Section LogsLeaderLogs.
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 {llsi : leaderLogs_sorted_interface}.
Context {rlmli : refined_log_matching_lemmas_interface}.
Context {llci : leaderLogs_contiguous_interface}.
Context {lllmi : leaderLogs_entries_match_interface}.
Context {lhllsi : leaders_have_leaderLogs_strong_interface}.
Context {nisi : nextIndex_safety_interface}.
Context {si : sorted_interface}.
Context {lpholli : log_properties_hold_on_leader_logs_interface}.
Definition weak_sanity pli ll ll' :=
pli = 0 ->
(exists e, eIndex e = 0 /\ In e ll) \/
ll = ll'.
Definition logs_leaderLogs_nw_weak net :=
forall p t n pli plt es ci e,
In p (nwPackets net) ->
pBody p = AppendEntries t n pli plt es ci ->
In e es ->
exists leader ll es' ll',
In (eTerm e, ll) (leaderLogs (fst (nwState net leader))) /\
Prefix ll' ll /\
removeAfterIndex es (eIndex e) = es' ++ ll' /\
(forall e', In e' es' -> eTerm e' = eTerm e) /\
weak_sanity pli ll ll'.
Proof using.
intros. unfold logs_leaderLogs_nw, logs_leaderLogs_nw_weak, weak_sanity in *.
intros.
eapply_prop_hyp In In; eauto.
break_exists_exists; intuition; subst; try lia.
break_exists; intuition; eauto.
Qed.
Definition logs_leaderLogs_inductive net :=
logs_leaderLogs net /\
logs_leaderLogs_nw net.
Proof using si rri.
intros.
eapply lift_prop; eauto using logs_sorted_invariant.
Qed.
Proof using si rri.
intros.
find_apply_lem_hyp lift_sorted.
unfold logs_sorted, logs_sorted_host in *.
intuition.
unfold deghost in *. simpl in *. break_match; eauto.
Qed.
Proof using nisi rri.
intros.
eapply lift_prop; eauto using nextIndex_safety_invariant.
Qed.
Proof using si nisi rlmli rri.
intros.
find_copy_apply_lem_hyp entries_contiguous_invariant.
find_copy_apply_lem_hyp lift_nextIndex_safety.
assert (pred (getNextIndex (snd (nwState net h)) h') > 0) by lia.
unfold nextIndex_safety in *.
match goal with
| H : forall _ _, type _ = _ -> _ |- _ => specialize (H h h')
end.
intuition.
unfold entries_contiguous in *. specialize (H2 h).
unfold contiguous_range_exact_lo in *. intuition.
match goal with
| H : forall _, _ < _ <= _ -> _ |- _ =>
specialize (H (pred (getNextIndex (snd (nwState net h)) h')))
end.
unfold raft_refined_base_params in *.
repeat find_rewrite_lem deghost_spec. intuition.
break_exists_exists. intuition.
apply findAtIndex_intro; eauto using lift_logs_sorted, sorted_uniqueIndices.
Qed.
Proof using.
unfold update_elections_data_client_request in *.
intros. repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
unfold update_elections_data_timeout.
intros.
repeat break_match; simpl in *; auto.
Qed.
Proof using.
intros.
unfold update_elections_data_appendEntries.
repeat break_match; subst; simpl in *; auto.
Qed.
Proof using.
unfold update_elections_data_requestVote.
intros.
repeat break_match; auto.
Qed.
Proof using.
unfold update_elections_data_requestVoteReply.
intros.
repeat break_match; auto.
simpl in *. intuition.
Qed.
Ltac prove_in :=
match goal with
| [ _ : nwPackets ?net = _,
_ : In (?p : packet) _ |- _] =>
assert (In p (nwPackets net)) by (repeat find_rewrite; do_in_app; intuition)
| [ _ : nwPackets ?net = _,
_ : pBody ?p = _ |- _] =>
assert (In p (nwPackets net)) by (repeat find_rewrite; intuition)
end.
Proof using rlmli.
red. intros.
apply entries_contiguous_invariant; auto.
