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Require Import LPTerm.
Require Import conversion.


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Inductive Sort : Set := SKind | SType | SObject.


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Theorem kind_not_typable2 : forall Γ T t, Γ  t : T -> ~ (t = kind).
Proof. intros. induction H; try easy. Qed.

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Theorem kind_not_typable : forall Γ T, Γ  kind : T -> False.
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Proof. intros. pose proof (kind_not_typable2 _ _ _ H). easy. Qed.
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Lemma not_var_kind_gamma : forall Γ x, Γ  -> x : kind  Γ -> False.
Proof.
  induction Γ ; intros.
  - inversion H0.
  - inversion H0 ; subst.
    + inversion H ; subst. inversion H3 ; subst. eapply no_type_kind ; eassumption.
    + inversion H ; subst ; eapply IHΓ ; eassumption.
  - inversion H0 ; subst ; inversion H ; subst ; eapply IHΓ ; eassumption.
Qed.

Lemma not_rel_kind_gamma_left : forall Γ t, Γ, kind  t  -> False.
Proof.
  intros Γ t H ; inversion H ; subst ; eapply no_type_kind ; eassumption.
Qed.

Lemma not_rel_kind_gamma_right : forall Γ t, Γ, t  kind  -> False.
Proof.
  intros Γ t H ; inversion H ; subst ; eapply no_type_kind ; eassumption.
Qed.

Lemma rel_kind_in_left : forall Γ t, Γ  -> kind  t  Γ -> False.
Proof.
  intros Γ.
  induction Γ ; intros u Γwf H.
  - inversion H.
  - eapply IHΓ ; inversion H ; subst.
    + now inversion Γwf.
    + eassumption.
  - inversion H ; subst.
    + eapply not_rel_kind_gamma_left ; eassumption.
    + eapply IHΓ.
      * now inversion Γwf.
      * eassumption.
Qed.

Lemma rel_kind_in_right : forall Γ t, Γ  -> t  kind  Γ -> False.
Proof.
  intros Γ.
  induction Γ ; intros u Γwf H.
  - inversion H.
  - eapply IHΓ ; inversion H ; subst.
    + now inversion Γwf.
    + eassumption.
  - inversion H ; subst.
    + eapply not_rel_kind_gamma_right ; eassumption.
    + eapply IHΓ.
      * now inversion Γwf.
      * eassumption.
Qed.

Inductive of_typed_kind : term -> Prop :=
| IKType : of_typed_kind type
| IKPiType : forall A B, of_typed_kind B -> of_typed_kind (Π A ~ B).



Lemma inhabit_kind : forall Γ t, Γ  -> Γ  t : kind -> of_typed_kind t.
Proof.
  intros.
  remember kind as x.
  induction H0 ; subst.
  - exfalso ; eapply not_var_kind_gamma ; eassumption.
  - constructor.
  - assert (Γ, x : A ).
    constructor ; eassumption.
    pose proof (IHtyping2 H0 eq_refl).
    now constructor.
  - inversion Heqx.
  -



Lemma inverse_kind : forall Γ t, Γ  -> forall u, Γ  u  t -> u = kind <-> t = kind.
Proof.
  intros.
  induction H0.
  - easy.
  - split ; intro.
    apply IHConv2 ; now apply IHConv1.
    apply IHConv1 ; now apply IHConv2.
  - split ; intro ; now apply IHConv.
  - split ; intro.
    + inversion H0.
    + (* FALSE because t and u might not be typable *) admit.
  - inversion H0 ; split ; intro ; subst.
    + exfalso ; eapply not_rel_kind_gamma_left ; eassumption.
    + exfalso ; eapply not_rel_kind_gamma_right ; eassumption.
    + exfalso ; eapply rel_kind_in_left.
      * inversion H ; subst ; eassumption.
      * eassumption.
    + exfalso ; eapply rel_kind_in_right.
      * inversion H ; subst ; eassumption.
      * eassumption.
    + exfalso ; eapply rel_kind_in_left.
      * inversion H ; subst ; eassumption.
      * eassumption.
    + exfalso ; eapply rel_kind_in_right.
      * inversion H ; subst ; eassumption.
      * eassumption.
  - split. admit. (* a better definition of context is needed here *)
Admitted.




Lemma type_kind : forall Γ A, Γ  type : A -> A = kind.
Proof.
  intros.
  remember type as x.
  induction H ; try inversion Heqx.
  - easy.
  - apply IHtyping1 in H2.
    pose proof (inversion_typing Γ t A H).
    subst.
    pose proof (inverse_kind Γ B H3 kind H1).
    now apply H2.
Qed.