Section PrefixWithinTerm.
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 {llli : logs_leaderLogs_interface}.
Context {rlmli : refined_log_matching_lemmas_interface}.
Context {ollpti : one_leaderLog_per_term_interface}.
Context {llsi : leaderLogs_sorted_interface}.
Context {llsli : leaderLogs_sublog_interface}.
Context {lsli : leader_sublog_interface}.
Context {nisi : nextIndex_safety_interface}.
Context {llci : leaderLogs_contiguous_interface}.
Context {aelmi : allEntries_log_matching_interface}.
Context {aertsi : append_entries_request_term_sanity_interface}.
Context {aelsi : allEntries_leader_sublog_interface}.
Definition lift_leader_sublog :
forall net leader e h,
refined_raft_intermediate_reachable net ->
type (snd (nwState net leader)) = Leader ->
In e (log (snd (nwState net h))) ->
eTerm e = currentTerm (snd (nwState net leader)) ->
In e (log (snd (nwState net leader))).
Proof using lsli rri.
intros. pose proof lift_prop leader_sublog_host_invariant.
conclude_using ltac:(apply leader_sublog_invariant_invariant).
find_apply_hyp_hyp.
match goal with
| H : leader_sublog_host_invariant _ |- _ =>
specialize (H leader e h)
end.
repeat find_rewrite_lem deghost_spec. intuition.
Qed.
Proof using.
unfold deghost.
simpl.
intros.
eexists; intuition eauto.
apply in_map_iff. eexists; eauto.
Qed.
Definition lift_leader_sublog_nw :
forall net leader p t leaderId prevLogIndex prevLogTerm entries leaderCommit e,
refined_raft_intermediate_reachable net ->
type (snd (nwState net leader)) = Leader ->
In p (nwPackets net) ->
pBody p = AppendEntries t leaderId prevLogIndex prevLogTerm
entries leaderCommit ->
In e entries ->
eTerm e = currentTerm (snd (nwState net leader)) ->
In e (log (snd (nwState net leader))).
Proof using lsli rri.
intros.
pose proof lift_prop leader_sublog_nw_invariant.
conclude_using ltac:(apply leader_sublog_invariant_invariant).
find_apply_hyp_hyp.
find_apply_lem_hyp exists_deghosted_packet.
match goal with
| H : exists _, _ |- _ => destruct H as [q]
end. break_and.
match goal with
| H : leader_sublog_nw_invariant _ |- _ =>
specialize (H leader q t leaderId prevLogIndex prevLogTerm entries leaderCommit e)
end.
repeat find_rewrite_lem deghost_spec. subst. simpl in *.
intuition.
Qed.
Definition append_entries_append_entries_prefix_within_term_nw net :=
forall p t n pli plt es ci p' t' n' pli' plt' es' ci' e e',
In p (nwPackets net) ->
pBody p = AppendEntries t n pli plt es ci ->
In p' (nwPackets net) ->
pBody p' = AppendEntries t' n' pli' plt' es' ci' ->
eTerm e = eTerm e' ->
eIndex e <= eIndex e' ->
In e es ->
In e' es' ->
(In e es' \/ (eIndex e = pli' /\ eTerm e = plt') \/ (eIndex e < pli' /\ eTerm e <= plt')).
Proof using ollpti rlmli llli.
red. red. intros.
match goal with
| H : In ?e _,
H' : In ?e' _ |- _ =>
copy_eapply logs_leaderLogs_invariant H; eauto;
copy_eapply logs_leaderLogs_invariant H'; eauto
end.
break_exists; intuition.
repeat find_rewrite.
find_eapply_lem_hyp one_leaderLog_per_term_log_invariant; eauto.
conclude_using eauto. subst.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' (log (snd (nwState net h')))) by
(eapply entries_contiguous_invariant; eauto; intuition;
[eapply entries_gt_0_invariant; eauto|];
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
apply entries_sorted_invariant; auto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _ _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end.
repeat find_rewrite. do_in_app. intuition.
- find_apply_hyp_hyp.
eapply entries_match_invariant; eauto; repeat find_rewrite; auto.
- cut (e = x0); intros; subst; intuition.
eapply uniqueIndices_elim_eq; eauto.
+ apply sorted_uniqueIndices.
apply entries_sorted_invariant.
auto.