Qed.
Proof using lpholli rlmli.
intros. pattern ll.
eapply log_properties_hold_on_leader_logs_invariant; eauto using contiguous_log_property.
Qed.
Proof using lpholli si lllmi llci rlmli llsi rri.
red. unfold logs_leaderLogs_inductive. intros.
subst. simpl in *. intuition.
- unfold logs_leaderLogs in *. intros.
simpl in *. repeat find_higher_order_rewrite.
update_destruct; subst; rewrite_update; eauto.
+ simpl in *. find_apply_lem_hyp handleAppendEntries_log.
intuition; repeat find_rewrite; eauto.
* find_apply_hyp_hyp. break_exists_exists; intuition.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
* { subst.
find_apply_lem_hyp logs_leaderLogs_nw_weaken.
unfold logs_leaderLogs_nw_weak in *.
copy_eapply_prop_hyp pBody pBody; eauto.
break_exists_exists.
break_exists. intuition.
- repeat find_higher_order_rewrite.
update_destruct; subst; rewrite_update; eauto.
simpl.
rewrite update_elections_data_appendEntries_leaderLogs.
auto.
- repeat find_rewrite. f_equal.
find_copy_apply_lem_hyp leaderLogs_sorted_invariant; eauto.
eapply sorted_Prefix_in_eq; eauto.
intros.
eapply prefix_contiguous with (i := 0); eauto.
+ unfold weak_sanity in *. concludes.
intuition; subst; simpl in *; intuition.
break_exists. intuition.
find_eapply_lem_hyp leaderLogs_contiguous_invariant; eauto.
lia.
+ eapply leaderLogs_contiguous_invariant; eauto.
+ assert (sorted (log d)) by (eapply entries_sorted_nw_invariant; eauto).
eapply contiguous_app with (l1 := x1).
* repeat find_reverse_rewrite.
apply removeAfterIndex_sorted. auto.
* repeat find_reverse_rewrite.
eapply removeAfterIndex_contiguous; eauto.
eapply entries_contiguous_nw_invariant; eauto.
}
* { break_exists. intuition. subst.
do_in_app. intuition.
- (* new entry *)
unfold logs_leaderLogs_nw in *.
prove_in.
copy_eapply_prop_hyp In In; eauto.
match goal with
| H:exists _, _ |- _ => destruct H as [leader]
end.
break_exists; intuition.
+ (* all of the new entries, as well as x, are in the
new term. This means we can apply the host invariant
to x and get our leader. *)
assert (x2 = []).
{
destruct x2; intuition. exfalso.
simpl in *. break_match; intuition.
simpl in *. subst.
find_copy_apply_lem_hyp entries_contiguous_nw_invariant.
eapply_prop_hyp entries_contiguous_nw pBody; eauto.
unfold contiguous_range_exact_lo in *. intuition.
match goal with
| _ : removeAfterIndex _ ?index = _ ++ ?e :: _ |- _ =>
assert (In e es) by
(apply removeAfterIndex_in with (i := index);
repeat find_rewrite; intuition)
end.
eapply_prop_hyp In In. lia.
} subst. rewrite app_nil_r in *.
match goal with
| H : forall _ _, In _ _ -> _ |- _ =>
specialize (H0 (pDst p) x)
end. intuition.
break_exists. intuition.
match goal with
| _ : In (_, ?ll) (leaderLogs (_ (_ _ ?h))),
_ : removeAfterIndex _ _ = ?es' ++ ?ll |- _ =>
exists h, ll, (removeAfterIndex es (eIndex e) ++ es')
end. intuition.
* repeat find_rewrite.
match goal with
| H : forall _, st' _ = _ |- _ =>
rewrite H
end.
update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
* rewrite removeAfterIndex_in_app; auto.
rewrite app_ass. repeat find_rewrite. auto.
* repeat find_rewrite. do_in_app. intuition; eauto.
