(* OLD Inductive Path_consistent : const -> tm -> tm -> Prop := PC_Const : forall T, Path_consistent T (a_Const T) (a_Const T) | PC_App   : forall T a1 rho a2 b1 b2, erased_tm a2 -> erased_tm b2 -> Path_consistent T a1 b1 -> Path_consistent T (a_App a1 rho a2) (a_App b1 rho b2) | PC_CApp  : forall T a1 b1, Path_consistent T a1 b1 -> Path_consistent T (a_CApp a1 g_Triv) (a_CApp b1 g_Triv). Hint Constructors Path_consistent. *)
Hint Constructors Path_consistent.

Lemma Par_Path_consistent_App : forall T G D a1 a2 b1 b2,
Path_consistent T (a_App a1 Rel a2) (a_App b1 Rel b2) ->
Par G D (a_App a1 Rel a2) ( a_App b1 Rel b2) ->
Par G D a1 b1.