Proof using.
intros.
unfold entries_match. intuition.
- match goal with
| [ H : _ |- _ ] => solve [eapply H; eauto]
end.
- match goal with
| [ H : forall _ _, _, H' : eIndex ?e1 = eIndex ?e2 |- _ ] =>
specialize (H e1 e2); do 4 concludes
end. intuition.
match goal with
| [ H : forall _, _ < _ <= _ -> _,
_ : eIndex ?e3 <= eIndex _
|- _ ] =>
specialize (H (eIndex e3));
conclude H
ltac:(split; [eauto|
eapply le_trans; eauto; apply maxIndex_is_max; eauto])
end.
break_exists. intuition.
match goal with
| [ _ : In ?x _,
_ : eIndex ?x = eIndex ?e3,
_ : eIndex ?e3 <= eIndex _ |- _ ] =>
eapply rachet with (x' := x); eauto using sorted_uniqueIndices
end.
match goal with
| [ H : _ |- _ ] => solve [ eapply H; eauto; congruence ]
end.
Qed.
Ltac use_entries_match :=
match goal with
| [ _ : eIndex ?e1 = eIndex ?e2,
H : context [entries_match]
|- _ ] =>
first [ solve [eapply H with (e:=e2)(e':=e1); eauto; congruence] |
solve [eapply H with (e:=e1)(e':=e2); eauto; congruence]]
end.

(* Specifications: *)
Lemma entries_match_append :
forall xs ys es ple pli plt,
sorted xs ->
sorted ys ->
sorted es ->
entries_match xs ys ->
(forall e1 e2,
eIndex e1 = eIndex e2 ->
eTerm e1 = eTerm e2 ->
In e1 es ->
In e2 ys ->
(forall e3,
eIndex e3 <= eIndex e1 ->
In e3 es ->
In e3 ys) /\
(pli <> 0 ->
exists e4,
eIndex e4 = pli /\
eTerm e4 = plt /\
In e4 ys)) ->
(forall i, pli < i <= maxIndex es -> exists e, eIndex e = i /\ In e es) ->
(forall e,
In e es ->
pli < eIndex e) ->
findAtIndex xs pli = Some ple ->
eTerm ple = plt ->
pli <> 0 ->
entries_match (es ++ removeAfterIndex xs pli) ys.