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_s10_cause_necessary_L17 : MP002 -> MP003 -> forall c d, clos_trans Thing Gives c d -> Through d c.
Proof.
unfold MP002, MP003. intros H2 H3 c d H. induction H as [x y Hxy | x y z Hxy IH1 Hyz IH2].
- apply H2. exact Hxy.
- exact (H3 z y x IH2 IH1).
Qed.

Lemma imkan_s10_cause_necessary_L22 : forall c, clos_trans Thing Gives c c -> exists y, Gives y c /\ (y = c \/ clos_trans Thing Gives c y).
Proof.
intros c H. apply clos_trans_tn1 in H. inversion H; subst.
- exists c. split; auto.
- eexists. split; [eassumption | right; apply clos_trans_tn1_iff; eassumption].
Qed.

Lemma imkan_s10_cause_necessary_L12 : MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c.
Proof.
intros H1 H2 H3 c Hc. destruct (imkan_s10_cause_necessary_L22 c Hc) as [y [Hg [He|Hy]]]. - subst. exact (H1 _ _ Hg (H2 _ _ Hg)). - exact (H1 _ _ Hg (imkan_s10_cause_necessary_L17 H2 H3 _ _ Hy)).
Qed.

Lemma imkan_s10_cause_necessary_L27 : MP001 -> MP002 -> forall A c, Gives c A -> c <> A.
Proof.
intros H1 H2 A c H E. subst. exact (H1 _ _ H (H2 _ _ H)).
Qed.

Lemma imkan_s10_cause_necessary_L32 : MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> c <> A -> ~ Ess c -> clos_trans Thing Gives c c.
Proof.
intros H11 H12 A c HAg Hg Hne Hn.
unfold AggregateOfPossibles in HAg.
destruct HAg as [HA _].
unfold MP011 in H11.
unfold MP012 in H12.
apply (H11 c A c Hg).
apply (proj2 (HA c Hne)).
unfold Possible.
split.
- exact (H12 c A Hg).
- exact Hn.
Qed.

Lemma imkan_s10_cause_necessary_L6 : MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c.
Proof.
intros H1 H2 H3 H11 H12 A c HA HG. unfold Necessary. split. - exact (H12 _ _ HG). - apply NNPP. intro HE. apply (imkan_s10_cause_necessary_L12 H1 H2 H3 c). exact (imkan_s10_cause_necessary_L32 H11 H12 A c HA HG (imkan_s10_cause_necessary_L27 H1 H2 A c HG) HE).
Qed.

End Pilot.

Print Assumptions imkan_s10_cause_necessary_L6.