Proof using.
intros. induction tr; simpl in *.
-unfold input_before_output. simpl in *.
left. break_if; repeat (do_bool; intuition).
- unfold input_before_output. simpl. right. intuition.
+ unfold key_in_output_trace in *.
apply Bool.not_true_iff_false. intuition.
find_false. unfold is_output_with_key in *.
break_match; subst. repeat break_match; try congruence.
exists l, n. intuition.
+ conclude_using
ltac:(intros; find_false; unfold key_in_output_trace in *;
break_exists_exists; intuition).
eauto.
Qed.
Program Instance TR : InverseTraceRelation step_failure :=
{
init := step_failure_init;
T := input_before_output client id;
R := fun s => in_applied_entries client id (snd s)
}.
Next Obligation.
destruct (client_id_in (applied_entries (nwState n))) eqn:?;
eauto using client_id_in_true_in_applied_entries, client_id_in_false_not_in_applied_entries.
Defined.
Next Obligation.
unfold input_before_output in *.
eauto using before_func_app.
Defined.
Next Obligation.
intuition. simpl in *.
unfold in_applied_entries, applied_entries in *. simpl in *.
break_match; simpl in *; break_exists; intuition.
Defined.
Next Obligation.
simpl in *.
find_eapply_lem_hyp in_applied_entries_step_applied_implies_input_state; eauto.
break_or_hyp.
- break_exists. intuition.
find_eapply_lem_hyp applied_implies_input; eauto.
apply before_func_app.
destruct (key_in_output_trace_dec client id tr);
[find_eapply_lem_hyp output_implies_applied; eauto; intuition|].
fold (input_before_output client id tr).
subst. eauto using in_input_not_in_output_input_before_output.
- destruct (key_in_output_trace_dec client id tr);
[find_eapply_lem_hyp output_implies_applied; eauto; intuition|].
break_exists. subst.
eauto using input_before_output_not_key_in_output_trace_snoc_key.
Defined.

Theorem output_implies_input_before_output :
forall failed net tr,
step_failure_star step_failure_init (failed, net) tr ->
key_in_output_trace client id tr ->
input_before_output client id tr.