+ (* we share an entry with the leader, so we can use
log matching to make sure our old entries match. we'll
get the new entries for free from the nw invariant. *)
break_exists; intuition;
match goal with
| [ _ : In (_, ?ll) (leaderLogs (_ (_ _ ?leader))),
_ : removeAfterIndex _ _ = ?es' ++ _ |- _ ] =>
exists leader, ll, es'
end; intuition.
* repeat find_rewrite.
match goal with
| H : forall _, st' _ = _ |- _ =>
rewrite H
end.
update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
* rewrite removeAfterIndex_in_app; auto.
repeat find_rewrite. rewrite app_ass.
find_copy_eapply_lem_hyp leaderLogs_sorted_invariant; eauto.
f_equal.
eapply thing; eauto using lift_logs_sorted;
[eauto using leaderLogs_contiguous|eapply leaderLogs_entries_match_invariant; eauto|].
assert (sorted es) by (eapply entries_sorted_nw_invariant; eauto).
find_copy_apply_lem_hyp entries_contiguous_nw_invariant.
unfold entries_contiguous_nw in *.
copy_eapply_prop_hyp contiguous_range_exact_lo pBody; eauto.
find_eapply_lem_hyp removeAfterIndex_contiguous; eauto.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
find_erewrite_lem H
end.
eapply contiguous_app; eauto.
repeat find_reverse_rewrite. eauto using removeAfterIndex_sorted.
* repeat find_rewrite.
match goal with
| H : forall _, st' _ = _ |- _ =>
rewrite H
end.
update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
* rewrite removeAfterIndex_in_app; auto.
repeat find_rewrite. rewrite app_ass.
f_equal.
assert (x2 = []).
{
destruct x2; intuition. exfalso.
simpl in *. break_match; intuition.
simpl in *. subst.
find_copy_apply_lem_hyp entries_contiguous_nw_invariant.
eapply_prop_hyp entries_contiguous_nw pBody; eauto.
unfold contiguous_range_exact_lo in *. break_and.
match goal with
| _ : removeAfterIndex _ ?index = _ ++ ?e :: _ |- _ =>
assert (In e es) by
(apply removeAfterIndex_in with (i := index);
repeat find_rewrite; intuition)
end.
clear H0 H7.
eapply_prop_hyp In In. lia.
} subst. simpl.
find_copy_eapply_lem_hyp leaderLogs_sorted_invariant; eauto.
match goal with
| |- _ = ?ll =>
rewrite removeAfterIndex_maxIndex_sorted with (l := ll) at 2
end; eauto.
eapply removeAfterIndex_same_sufficient; eauto using lift_logs_sorted;
intros;
match goal with
| _ : eIndex ?e = maxIndex _,
_ : In ?e (log _),
_ : eIndex ?e' = maxIndex _ |- In ?e'' _ =>
assert (eIndex e = eIndex e') as Heq by (repeat find_rewrite; auto);
assert (eIndex e'' <= eIndex e) by lia;
eapply leaderLogs_entries_match_invariant in Heq; eauto;
repeat conclude Heq ltac:(eauto; intuition)
end; intuition.
- (* old entry *)
(* can use host invariant, since we don't change removeAfterIndex *)
find_copy_apply_lem_hyp removeAfterIndex_in.
find_copy_apply_lem_hyp removeAfterIndex_In_le; eauto using lift_logs_sorted.
find_apply_hyp_hyp. break_exists_exists; intuition.
+ find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
+ rewrite removeAfterIndex_in_app_l'; eauto;
[rewrite <- removeAfterIndex_le; auto|].
intros. eapply gt_le_trans; [|eauto].
eapply entries_contiguous_nw_invariant; eauto.
}
+ find_apply_hyp_hyp. break_exists_exists; intuition.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
simpl in *.
rewrite update_elections_data_appendEntries_leaderLogs. auto.
- unfold logs_leaderLogs_nw in *.
intros. simpl in *. find_apply_hyp_hyp.
intuition.
+ prove_in.
copy_eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto;
simpl in *;
rewrite update_elections_data_appendEntries_leaderLogs; auto.
+ exfalso. subst. simpl in *.
unfold handleAppendEntries in *; repeat break_match; find_inversion; congruence.
