Proof with auto.
pose proof I. (* hack: shift up auto-named hyps *)
split; intro.
intros a'' ?.
destruct H1 as [a' ?].
specialize (H0 a').
apply H0. eauto. apply join_comm...
intros a' ?.
destruct H1 as [a'' ?].
rewrite <- (H0 a'') in *; eauto; clear a'' H0.
eapply unit_identity; eauto.
Qed.
(** The elements of a multi-unit separation algebra can be partitioned into equivalance classes, where two elements are in the class iff they have the same unit. *)
Definition comparable {A} `{Sep_alg A}  (a b:A)
:= core a = core b.

(* Specifications: *)
Lemma comparable_refl {A} `{Sep_alg A} : forall a, comparable a a.
Lemma common_unit_comparable {A} `{Sep_alg A} : forall a b,
(exists e, join e a a /\ join e b b) ->
comparable a b.
Lemma join_comparable  {A} `{Sep_alg A}:
forall phi1 phi2 phi3, join phi1 phi2 phi3 -> comparable phi1 phi3.
Lemma join_comparable2  {A} {J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}:
forall phi1 phi2 phi3, join phi1 phi2 phi3 -> comparable phi1 phi2.
Lemma join_sub_comparable  {A} `{Sep_alg A} : forall a c,
join_sub a c -> comparable a c.
Lemma joins_comparable  {A} {J: Join A}{PA: Perm_alg A}{SA: Sep_alg A} : forall a c,
joins a c -> comparable a c.
Lemma join_unit1 {A} `{Perm_alg A}:
forall x y z, unit_for x z -> y = z -> join x y z.
Lemma join_unit2 {A} `{Perm_alg A}:
forall x y z, unit_for y z -> x = z -> join x y z.
Lemma join_unit1_e {A} `{Perm_alg A}:
forall x y z, identity x -> join x y z -> y = z.
Lemma join_unit2_e {A} `{Perm_alg A}:
forall x y z, identity y -> join x y z -> x = z.
Lemma PermAlg_ext:
forall (T: Type) (J: @Join T) (sa1 sa2: @Perm_alg T J), sa1=sa2.
Lemma Sep_alg_ext {T} {J} {PA: @Perm_alg _ J}:
forall (sa1 sa2: @Sep_alg T J), sa1=sa2.