Definition midObj {A} {JA: Join A} (a : A) : Prop := ~identity a /\ ~ full a.Definition ijoinable A {JA: Join A} : Type := {sh : A & midObj sh}.Definition ijoin {A} {JA: Join A} (j1 j2 j3 : ijoinable A) : Prop :=
  match (j1, j2, j3) with
  (existT _ t1 _, existT _ t2 _, existT _ t3 _) => join t1 t2 t3
  end.

Lemma ijoin_assoc {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A} : forall a b c d e, ijoin a b d -> ijoin d c e -> {f : ijoinable A | ijoin b c f /\ ijoin a f e}.