+ match goal with
| H : context [ removeAfterIndex ?l ?index ] |- In _ ?l =>
apply removeAfterIndex_in with (i := index)
end.
repeat find_rewrite. apply in_app_iff; intuition.
Qed.
Definition locked_or x y :=
x \/ y.
Proof using llsi ollpti rlmli llli.
red. intros.
match goal with
| H : context [pBody],
H' : context [pBody] |- _ =>
copy_eapply logs_leaderLogs_nw_invariant H; eauto;
copy_eapply logs_leaderLogs_nw_invariant H'; eauto
end; repeat conclude_using eauto.
break_exists; break_and.
repeat find_rewrite.
match goal with
| H : refined_raft_intermediate_reachable _ |- _ =>
copy_apply leaderLogs_sorted_invariant H
end.
find_eapply_lem_hyp one_leaderLog_per_term_log_invariant; eauto.
conclude_using eauto. subst.
intuition; subst.
- destruct (lt_eq_lt_dec (eIndex e) pli'); intuition.
left.
match goal with
| H : removeAfterIndex ?l ?index = ?es ++ ?ll |- _ =>
eapply app_contiguous_maxIndex_le_eq in H
end;
[|idtac|eapply removeAfterIndex_contiguous; [eapply entries_sorted_nw_invariant; eauto|eapply entries_contiguous_nw_invariant; eauto]|idtac]; eauto; [|lia].
assert (exists e'', eIndex e'' = eIndex e /\ In e'' es') by
(eapply entries_contiguous_nw_invariant; eauto; intuition;
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
eapply entries_sorted_nw_invariant; eauto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
find_apply_hyp_hyp.
eapply entries_match_nw_1_invariant in H1; eauto.
apply H1; eauto.
repeat find_rewrite; auto.
- break_exists. break_and.
match goal with
| H : _ \/ _ |- _ => clear H
end.
subst. left.
assert (pli < eIndex e) by
(eapply entries_contiguous_nw_invariant; [idtac|idtac|eauto|]; eauto).
assert (maxIndex x4 < eIndex e) by lia.
assert (eIndex x0 < eIndex e).
eapply le_lt_trans; [|eauto].
eapply maxIndex_is_max; eauto.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' es') by
(eapply entries_contiguous_nw_invariant; eauto; intuition;
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
eapply entries_sorted_nw_invariant; eauto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat find_rewrite. do_in_app. intuition.
+ find_apply_hyp_hyp.
eapply entries_match_nw_1_invariant.
10: { eauto. }
5: { eauto. }
4: { eauto. }
all:eauto; try lia. repeat find_rewrite; auto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; eauto.
lia.
- left.
assert (0 < eIndex e) by
(eapply entries_gt_0_nw_invariant; [|idtac|idtac|eauto]; [|idtac|eauto]; eauto).
assert (pli < eIndex e) by
(eapply entries_contiguous_nw_invariant; [|idtac|idtac|eauto]; [|idtac|eauto]; eauto).
assert (exists e'', eIndex e'' = eIndex e /\ In e'' es') by
(eapply entries_contiguous_nw_invariant; eauto; intuition;
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
eapply entries_sorted_nw_invariant; eauto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat find_rewrite. do_in_app. intuition.
+ find_apply_hyp_hyp.
eapply entries_match_nw_1_invariant.
10: { eauto. }
5: { eauto. }
4: { eauto. }
all:eauto; try lia. repeat find_rewrite; auto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; eauto.
lia.
- break_exists. break_and.
match goal with
| H : _ \/ _ |- _ => clear H
end.
subst.
destruct (lt_eq_lt_dec (eIndex e) pli'); intuition.
left.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' es') by
(eapply entries_contiguous_nw_invariant; eauto; intuition;
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
eapply entries_sorted_nw_invariant; eauto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat find_rewrite. do_in_app. intuition.
+ find_apply_hyp_hyp.
eapply entries_match_nw_1_invariant.
10: { eauto. }
5: { eauto. }
4: { eauto. }
all:eauto; try lia. repeat find_rewrite; auto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; eauto.
lia.
- destruct (lt_eq_lt_dec (eIndex e) pli'); intuition.
left.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' es') by
(eapply entries_contiguous_nw_invariant; eauto; intuition;
eapply le_trans; eauto;
eapply maxIndex_is_max; eauto;
eapply entries_sorted_nw_invariant; eauto).
break_exists. intuition.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In ?x ?l,
_ : eIndex ?x = _ |- _ =>
assert (In x (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat find_rewrite. do_in_app. intuition.