Qed.
Proof using.
red. unfold logs_leaderLogs_inductive. intros. intuition.
- find_apply_lem_hyp handleAppendEntriesReply_log. subst.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *;
repeat find_rewrite;
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
- find_eapply_lem_hyp handleAppendEntriesReply_packets. subst.
intuition. unfold logs_leaderLogs_nw in *. intros.
simpl in *. find_apply_hyp_hyp. intuition.
prove_in. copy_eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto.
Qed.
Proof using.
red. unfold logs_leaderLogs_inductive. intros. intuition.
- find_apply_lem_hyp handleRequestVote_log. subst.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *;
repeat find_rewrite;
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_requestVote_leaderLogs; eauto.
- find_eapply_lem_hyp handleRequestVote_no_append_entries. subst.
intuition. unfold logs_leaderLogs_nw in *. intros.
simpl in *. find_apply_hyp_hyp. intuition.
+ prove_in. copy_eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_requestVote_leaderLogs; eauto.
+ find_false. subst. simpl in *. subst.
repeat eexists; eauto.
Qed.
Proof using.
red. unfold logs_leaderLogs_inductive. intros. intuition.
- subst.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *;
repeat find_rewrite;
[match goal with
| H : In _ (log _ ) |- _ =>
erewrite handleRequestVoteReply_log; eauto;
erewrite handleRequestVoteReply_log in H; eauto
end|];
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite ; update_destruct; subst; rewrite_update; eauto;
simpl in *; apply update_elections_data_requestVoteReply_leaderLogs; eauto.
- unfold logs_leaderLogs_nw in *. intros.
simpl in *. find_apply_hyp_hyp. intuition.
prove_in. copy_eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto;
simpl in *; apply update_elections_data_requestVoteReply_leaderLogs; eauto.
Qed.
Proof using si lhllsi rri.
red. unfold logs_leaderLogs_inductive. intros. intuition.
- subst.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *.
+ find_apply_lem_hyp handleClientRequest_log. intuition; subst; repeat find_rewrite.
* find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_client_request_leaderLogs; eauto.
* {break_exists. intuition. repeat find_rewrite. simpl in *.
intuition; subst.
- find_apply_lem_hyp leaders_have_leaderLogs_strong_invariant; eauto.
break_exists. intuition.
unfold ghost_data in *. simpl in *. repeat find_rewrite.
match goal with
| h : name, _ : _ = ?e :: ?es ++ ?ll |- _ =>
exists h, ll, (e :: x0)
end; intuition.
+ find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_client_request_leaderLogs; eauto.
+ break_if; eauto using app_ass; do_bool; lia.
+ simpl in *. intuition; subst; eauto.
- find_copy_apply_lem_hyp maxIndex_is_max; eauto using lift_logs_sorted.
find_apply_hyp_hyp. break_exists_exists; intuition;
[find_higher_order_rewrite ; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_client_request_leaderLogs; eauto|].
break_if; do_bool; intuition; lia.
}
+ find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_client_request_leaderLogs; eauto.
- unfold logs_leaderLogs_nw in *.
intros. simpl in *. find_apply_hyp_hyp. intuition.
+ eapply_prop_hyp In pBody; eauto; break_exists_exists; intuition; subst;
repeat find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_client_request_leaderLogs; eauto.
+ do_in_map. subst. simpl in *. find_eapply_lem_hyp handleClientRequest_no_append_entries;
eauto. intuition. find_false. repeat find_rewrite. repeat eexists; eauto.
Qed.
Proof using.
red. unfold logs_leaderLogs_inductive. intros. intuition.
- find_apply_lem_hyp handleTimeout_log_same. subst.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *;
repeat find_rewrite;
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_timeout_leaderLogs; eauto.
- intuition. unfold logs_leaderLogs_nw in *. intros.
simpl in *. find_apply_hyp_hyp. intuition.
+ copy_eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto;
simpl in *; rewrite update_elections_data_timeout_leaderLogs; eauto.
