Inductive wf : env -> Prop :=
| wf_nil : wf nil
| wf_var :
forall (e : env) (T : term) (s : sort),
typ e T (Srt s) -> wf (Ax T :: e)
| wf_cst :
forall (e : env) (t T : term) (s : sort),
typ e t T -> typ e T (Srt s) -> wf (Def t T :: e)
with typ : env -> term -> term -> Prop :=
| type_ax :
forall (e : env) (s1 s2 : sort),
wf e -> axiom s1 s2 -> typ e (Srt s1) (Srt s2)
| type_var :
forall (e : env) (v : nat) (t : term),
wf e -> item_lift t e v -> typ e (Ref v) t
| type_abs :
forall (e : env) (T U M : term) (s : sort),
typ e (Prod T U) (Srt s) ->
typ (Ax T :: e) M U -> typ e (Abs T M) (Prod T U)
| type_app :
forall (e : env) (u v V Ur : term),
typ e v V -> typ e u (Prod V Ur) -> typ e (App u v) (subst v Ur)
| type_prod :
forall (e : env) (T U : term) (s1 s2 s3 : sort),
typ e T (Srt s1) ->
typ (Ax T :: e) U (Srt s2) ->
rule s1 s2 s3 -> typ e (Prod T U) (Srt s3)
| type_conv :
forall (e : env) (t U V : term) (s : sort),
typ e t U ->
le_type e U V ->
typ e V (Srt s) ->
typ e t V
(* sert uniquement quand il y a des sortes non typees: *)
| type_conv_srt :
forall (e : env) (t U : term) (s : sort),
typ e t U -> le_type e U (Srt s) -> typ e t (Srt s).
Hint Resolve wf_nil type_ax type_var: pts.
Definition rule_sat (s1 s2 s3 : sort) : Prop :=
exists2 x1 : _,
le_sort s1 x1 & (exists2 x2 : _, le_sort s2 x2 & rule x1 x2 s3).
intros.
inversion_clear H1.
inversion_clear H3.
apply type_prod with x x0; trivial with arith pts.
apply type_conv_srt with (Srt s1); auto with arith pts.
apply type_conv_srt with (Srt s2); auto with arith pts.
Qed.
simple induction 1; auto with arith pts.
Qed.
intros.
inversion_clear H.
apply typ_wf with T (Srt s); auto with arith pts.
apply typ_wf with T (Srt s); auto with arith pts.
Qed.
intros.
inversion_clear H; exists s; auto with arith pts.
Qed.
Inductive wf_type (e : env) : term -> Prop :=
| wft_typed :
forall (T : term) (s : sort), typ e T (Srt s) -> wf_type e T
| wft_top : forall s : sort, wf_type e (Srt s).
Hint Resolve wft_top: pts.
Definition le_type_wf : red_rule :=
fun e => clos_refl_trans _ (fun x y => le_type e x y /\ wf_type e y).
(* Lemma le_type_wf_le: (e:env)(x,y:term)(le_type_wf e x y)->(le_type e x y). *)
Definition inv_typ (e : env) (t T : term) (P : Prop) : Prop :=
match t with
| Srt s1 => forall s2 : sort, axiom s1 s2 -> le_type e (Srt s2) T -> P
| Ref n =>
forall x : decl,
item x e n -> le_type e (lift (S n) (typ_of_decl x)) T -> P
| Abs A M =>
forall (s : sort) (U : term),
typ (Ax A :: e) M U ->
typ e (Prod A U) (Srt s) -> le_type e (Prod A U) T -> P
| App u v =>
forall Ur V : term,
typ e v V -> typ e u (Prod V Ur) -> le_type e (subst v Ur) T -> P
| Prod A B =>
forall s1 s2 s3 : sort,
rule s1 s2 s3 ->
typ e A (Srt s1) ->
typ (Ax A :: e) B (Srt s2) -> le_type e (Srt s3) T -> P
end.
do 5 intro.
cut (forall x : term, le_type e x U -> le_type e x V).
case t; simpl in |- *; intros.
apply H1 with s2; auto with arith pts.
apply H1 with x; auto with arith pts.
apply H1 with s U0; auto with arith pts.
apply H1 with Ur V0; auto with arith pts.
apply H1 with s1 s2 s3; auto with arith pts.
intros.
apply le_type_trans with U; auto with arith pts.
Qed.
intros e t T P inv_P j.
generalize P inv_P; clear inv_P P.
elim j; simpl in |- *; intros.
apply inv_P with s2; trivial with arith pts.
inversion_clear H0.
apply inv_P with x; trivial with arith pts.
elim H1; trivial with arith pts.
apply inv_P with s U; auto with arith pts.
apply inv_P with Ur V; auto with arith pts.
apply inv_P with s1 s2 s3; auto with arith pts.
apply H0.
apply inv_typ_conv with V; auto with arith pts.
apply H0.
apply inv_typ_conv with (Srt s); auto with arith pts.
Qed.
Ltac Inversion_typ T :=
let INVTYP := fresh "INVTYP" in
(generalize T; intro INVTYP; inversion INVTYP using inversion_lemma;
unfold inv_typ in |- *; clear INVTYP; intros).
simple destruct 1; try constructor; intros.
apply typ_free_db with (1 := H0).
Qed.