+ find_apply_hyp_hyp.
eapply entries_match_nw_1_invariant.
10: { eauto. }
5: { eauto. }
4: { eauto. }
all:eauto; try lia. repeat find_rewrite; auto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; eauto.
lia.
- break_exists. break_and.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e ?l |- _ =>
assert (In e (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e' ?l |- _ =>
assert (In e' (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat match goal with
| _ : removeAfterIndex ?es _ = ?x ++ ?y,
Hp : pBody _ = AppendEntries _ _ ?i _ ?es _ |- _ =>
try match goal with
| _ : sorted (x ++ y) |- _ =>
fail 2
end;
assert (sorted (x ++ y)) by
(repeat find_reverse_rewrite; eapply removeAfterIndex_sorted;
eapply entries_sorted_nw_invariant; try apply Hp; eauto);
assert (contiguous_range_exact_lo (x ++ y) i) by
(repeat find_reverse_rewrite;
eapply removeAfterIndex_contiguous;
[eapply entries_sorted_nw_invariant; try apply Hp; eauto|
eapply entries_contiguous_nw_invariant; try apply Hp; eauto])
end.
repeat match goal with
| H : ?x \/ ?y |- _ =>
copy_eapply contiguous_app_prefix_contiguous H; eauto; fold (locked_or x y) in H
end.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
assert (forall e, In e x1 -> In e es') by (intros; eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
assert (forall e, In e x2 -> In e es') by (intros; eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
assert (forall e, In e x5 -> In e es) by (intros; eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
repeat find_rewrite. repeat do_in_app. intuition.
+ subst.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' x1).
{ eapply_prop (contiguous_range_exact_lo x1). intuition.
match goal with
| H : contiguous_range_exact_lo ?l ?i , _ : In ?e ?l |- ?i < eIndex ?e =>
eapply H
end; eauto.
eapply le_trans; eauto.
eapply maxIndex_is_max; eauto using sorted_app_1.
}
break_exists. break_and.
left.
find_copy_apply_hyp_hyp.
match goal with
| H : refined_raft_intermediate_reachable _ |- _ =>
copy_apply entries_match_nw_1_invariant H
end.
match goal with
| H : entries_match_nw_1 _ |- _ =>
eapply H with (es := es) (p := p) (p' := p')
end. all:eauto. all:intuition.
* eapply_prop_hyp eTerm In. congruence.
* replace (eIndex e) with (eIndex x8).
eapply_prop (contiguous_range_exact_lo (x1 ++ x2)); eauto. in_crush.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; [|idtac|eauto]; eauto.
match goal with
| H: contiguous_range_exact_lo ?l ?i, _ : In ?e ?l, _ : eIndex _ <= ?i |- _ =>
assert (i < eIndex e) by (eapply H; eauto)
end.
lia.
+ assert (In e x4) by eauto using Prefix_In.
match goal with
| _ : removeAfterIndex ?es _ = ?l, _ : contiguous_range_exact_lo ?l ?i |-
context [In ?e ?es] =>
destruct (lt_eq_lt_dec (eIndex e) i); intuition
end.
* right. right. intuition.
repeat find_reverse_rewrite.
match goal with
| _ : In (_, ?ll) (leaderLogs _) |- _ =>
eapply sorted_term_index_lt with (l := ll)
end; eauto.
* match goal with
| _ : eIndex ?e = eIndex ?x, _ : In (_, ?ll) (leaderLogs _) |- _ =>
assert (e = x); subst; repeat find_rewrite; intuition;
eapply uniqueIndices_elim_eq with (xs := ll); eauto using sorted_uniqueIndices
end.
* {
left.
repeat match goal with
| H : contiguous_range_exact_lo _ _ |- _ =>
eapply contiguous_app in H; eauto; [idtac]
end.
match goal with
| _ : eIndex ?x < eIndex ?e , H : locked_or _ (eIndex ?x = _) |- _ =>
unfold locked_or in H
end. intuition.
- match goal with
| _ : ?x = [] -> False, _ : Prefix ?x ?ll |- In ?e _ =>
cut (In e x); eauto;
eapply prefix_contiguous; eauto
end.
- exfalso. repeat find_rewrite.
match goal with
| H : In ?e ?l, _ : maxIndex ?l < eIndex ?e |- _ =>
eapply maxIndex_is_max in H
end; eauto; lia.