+ do_in_map. subst. simpl in *.
find_eapply_lem_hyp handleTimeout_packets; eauto. intuition.
find_false. repeat find_rewrite.
repeat eexists; eauto.
Qed.
Proof using.
red. unfold logs_leaderLogs_inductive. intros.
match goal with
| H : nwState ?net ?h = (?gd, ?d) |- _ =>
replace gd with (fst (nwState net h)) in * by (rewrite H; reflexivity);
replace d with (snd (nwState net h)) in * by (rewrite H; reflexivity);
clear H
end.
intuition.
- find_apply_lem_hyp doGenericServer_log.
unfold logs_leaderLogs in *. intros.
simpl in *.
find_higher_order_rewrite; update_destruct; subst; rewrite_update; simpl in *;
repeat find_rewrite;
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
- find_apply_lem_hyp doGenericServer_packets. subst.
unfold logs_leaderLogs_nw in *. intros. simpl in *.
find_apply_hyp_hyp. intuition.
eapply_prop_hyp In In; eauto.
break_exists_exists; intuition;
repeat find_higher_order_rewrite;
update_destruct; subst; rewrite_update; eauto.
Qed.
Proof using.
intros. unfold doLeader, advanceCommitIndex in *.
break_match; try solve [find_inversion; simpl in *; intuition].
break_if; try solve [find_inversion; simpl in *; intuition].
find_inversion. simpl. do_in_map. subst.
simpl in *. find_inversion. intuition.
match goal with
| |- context [pred ?x] =>
remember (pred x) as index
end. break_match; simpl in *.
- right. left. eauto.
-  destruct index; intuition.
right. right. exists x.
match goal with
| _ : S _ = pred ?x |- context [pred ?y] =>
assert (pred x = pred y) by auto
end.
repeat find_rewrite. intuition.
Qed.
Proof using si nisi rlmli llsi rri.
red. unfold logs_leaderLogs_inductive. intros.
match goal with
| H : nwState ?net ?h = (?gd, ?d) |- _ =>
replace gd with (fst (nwState net h)) in * by (rewrite H; reflexivity);
replace d with (snd (nwState net h)) in * by (rewrite H; reflexivity);
clear H
end.
intuition.
- unfold logs_leaderLogs in *.
intros.
simpl in *.
find_apply_lem_hyp doLeader_log.
find_higher_order_rewrite. simpl in *.
update_destruct; subst; rewrite_update; simpl in *; repeat find_rewrite;
find_apply_hyp_hyp; break_exists_exists; intuition;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
- unfold logs_leaderLogs_nw.
intros. simpl in *. find_apply_hyp_hyp.
break_or_hyp.
+ unfold logs_leaderLogs_nw in *.
eapply_prop_hyp pBody pBody; eauto.
break_exists_exists; intuition; subst;
find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
+ { do_in_map. subst. simpl in *.
find_eapply_lem_hyp doLeader_spec; eauto. intuition.
- subst. (* just use host invariant *)
unfold logs_leaderLogs in *.
find_apply_lem_hyp findGtIndex_necessary.
intuition.
eapply_prop_hyp In In.
break_exists_exists. intuition.
match goal with
| _ : In (_, ?ll) _ |- _ =>
exists ll
end.
intuition; eauto using Prefix_refl;
[find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto|].
rewrite sorted_findGtIndex_0; eauto using lift_logs_sorted.
intros. eapply entries_gt_0_invariant; eauto.
- break_exists. break_and.
subst.
unfold logs_leaderLogs in *.
find_apply_lem_hyp findGtIndex_necessary.
break_and. find_apply_hyp_hyp.
match goal with
| H : exists _ _ _, _ |- _ =>
destruct H as [leader];
destruct H as [ll];
destruct H as [es]
end.
break_and.
destruct (lt_eq_lt_dec (maxIndex ll) pli); intuition.
+ exists leader, ll, (findGtIndex es pli), []. intuition.
* find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
* simpl; auto.
