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; omega.
* 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; omega)
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; omega.
+ 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. omega.
* 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; omega)
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 omega. 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|omega]
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 omega. 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.