(* -----  inversion lemmas for some typing judgments (with maximal co-finite quantification) ----------- *)

Axiom An_Pi_inversion :
∀ (G:context) rho A B T,
AnnTyping G (a_Pi rho A B) T ->
T = a_Star /\
AnnTyping G A a_Star /\
∀ x, x \notin dom G -> AnnTyping (( x ~ Tm  A) ++ G) (open_tm_wrt_tm B (a_Var_f x)) a_Star.