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_s8_no_internal_cause_L7 : MP001 -> MP002 -> forall x, ~ Gives x x.
Proof.
intros; firstorder.
Qed.

Lemma imkan_s8_no_internal_cause_L12 : MP001 -> MP002 -> MP003 -> MP011 -> forall A c, MemberOf c A -> ~ Gives c A.
Proof.
intros H1 H2 H3 H11 A c Hm Hg.
assert (K: forall a b, clos_trans Thing Gives a b -> exists m, (a=m \/ clos_trans Thing Gives a m) /\ Gives m b).
{ intros a b H. induction H as [a b Hab | a y b Hay IH1 Hyb IH2].
  - exists a. split; [left; reflexivity | exact Hab].
  - destruct IH2 as [m [[E|Hm'] Hg']].
    + exists m. split; [right; rewrite <- E; exact Hay | exact Hg'].
    + exists m. split; [right; eapply t_trans; eauto | exact Hg']. }
assert (T: forall a b, clos_trans Thing Gives a b -> Through b a).
{ intros a b H. induction H as [a b Hab | a y b Hay IH1 Hyb IH2].
  - apply H2; exact Hab.
  - eapply H3; eauto. }
assert (Hc := H11 c A c Hg Hm).
destruct (K c c Hc) as [m [[E|Hm'] Hg']].
- rewrite <- E in Hg'. exact (imkan_s8_no_internal_cause_L7 H1 H2 c Hg').
- apply (H1 c m Hg'). apply T. exact Hm'.
Qed.

Lemma imkan_s8_no_internal_cause_L17 : MP010 -> forall A, AggregateOfPossibles A -> forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> MemberOf B A.
Proof.
intros H10 A [HA HnA] B HBA HEB Hex Hsub.
apply (proj2 (HA B HBA)).
split.
- exact HEB.
- apply H10.
  + exact Hex.
  + intros p Hp.
    assert (H1 : MemberOf p A) by (apply Hsub; exact Hp).
    assert (H2 : p <> A) by (intro e; subst; exact (HnA H1)).
    apply (proj1 (HA p H2)) in H1.
    destruct H1 as [_ H3]; exact H3.
Qed.

Lemma imkan_s8_no_internal_cause_L6 : MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\ ~ Gives A A /\ (forall c, MemberOf c A -> ~ Gives c A) /\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A).
Proof.
intros H1 H2 H3 H10 H11 A HA. refine (conj (imkan_s8_no_internal_cause_L7 H1 H2) (conj _ (conj _ _))).
- intro h. exact (imkan_s8_no_internal_cause_L7 H1 H2 A h).
- intros c hc. exact (imkan_s8_no_internal_cause_L12 H1 H2 H3 H11 A c hc).
- intros B hne hex hm hs hg. exact (imkan_s8_no_internal_cause_L12 H1 H2 H3 H11 A B (imkan_s8_no_internal_cause_L17 H10 A HA B hne hex hm hs) hg).
Qed.

End Pilot.

Print Assumptions imkan_s8_no_internal_cause_L6.