}
+ match goal with
| H : In ?e ?l, H' : In ?e' ?l', Hp : Prefix ?l _, Hp' : Prefix ?l' _ |- _ =>
copy_eapply Prefix_In H; try apply Hp; eauto;
copy_eapply Prefix_In H'; try apply Hp'; eauto
end.
match goal with
| _ : removeAfterIndex ?es _ = ?l, _ : contiguous_range_exact_lo ?l ?i |-
context [In ?e ?es] =>
destruct (lt_eq_lt_dec (eIndex e) i); intuition
end.
* right. right. intuition.
repeat find_reverse_rewrite.
match goal with
| _ : In (_, ?ll) (leaderLogs _) |- _ =>
eapply sorted_term_index_lt with (l := ll)
end; eauto.
* match goal with
| _ : eIndex ?e = eIndex ?x, _ : In (_, ?ll) (leaderLogs _) |- _ =>
assert (e = x); subst; repeat find_rewrite; intuition;
eapply uniqueIndices_elim_eq with (xs := ll); eauto using sorted_uniqueIndices
end.
* left.
repeat match goal with
| H : contiguous_range_exact_lo _ _ |- _ =>
eapply contiguous_app in H; eauto; [idtac]
end.
match goal with
| _ : Prefix ?x ?ll |- In ?e _ =>
cut (In e x); eauto
end.
eapply prefix_contiguous; eauto.
match goal with
| [ H : locked_or (x2 = [] -> _) _ |- _ ] =>
unfold locked_or in H; invc H; auto
end.
match goal with
| [ H : In ?e _, H' : _ < eIndex ?e |- _ ] =>
apply maxIndex_is_max in H; [|solve[eapply_prop leaderLogs_sorted; eauto]]
end.
lia.
- left.
break_exists.
break_and.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e ?l |- _ =>
assert (In e (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e' ?l |- _ =>
assert (In e' (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat match goal with
| _ : removeAfterIndex ?es _ = ?x ++ ?y,
Hp : pBody _ = AppendEntries _ _ ?i _ ?es _ |- _ =>
try match goal with
| _ : sorted (x ++ y) |- _ =>
fail 2
end;
assert (sorted (x ++ y)) by
(repeat find_reverse_rewrite; eapply removeAfterIndex_sorted;
eapply entries_sorted_nw_invariant; try apply Hp; eauto);
assert (contiguous_range_exact_lo (x ++ y) i) by
(repeat find_reverse_rewrite;
eapply removeAfterIndex_contiguous;
[eapply entries_sorted_nw_invariant; try apply Hp; eauto|
eapply entries_contiguous_nw_invariant; try apply Hp; eauto])
end.
repeat find_rewrite.
repeat match goal with
| H : ?x \/ ?y |- _ =>
copy_eapply contiguous_app_prefix_contiguous H; eauto; fold (locked_or x y) in H
end.
repeat do_in_app. intuition.
+ subst.
match goal with
| _ : In e' ?x, _ : sorted (?x ++ _), H : contiguous_range_exact_lo ?x' _, _ : In e ?x'
|- _ =>
assert (exists e'', eIndex e'' = eIndex e /\ In e'' x) by
(eapply contiguous_app_prefix_2; eauto;
intuition; [eapply H; eauto|eapply le_trans; eauto; eapply maxIndex_is_max; eauto using sorted_app_1])
end.
break_exists. intuition.
match goal with
| H : refined_raft_intermediate_reachable _ |- _ =>
copy_apply entries_match_nw_1_invariant H
end.
match goal with
| He : eIndex ?x = eIndex e, H : removeAfterIndex _ _ = _ |- _ =>
symmetry in He;
assert (In x es') by (eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
repeat find_apply_hyp_hyp.
match goal with
| H : entries_match_nw_1 _ |- _ =>
eapply H with (es := es) (p := p) (p' := p')
end.
7: { eauto. }
all:repeat find_rewrite. all:eauto. intuition.
eapply entries_gt_0_nw_invariant; eauto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; [|idtac|eauto]; eauto.
match goal with
| H: contiguous_range_exact_lo ?l ?i, _ : In ?e ?l, _ : eIndex _ <= ?i |- _ =>
assert (i < eIndex e) by (eapply H; eauto)
end.
lia.
+ match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; [idtac]
end. eauto using Prefix_In.
