Section OutputGreatestId.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Context {lmi : log_matching_interface}.
Context {si : sorted_interface}.
Context {aemi : applied_entries_monotonic_interface}.
Context {smsi : state_machine_safety_interface}.
Context {misi : max_index_sanity_interface}.
Context {smci : state_machine_correct_interface}.
Context {lacimi : lastApplied_commitIndex_match_interface}.
Proof using.
intros. unfold key_in_output_trace in *.
break_exists_exists.
intuition. do_in_app; intuition.
exfalso. eauto.
Qed.
Proof using.
intros.
unfold key_in_output_list in *.
break_exists; do_in_app; intuition eauto.
Qed.
Proof using.
intuition.
unfold key_in_output_list in *.
break_exists; intuition.
Qed.
Proof using.
intros. unfold doLeader, advanceCommitIndex in *.
repeat break_match; find_inversion; intuition;
find_apply_lem_hyp key_in_output_list_empty; auto.
Qed.
Proof using.
intros. unfold handleInput, handleTimeout, handleClientRequest, tryToBecomeLeader in *.
repeat break_match; find_inversion; intuition eauto using key_in_output_list_empty;
unfold key_in_output_list in *; break_exists; simpl in *; intuition; congruence.
Qed.
Proof using.
intros. unfold has_key. break_match; subst; simpl in *.
break_if; repeat (do_bool; intuition).
Qed.

Lemma has_key_true_necessary :
forall client id e,
has_key client id e = true ->
eClient e = client /\ eId e = id.