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 pr003_no_circle_L11 : MP002 -> MP003 -> forall x y, clos_trans Thing DependsOn x y -> Through x y.
Proof.
intros A2 A3 x y H. induction H as [x y H | x y z H1 IH1 H1' IH2].
- unfold MP002 in A2. apply A2. exact H.
- unfold MP003 in A3. exact (A3 x y z IH1 IH2).
Qed.

Lemma pr003_no_circle_L23 : forall x y, clos_trans Thing DependsOn x y -> clos_trans_1n Thing DependsOn x y.
Proof.
intros x y H. apply clos_trans_t1n_iff. exact H.
Qed.

Lemma pr003_no_circle_L35 : forall x y, clos_trans_1n Thing DependsOn x y -> Gives y x \/ (exists z, Gives z x /\ clos_trans_1n Thing DependsOn z y).
Proof.
intros x y H. destruct H as [y0 H | y0 z H H0].
- left. exact H.
- right. exists y0. split; assumption.
Qed.

Lemma pr003_no_circle_L40 : forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y.
Proof.
intros x y H. induction H as [x y H | x y z H1 H2 IH].
- apply t_step. exact H.
- eapply t_trans.
  + apply t_step. exact H1.
  + exact IH.
Qed.

Lemma pr003_no_circle_L30 : forall x y, clos_trans_1n Thing DependsOn x y -> exists z, Gives z x /\ (z = y \/ clos_trans Thing DependsOn z y).
Proof.
intros x y H. destruct (pr003_no_circle_L35 x y H) as [Hg | [z [Hz Hc]]].
- exists y. split; [exact Hg | left; reflexivity].
- exists z. split; [exact Hz | right; apply pr003_no_circle_L40; exact Hc].
Qed.

Lemma pr003_no_circle_L18 : forall x, clos_trans Thing DependsOn x x -> exists y, Gives y x /\ (y = x \/ clos_trans Thing DependsOn y x).
Proof.
intros x H. destruct (pr003_no_circle_L30 x x (pr003_no_circle_L23 x x H)) as [z [Hz Hr]]. exists z. split; [exact Hz | exact Hr].
Qed.

Lemma pr003_no_circle_L6 : MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x.
Proof.
intros H1 H2 H3 x H. destruct (pr003_no_circle_L18 x H) as [y [Hg [E|Hc]]]. - subst. exact (H1 _ _ Hg (H2 _ _ Hg)). - exact (H1 x y Hg (pr003_no_circle_L11 H2 H3 y x Hc)).
Qed.

End Pilot.

Print Assumptions pr003_no_circle_L6.