Proof using.
intros.
destruct (pBody p) eqn:?; simpl in *; repeat break_let; repeat find_inversion;
simpl in *; intuition; subst; simpl in *.
- exfalso. eapply handleRequestVote_doesn't_send_AE; eauto using mEntries_some_is_applied_entries.
- exfalso. eapply handleAppendEntries_doesn't_send_AE; eauto using mEntries_some_is_applied_entries.
- exfalso.
eapply handleAppendEntriesReply_doesn't_send_AE;
eauto using mEntries_some_is_applied_entries.
Qed.

Lemma applied_implies_input_in_input_trace :
forall net failed net' failed' tr,
raft_intermediate_reachable net ->
@step_failure _ _ failure_params (failed, net) (failed', net') tr ->
~ applied_implies_input_state client id i net ->
applied_implies_input_state client id i net' ->
in_input_trace client id i tr.