Module unique := fc_unique wf subst.
Set Bullet Behavior "Strict Subproofs".
Set Implicit Arguments.
(* ------------------------------------------------------------- *)
(* -------------------------------------------------------------- *)
(*  Interlude: some lemmas about how erasure and typing interact. *)
(* -------------------------------------------------------------- *)
(* Lemma erase_abs : forall AB0 rho B, erase AB0 = (a_UAbs rho B) -> exists A1 B0, erase AB0 = erase (a_Abs rho A1 B0) /\ erase B0 = B. *)
(* Lemma erase_capp : forall AB0 A C G, erase AB0 = (a_CApp A g_Triv) -> AnnTyping G AB0 C -> exists A1 g0 g D, erase AB0 = erase (a_CApp A1 g0) /\ erase A1 = A /\ AnnTyping G (a_CApp A1 g0) D /\ AnnDefEq G (dom G) g C D. *)
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(* ------------------------------------------------------------- *)
Definition  AnnDefEq_regularity :
(forall G D g A B,   AnnDefEq G D g A B ->
exists C1 C2 g', AnnTyping G A C1 /\ AnnTyping G B C2
/\ AnnDefEq G (dom G) g' C1 C2) :=
fourth AnnDefEqAnnIso_regularity.
Definition AnnIso_regularity :
forall G D g phi1 phi2, AnnIso G D g phi1 phi2 ->
AnnPropWff G phi1 /\ AnnPropWff G phi2 :=
third AnnDefEqAnnIso_regularity.
(* --------------------------------------------------------- *)
(* Smart constructors for the annotated language *)
(* ---------------------------------------------- *)
(* ---------------------------------------------- *)

Lemma An_Pi_exists2
: ∀ (x : atom) (G : list (atom * sort)) (rho : relflag) (A B : tm),
x `notin` union (dom G) (fv_tm_tm_tm B)
→ AnnTyping ([(x, Tm A)] ++ G) (open_tm_wrt_tm B (a_Var_f x)) a_Star
→ AnnTyping G (a_Pi rho A B) a_Star.