* rewrite app_nil_r.
rewrite findGtIndex_removeAfterIndex_commute; eauto using lift_logs_sorted.
unfold ghost_data in *. simpl in *.
find_rewrite.
eapply findGtIndex_app_1; lia.
* eauto using findGtIndex_in.
* left. intuition.
match goal with
| H : forall _, In _ _ -> _ |- _ =>
apply H
end.
find_apply_lem_hyp findAtIndex_elim. intuition.
subst.
match goal with
| _ : In ?x ?l, _ : eIndex ?e > eIndex ?x |- _ =>
assert (In x (removeAfterIndex l (eIndex e))) by
(apply removeAfterIndex_le_In; eauto; lia)
end.
unfold ghost_data in *. simpl in *.
find_rewrite.
do_in_app; intuition.
find_copy_apply_lem_hyp leaderLogs_sorted_invariant; eauto.
find_apply_lem_hyp maxIndex_is_max; eauto; lia.
+ exists leader, ll, (findGtIndex es pli), []. intuition.
* find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
* simpl; auto.
* rewrite app_nil_r.
rewrite findGtIndex_removeAfterIndex_commute; eauto using lift_logs_sorted.
unfold ghost_data in *. simpl in *.
find_rewrite.
apply findGtIndex_app_1. lia.
* eauto using findGtIndex_in.
* right. find_apply_lem_hyp findAtIndex_elim.
left. exists x0. intuition. subst.
match goal with
| _ : In ?x ?l, _ : eIndex ?e > _ |- _ =>
assert (In x (removeAfterIndex l (eIndex e))) by
(apply removeAfterIndex_le_In; eauto; lia)
end.
unfold ghost_data in *. simpl in *.
find_rewrite.
assert (sorted (es ++ ll)) by (repeat find_reverse_rewrite;
apply removeAfterIndex_sorted;
repeat find_rewrite; eauto using lift_logs_sorted).
eapply thing3; eauto; try lia. intros.
match goal with
| |- eIndex ?e > _ =>
assert (In e (log (snd (nwState net h)))) by
(unfold ghost_data in *; simpl in *;
eapply removeAfterIndex_in; repeat find_reverse_rewrite; eauto)
end.
eapply entries_gt_0_invariant; eauto.
+ exists leader, ll, es, (findGtIndex ll pli). intuition.
* find_higher_order_rewrite; update_destruct; subst; rewrite_update; eauto.
* (* prefix_findGtIndex *)
apply findGtIndex_Prefix.
* rewrite findGtIndex_removeAfterIndex_commute; eauto using lift_logs_sorted.
unfold ghost_data in *. simpl in *.
find_rewrite.
assert (sorted (es ++ ll)) by (repeat find_reverse_rewrite;
apply removeAfterIndex_sorted;
repeat find_rewrite; eauto using lift_logs_sorted).
apply findGtIndex_app_2; auto.
* { right. left.
find_apply_lem_hyp findAtIndex_elim.
exists x0.  intuition.
- subst.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
find_eapply_lem_hyp removeAfterIndex_le_In;
[unfold ghost_data in *; simpl in *; find_rewrite_lem H|lia]
end.
assert (sorted (es ++ ll)) by (repeat find_reverse_rewrite;
apply removeAfterIndex_sorted;
repeat find_rewrite; eauto using lift_logs_sorted).
eapply thing3; eauto; try lia. intros.
match goal with
| |- eIndex ?e > _ =>
assert (In e (log (snd (nwState net h)))) by
(unfold ghost_data in *; simpl in *;
eapply removeAfterIndex_in; repeat find_reverse_rewrite; eauto)
end.
eapply entries_gt_0_invariant; eauto.
- left. intros.
eapply findGtIndex_non_empty; eauto.
}
- exfalso. (* use nextIndex_sanity *)
break_exists. intuition.
find_eapply_lem_hyp nextIndex_sanity; eauto. break_exists.
unfold ghost_data in *. simpl in *. congruence.
}
Qed.

Lemma logs_leaderLogs_inductive_init :
refined_raft_net_invariant_init logs_leaderLogs_inductive.