(* -----  exists forms of the binding constructors ----------- *)

Axiom An_PiCong_inversion : forall (G:context) (D:available_props) (rho:relflag) (g1 g2:co) (C1 C2 :tm),
AnnDefEq G D (g_PiCong rho g1 g2) C1 C2 ->
exists A1 B1 A2 B2 B3,
C1 = (a_Pi rho A1 B1) /\
C2 = (a_Pi rho A2 B3) /\
AnnTyping G (a_Pi rho A1 B1) a_Star /\
AnnTyping G (a_Pi rho A2 B3) a_Star /\
AnnTyping G (a_Pi rho A1 B2) a_Star /\
AnnDefEq G D g1 A1 A2 /\
(forall x , x \notin dom G  ->
AnnDefEq  ((x ~ Tm  A1) ++ G) D (open_co_wrt_tm g2 (a_Var_f x)) (open_tm_wrt_tm B1 (a_Var_f x)) ((open_tm_wrt_tm B2 (a_Var_f x)))  /\
(open_tm_wrt_tm B3 (a_Var_f x)  = (open_tm_wrt_tm  B2 (a_Conv (a_Var_f x) (g_Sym g1))))).