+ match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; [idtac]
end. eauto using Prefix_In.
- break_exists.
break_and.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e ?l |- _ =>
assert (In e (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e' ?l |- _ =>
assert (In e' (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. subst.
repeat match goal with
| _ : removeAfterIndex ?es _ = ?x ++ ?y,
Hp : pBody _ = AppendEntries _ _ ?i _ ?es _ |- _ =>
try match goal with
| _ : sorted (x ++ y) |- _ =>
fail 2
end;
assert (sorted (x ++ y)) by
(repeat find_reverse_rewrite; eapply removeAfterIndex_sorted;
eapply entries_sorted_nw_invariant; try apply Hp; eauto);
assert (contiguous_range_exact_lo (x ++ y) i) by
(repeat find_reverse_rewrite;
eapply removeAfterIndex_contiguous;
[eapply entries_sorted_nw_invariant; try apply Hp; eauto|
eapply entries_contiguous_nw_invariant; try apply Hp; eauto])
end.
repeat find_rewrite.
repeat match goal with
| H : ?x \/ ?y |- _ =>
copy_eapply contiguous_app_prefix_contiguous H; eauto; fold (locked_or x y) in H
end.
repeat do_in_app. intuition.
+ left.
assert (exists e'', eIndex e'' = eIndex e /\ In e'' x1).
{ eapply_prop (contiguous_range_exact_lo x1). intuition.
- eapply contiguous_0_app; eauto.
- eapply le_trans; eauto.
eapply maxIndex_is_max; eauto using sorted_app_1.
}
break_exists. intuition.
match goal with
| H : refined_raft_intermediate_reachable _ |- _ =>
copy_apply entries_match_nw_1_invariant H
end.
match goal with
| He : eIndex ?x = eIndex e, H : removeAfterIndex _ _ = _ |- _ =>
symmetry in He;
assert (In x es') by (eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
repeat find_apply_hyp_hyp.
match goal with
| H : entries_match_nw_1 _ |- _ =>
eapply H with (es := es) (p := p) (p' := p')
end.
7: { eauto. }
all:repeat find_rewrite. all:eauto. intuition.
eapply entries_contiguous_nw_invariant; eauto.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; [|idtac|eauto]; eauto.
find_eapply_lem_hyp contiguous_0_app; eauto. lia.
+ match goal with
| _ : removeAfterIndex ?es _ = ?l, _ : contiguous_range_exact_lo ?l ?i |-
context [In ?e ?es] =>
destruct (lt_eq_lt_dec (eIndex e) i); intuition
end.
* right. right. intuition.
repeat find_reverse_rewrite.
match goal with
| _ : In (_, ?ll) (leaderLogs _) |- _ =>
eapply sorted_term_index_lt with (l := ll)
end; eauto.
* match goal with
| _ : eIndex ?e = eIndex ?x, _ : In (_, ?ll) (leaderLogs _) |- _ =>
assert (e = x); subst; repeat find_rewrite; intuition;
eapply uniqueIndices_elim_eq with (xs := ll); eauto using sorted_uniqueIndices
end.
* {
left.
repeat match goal with
| H : contiguous_range_exact_lo _ _ |- _ =>
eapply contiguous_app in H; eauto; [idtac]
end.
match goal with
| _ : eIndex ?x < eIndex ?e , H : locked_or _ (eIndex ?x = _) |- _ =>
unfold locked_or in H
end. intuition.
- match goal with
| _ : ?x = [] -> False, _ : Prefix ?x ?ll |- In ?e _ =>
assert (In e x) by (eapply prefix_contiguous; eauto)
end.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition
end.
- exfalso. repeat find_rewrite.
match goal with
| H : In ?e ?l, _ : maxIndex ?l < eIndex ?e |- _ =>
eapply maxIndex_is_max in H
end; eauto; lia.
}
+ match goal with
| H : In ?e ?l, H' : In ?e' ?l', Hp : Prefix ?l _, Hp' : Prefix ?l' _ |- _ =>
copy_eapply Prefix_In H; try apply Hp; eauto;
copy_eapply Prefix_In H'; try apply Hp'; eauto
end.
match goal with
| _ : removeAfterIndex ?es _ = ?l, _ : contiguous_range_exact_lo ?l ?i |-
context [In ?e ?es] =>
destruct (lt_eq_lt_dec (eIndex e) i); intuition
end.
