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_s7_cause_not_member_L11 : MP001 -> MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> exists x, Gives a x /\ (x = b \/ Through b x).
Proof.
intros H1 H2 H3 a b H. induction H as [a b Hab | a b c Hab IH1 Hbc IH2].
- exists b; split; [exact Hab | left; reflexivity].
- destruct IH1 as [x [Hax Hx]]. destruct IH2 as [y [Hby Hy]].
  assert (Tcb : Through c b).
  { destruct Hy as [Hy|Hy].
    - subst y. apply H2; exact Hby.
    - apply H3 with y; [exact Hy | apply H2; exact Hby]. }
  exists x; split; [exact Hax|]. right.
  destruct Hx as [Hx|Hx].
  + subst x. exact Tcb.
  + apply H3 with b; assumption.
Qed.

Lemma imkan_s7_cause_not_member_L16 : MP001 -> MP002 -> forall a x, Gives a x -> (x = a \/ Through a x) -> False.
Proof.
intros H1 H2 a x G [E|T].
- subst. exact (H1 _ _ G (H2 _ _ G)).
- exact (H1 _ _ G T).
Qed.

Lemma imkan_s7_cause_not_member_L6 : MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A.
Proof.
intros h1 h2 h3 h11 c A hg hm. pose proof (h11 c A c hg hm) as H. destruct (imkan_s7_cause_not_member_L11 h1 h2 h3 c c H) as [x [hx hor]]. exact (imkan_s7_cause_not_member_L16 h1 h2 c x hx hor).
Qed.

End Pilot.

Print Assumptions imkan_s7_cause_not_member_L6.