Ltac no_paths:=
match goal with
[ EE : erase (a_App ?b0 ?rho ?a) = a_UAbs _ ?b, H : Path ?T ?b0 |- _ ] =>
destruct rho; simpl in EE;
match goal with
[ FF : ?a = a_UAbs _ ?b |- _ ] =>
have P: (Path T a); by eauto using lc_tm_erase end end.
(* Each of the subcases are by induction on a0 to account for top-level coercions on the term. This tactic handles all of the inductive cases. *)
Ltac induction_a0 :=
let IR := fresh  in
let a0' := fresh in
let y := fresh in
let AA := fresh in
let b0 := fresh in
let h0 := fresh in
match goal with
[ IHa0 : ∀ A0 : tm, irrelevant ?G (dom ?G) ?a0 → _,
H1 : irrelevant ?G _ (a_Conv ?a0 ?g),
H2 : AnnTyping ?G (a_Conv ?a0 ?g) ?A0 |- _ ] =>
inversion H2; subst;
(have IR: irrelevant G (dom G) a0 by solve_irrelevant y AA b0 h0);
move: (IHa0 _ IR ltac:(eauto) ltac:(auto)) =>
[a0' [? ?]];
exists (a_Conv a0' g);
split; [ eapply multi_An_ConvTerm; eauto with lc | simpl; auto ]
end.

(* Specifications: *)
Lemma reduction_annotation : forall a a',
reduction_in_one a a' ->
forall G a0 A0, irrelevant G (dom G) a0 -> AnnTyping G a0 A0 -> erase a0 = a ->
exists a0', multi (head_reduction G) a0 a0' /\ erase a0' = a'.