Lemma multipar_CAbs : forall S D A C, multipar S D A C -> forall A1 A2 A3 B, A = a_CAbs (Eq A1 A2 A3) B -> exists B1 B2 B3 C2,
        C = (a_CAbs (Eq B1 B2 B3) C2).
Proof.
  intros S D A C H. induction H; intros; subst.lc_inversion c. destruct IHa as [[T h0]|n].
    auto.
    left; eauto.
    right. intros T h; inversion h; subst; unfold not in n; eauto.
  - lc_inversion c. destruct IHa as [[T h0]|n].
    auto.
    left. exists T. auto.
    right. intros T h; inversion h; subst; unfold not in n; eauto.Qed.(* --------------------------------------------------- *)


(* Proof that joinability implies consistency. *)

Ltac step_left := apply consistent_a_Step_R; [auto |intro N; inversion N; inversion H0]; fail.
Ltac step_right := apply consistent_a_Step_L; [auto | intro N; inversion N; inversion H0]; fail.

(* look for a multipar involvong a head form and apply the appropriate lemma for that
   head form. Note: for paths, the lemma has already been applied so we only need
   to look for a hypothesis about path consistency. *)
Ltac multipar_step SIDE EQ :=
  match goal with
  | [ SIDE : multipar _ _ a_Star _ |- _ ] =>
    apply multipar_Star in SIDE; auto; rename SIDE into EQ
  | [ SIDE : multipar _ _ (a_Pi _ _ _) _ |- _ ] =>
    de

a --> b -->* c d - by prior lemma | | | e - by induction.