* right. right. intuition.
repeat find_reverse_rewrite.
match goal with
| _ : In (_, ?ll) (leaderLogs _) |- _ =>
eapply sorted_term_index_lt with (l := ll)
end; eauto.
* match goal with
| _ : eIndex ?e = eIndex ?x, _ : In (_, ?ll) (leaderLogs _) |- _ =>
assert (e = x); subst; repeat find_rewrite; intuition;
eapply uniqueIndices_elim_eq with (xs := ll); eauto using sorted_uniqueIndices
end.
* left.
repeat match goal with
| H : contiguous_range_exact_lo _ _ |- _ =>
eapply contiguous_app in H; eauto; [idtac]
end.
assert (x2 <> []).
{
unfold locked_or in *. intuition.
find_apply_lem_hyp maxIndex_is_max; [|solve[eapply_prop leaderLogs_sorted; eauto]].
lia.
}
match goal with
|  _ : Prefix ?x ?ll |- In ?e _ =>
assert (In e x2) by (eapply prefix_contiguous; eauto)
end.
match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition
end.
- left.
repeat match goal with
| _ : removeAfterIndex ?es _ = ?x ++ ?y,
Hp : pBody _ = AppendEntries _ _ ?i _ ?es _ |- _ =>
try match goal with
| _ : sorted (x ++ y) |- _ =>
fail 2
end;
assert (sorted (x ++ y)) by
(repeat find_reverse_rewrite; eapply removeAfterIndex_sorted;
eapply entries_sorted_nw_invariant; try apply Hp; eauto);
assert (contiguous_range_exact_lo (x ++ y) i) by
(repeat find_reverse_rewrite;
eapply removeAfterIndex_contiguous;
[eapply entries_sorted_nw_invariant; try apply Hp; eauto|
eapply entries_contiguous_nw_invariant; try apply Hp; eauto])
end.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e ?l |- _ =>
assert (In e (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end.
match goal with
| _ : removeAfterIndex ?l ?i = _,
_ : In e' ?l |- _ =>
assert (In e' (removeAfterIndex l i)) by (apply removeAfterIndex_le_In; eauto; lia)
end. repeat find_rewrite. repeat do_in_app. intuition.
+ match goal with
| H : In e _ |- _ =>
copy_eapply contiguous_0_app H; eauto; [idtac]
end.
match goal with
| _ : In e' ?l, H : contiguous_range_exact_lo (?l ++ _) 0 |- _ =>
eapply contiguous_app_prefix_2 with (i := eIndex e) in H
end; eauto; intuition;
[|eapply le_trans; eauto; eapply maxIndex_is_max; eauto using sorted_app_1]; [idtac].
break_exists. intuition.
match goal with
| H : refined_raft_intermediate_reachable _ |- _ =>
copy_apply entries_match_nw_1_invariant H
end.
match goal with
| He : eIndex ?x = eIndex e, H : removeAfterIndex _ _ = _ |- _ =>
symmetry in He;
assert (In x es') by (eapply removeAfterIndex_in; rewrite H; apply in_app_iff; intuition)
end.
repeat find_apply_hyp_hyp.
match goal with
| H : entries_match_nw_1 _ |- _ =>
eapply H with (es := es) (p := p) (p' := p')
end.
7: { eauto. }
all:repeat find_rewrite. all:eauto. intuition.
+ exfalso.
find_eapply_lem_hyp Prefix_maxIndex; [|idtac|eauto]; eauto.
find_eapply_lem_hyp contiguous_0_app; eauto. lia.
+ match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; [idtac]
end. eauto using Prefix_In.
+ match goal with
| H : removeAfterIndex _ _ = _ |- _ =>
eapply removeAfterIndex_in; rewrite H; apply in_app_iff; [idtac]
end. eauto using Prefix_In.
Qed.
Definition log_leaderLogs_prefix_within_term net :=
forall h t ll leader,
In (t, ll) (leaderLogs (fst (nwState net leader))) ->
prefix_within_term (log (snd (nwState net h))) ll.
Definition allEntries_log_prefix_within_term net :=
forall h h',
prefix_within_term (map snd (allEntries (fst (nwState net h)))) (log (snd (nwState net h'))).
