp. exact (proj2 Pp). }
destruct (classic (ThroughAnother A)) as [[c Hc]|N].
- exists c.
  assert (ne: c <> A) by (intro e; subst; eapply H1; [exact Hc | eapply H2; exact Hc]).
  assert (Ex: Exists c) by (eapply H12; exact Hc).
  assert (Es: Ess c).
  { destruct (classic (Ess c)) as [E|NE]; [exact E|].
    exfalso.
    assert (Pc: Possible c) by (split; assumption).
    pose proof (proj2 (HA c ne) Pc) as Mc.
    exact (NC c (H11 c A c Hc Mc)). }
  assert (NM: ~ MemberOf c A).
  { intro Mc. exact (NC c (H11 c A c Hc Mc)). }
  repeat split; try assumption.
- exfalso. apply NEss. apply H8; assumption.
Qed.

End Pilot.

Print Assumptions imkan_s11_necessary_cause_L5.