Section LeaderCompleteness.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Context {rri : raft_refinement_interface}.
Context {pwti : prefix_within_term_interface}.
Context {lltsi : leaderLogs_term_sanity_interface}.
Context {llpi : leaderLogs_preserved_interface}.
Context {eewci : every_entry_was_created_interface}.
Context {llvwli : leaderLogs_votesWithLog_interface}.
Context {aevwli : allEntries_votesWithLog_interface}.
Context {vwlsi : votesWithLog_sorted_interface}.
Context {taifoi : terms_and_indices_from_one_interface}.
Context {lllmi : leaderLogs_entries_match_interface}.
Fixpoint contradicting_leader_logs_on_leader l t e :=
match l with
| (t', log') :: l' =>
if (sumbool_and _ _ _ _
(lt_dec t t')
(sumbool_not _ _
(in_dec entry_eq_dec e log'))) then
(t', log') :: contradicting_leader_logs_on_leader l' t e
else
contradicting_leader_logs_on_leader l' t e
| [] => []
end.
Fixpoint contradicting_leader_logs net nodes t e : list (term * name * list entry) :=
match nodes with
| h :: nodes' => (map (fun l => (fst l, h, snd l))
(contradicting_leader_logs_on_leader (leaderLogs (fst (nwState net h))) t e))
++ contradicting_leader_logs net nodes' t e
| [] => []
end.
Definition minimal_contradicting_leader_log net t e :=
argmin (fun l => fst (fst l)) (contradicting_leader_logs net nodes t e).
Proof using.
induction l; intros; simpl in *; intuition; subst;
repeat break_match; subst; simpl in *; intuition eauto;
congruence.
Qed.
Proof using.
intros.
induction nodes; simpl in *; intuition; subst;
find_apply_lem_hyp app_eq_nil; intuition;
find_apply_lem_hyp map_eq_nil; auto.
Qed.
Proof using.
intros.
find_apply_lem_hyp argmin_None.
eapply contradicting_leader_logs_on_leader_empty; eauto.
eapply contradicting_leader_logs_empty; eauto.
apply all_fin_all.
Qed.

Lemma In_contradicting_logs_on_leader_elim :
forall l t e t' l',
In (t', l') (contradicting_leader_logs_on_leader l t e) ->
In (t', l') l /\ t < t' /\ ~ In e l'.