Proof using uii misi smsi lmi lacimi si.
intros.
destruct s as (failed, net).
destruct s' as (failed', net'). simpl in *.
find_apply_lem_hyp step_failure_star_raft_intermediate_reachable.
eauto using in_applied_entries_step_applied_implies_input_state'.
Qed.

Lemma in_input_not_in_output_input_before_output :
forall client id i tr,
in_input_trace client id i tr ->
~ key_in_output_trace client id tr ->
input_before_output client id tr.