Hint Resolve wft_free_db: pts.
simple induction 1; simpl in |- *; intros; auto with arith pts.
elim (le_gt_dec n v); intros; apply type_var; auto with arith pts.
elim H1; intros.
rewrite H4.
unfold lift in |- *.
rewrite simpl_lift_rec; simpl in |- *; auto with arith pts.
exists x; auto with arith pts.
apply ins_item_ge with (1 := H2); auto with arith pts.
apply ins_item_lift_lt with (1 := H2); auto with arith pts.
cut (wf (Ax (lift_rec 1 T0 n) :: f)).
intro.
apply type_abs with s; auto with arith pts.
fold (lift_decl 1 (Ax T0) n) in |- *; auto with arith pts.
Inversion_typ (H1 _ _ H4 H5).
apply wf_var with s1; auto with arith pts.
rewrite distr_lift_subst.
apply type_app with (lift_rec 1 V n); auto with arith pts.
cut (wf (Ax (lift_rec 1 T0 n) :: f)).
intro.
apply type_prod with s1 s2; auto with arith pts.
fold (lift_decl 1 (Ax T0) n) in |- *; auto with arith pts.
apply wf_var with s1; auto with arith pts.
apply type_conv with (lift_rec 1 U n) s; auto with arith pts.
apply le_type_lift with (2 := H5); trivial with arith pts.
apply type_conv_srt with (lift_rec 1 U n); auto with arith pts.
fold (lift_rec 1 (Srt s) n) in |- *.
apply le_type_lift with (2 := H3); auto with arith pts.
Qed.
unfold lift in |- *.
intros.
apply typ_weak with e d e; auto with arith pts.
Qed.
simple induction n.
intros.
rewrite lift0.
rewrite lift0.
replace e with f; auto with arith pts.
inversion_clear H; auto with arith pts.
do 8 intro.
inversion_clear H0.
intro.
rewrite simpl_lift; auto with arith pts.
pattern (lift (S n0) T) in |- *.
rewrite simpl_lift; auto with arith pts.
inversion_clear H0.
change (typ (Ax T0 :: e0) (lift 1 (lift n0 t)) (lift 1 (lift n0 T))) in |- *.
apply thinning; auto with arith pts.
apply H with f; auto with arith pts.
apply typ_wf with T0 (Srt s); auto with arith pts.
apply wf_var with s; auto with arith pts.
change (typ (Def t0 T0 :: e0) (lift 1 (lift n0 t)) (lift 1 (lift n0 T)))
in |- *.
apply thinning; auto with arith pts.
apply H with f; auto with arith pts.
apply typ_wf with t0 T0; auto with arith pts.
apply wf_cst with s; auto with arith pts.
Qed.
simple induction n.
simple destruct e; intros.
elim H0; intros.
inversion_clear H2.
elim H0; intros.
rewrite H1.
inversion_clear H2.
elim inv_wf_sort with l d; intros; auto with arith pts.
exists x0.
replace (Srt x0) with (lift 1 (Srt x0)); auto with arith pts.
apply thinning; auto with arith pts.
intros.
elim H1; intros.
rewrite H2.
generalize H0.
inversion_clear H3; intros.
rewrite simpl_lift; auto with arith pts.
elim H with l (lift (S n0) (typ_of_decl x)); intros; auto with arith pts.
exists x0.
replace (Srt x0) with (lift 1 (Srt x0)); auto with arith pts.
apply thinning; auto with arith pts.
apply inv_wf with y; auto with arith pts.
exists x; auto with arith pts.
Qed.
intros.
elim item_trunc with (1 := H); intros.
apply thinning_n with x; auto with arith pts.
apply wf_definition with (1 := H); trivial with arith pts.
Qed.
simple induction 2; simpl in |- *; intros; auto with arith pts.
elim (lt_eq_lt_dec n v); intros.
elim H2.
elim a.
case v; intros.
inversion_clear a0.
simpl in |- *.
rewrite H5.
rewrite simpl_subst; auto with arith pts.
apply type_var; auto with arith pts.
exists x0; auto with arith pts.
apply nth_sub_sup with (1 := H3); auto with arith pts.
intros.
rewrite H5.
elim b.
rewrite simpl_subst; auto with arith pts.
apply thinning_n with g; auto with arith pts.
elim H3; intros; auto with arith pts.
rewrite <- b in H6.
elim nth_sub_eq with (1 := H3) (2 := H6).
trivial with arith pts.
apply type_var; auto with arith pts.
apply nth_sub_item_inf with (1 := H3); auto with arith pts.
cut (wf (subst_decl x (Ax T) n :: f)); intros.
apply type_abs with s; auto with arith pts.
replace (Ax (subst_rec x T n)) with (subst_decl x (Ax T) n);
auto with arith pts.
apply
inversion_lemma
with f (Prod (subst_rec x T n) (subst_rec x U0 (S n))) (Srt s);
simpl in |- *; intros; auto with arith pts.
apply wf_var with s1; auto with arith pts.
rewrite distr_subst.
apply type_app with (subst_rec x V n); auto with arith pts.
cut (wf (Ax (subst_rec x T n) :: f)); intros; auto with arith pts.
apply type_prod with s1 s2; auto with arith pts.
replace (Ax (subst_rec x T n)) with (subst_decl x (Ax T) n);
auto with arith pts.
apply wf_var with s1; auto with arith pts.
apply type_conv with (subst_rec x U0 n) s; auto with arith pts.
apply le_type_subst with g d e0; auto with arith pts.
apply type_conv_srt with (subst_rec x U0 n); auto with arith pts.
fold (subst_rec x (Srt s) n) in |- *.
apply le_type_subst with g d e0; auto with arith pts.
Qed.

Theorem substitution :
forall (e : env) (d t u U : term),
typ (Ax t :: e) u U -> typ e d t -> typ e (subst d u) (subst d U).