Module red := ext_red invert.
Export red.
Set Implicit Arguments.
Set Bullet Behavior "Strict Subproofs".
Definition Good (G : context) (D : available_props):=
erased_context G /\
forall c1 A B1 T1,
binds c1 (Co (Eq A B1 T1)) G
-> c1 `in` D
-> exists C, Par G D A C /\ Par G D B1 C.
(* ---------------------------------------- *)

Lemma Par_Abs_inversion : forall G D a b rho,
Par G D (a_UAbs rho a) b ->
(exists a', b = (a_UAbs rho a') /\
forall x, x `notin` fv_tm_tm_tm a \u fv_tm_tm_tm a' ->
Par G D (open_tm_wrt_tm a (a_Var_f x)) (open_tm_wrt_tm a' (a_Var_f x)))
\/
(exists a', Par G D a' b /\ (forall x, x `notin`  fv_tm_tm_tm a ->
open_tm_wrt_tm a (a_Var_f x) = a_App a' rho (a_Var_f x)) /\ rho = Rel)
\/ (exists a', Par G D a' b /\ (forall x, x `notin`  fv_tm_tm_tm a ->
open_tm_wrt_tm a (a_Var_f x) = a_App a' rho a_Bullet) /\ rho = Irrel).