Definition allEntries_append_entries_prefix_within_term_nw net :=
forall p t n pli plt es ci h e e',
In p (nwPackets net) ->
pBody p = AppendEntries t n pli plt es ci ->
eTerm e = eTerm e' ->
eIndex e <= eIndex e' ->
In e (map snd (allEntries (fst (nwState net h)))) ->
In e' es ->
(In e es \/ (eIndex e = pli /\ eTerm e = plt) \/ (eIndex e < pli /\ eTerm e <= plt)).
Definition append_entries_leaderLogs_prefix_within_term net :=
forall p t n pli plt es ci h t' ll,
In p (nwPackets net) ->
pBody p = AppendEntries t n pli plt es ci ->
In (t', ll) (leaderLogs (fst (nwState net h))) ->
prefix_within_term es ll.
Definition append_entries_log_prefix_within_term net :=
forall p t n pli plt es ci h,
In p (nwPackets net) ->
pBody p = AppendEntries t n pli plt es ci ->
prefix_within_term es (log (snd (nwState net h))).
Definition prefix_within_term_inductive net :=
allEntries_leaderLogs_prefix_within_term net /\
log_leaderLogs_prefix_within_term net /\
allEntries_log_prefix_within_term net /\
allEntries_append_entries_prefix_within_term_nw net /\
append_entries_leaderLogs_prefix_within_term net /\
append_entries_log_prefix_within_term net.
Proof using.
unfold prefix_within_term. intros.
find_apply_lem_hyp findGtIndex_in. eauto.
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.
Proof using.
intros. unfold handleAppendEntries in *.
repeat break_match; find_inversion; congruence.
Qed.
Proof using.
intros.
unfold update_elections_data_appendEntries in *.
repeat break_match; subst; simpl in *; auto.
find_apply_lem_hyp handleAppendEntriesReply_entries. subst.
do_in_map. do_in_app. subst. intuition.
- do_in_map. subst. simpl in *. auto.
- left. apply in_map_iff. eexists; eauto.
Qed.
Proof using.
intros.
unfold update_elections_data_client_request in *.
repeat break_match; subst; simpl in *; auto. intuition. subst.
do_bool. find_apply_lem_hyp handleClientRequest_log.
intuition.
- match goal with
| H : log _ = log (snd _) |- _ => symmetry in H
end. repeat find_rewrite. simpl in *. lia.
- break_exists. intuition. repeat find_rewrite.
find_inversion. intuition.
Qed.
Proof using.
intros.
unfold update_elections_data_client_request in *.
repeat break_match; subst; simpl in *; auto.
Qed.
Proof using.
intros.
unfold prefix_within_term in *.
intros.
find_apply_hyp_hyp. intuition; eauto.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
- subst. break_if; eauto.
do_bool; lia.
- break_if; eauto. do_bool.
specialize (H1 e); intuition. lia.
Qed.
Proof using aertsi aelmi llsi ollpti rlmli llli.
red. unfold prefix_within_term_inductive. intuition.
- unfold allEntries_leaderLogs_prefix_within_term. intros.
simpl in *. subst. repeat find_higher_order_rewrite.
repeat destruct_update; simpl in *; eauto;
try find_rewrite_lem update_elections_data_appendEntries_leaderLogs; eauto;
solve [
eapply prefix_within_term_union;
[|idtac|eapply update_elections_data_appendEntries_allEntries; eauto]; eauto].
- unfold log_leaderLogs_prefix_within_term. intros.
simpl in *. subst. repeat find_higher_order_rewrite.
repeat destruct_update; simpl in *; eauto;
try find_rewrite_lem update_elections_data_appendEntries_leaderLogs; eauto;
find_apply_lem_hyp handleAppendEntries_log; intuition;
repeat find_rewrite; eauto;
match goal with
| |- context [?es ++ removeAfterIndex ?l _] =>
eapply prefix_within_term_union with (l1' := es) (l1'' := l)
end; eauto;
intros;
do_in_app; intuition; eauto using removeAfterIndex_in.
- unfold allEntries_log_prefix_within_term. intros.
simpl in *. subst. repeat find_higher_order_rewrite.
repeat destruct_update; simpl in *; eauto;
try find_rewrite_lem update_elections_data_appendEntries_leaderLogs; eauto;
find_apply_lem_hyp handleAppendEntries_log; intuition; subst; repeat find_rewrite;
try solve [
eapply prefix_within_term_union;
[|idtac|solve[eapply update_elections_data_appendEntries_allEntries; eauto]]; eauto].
Unshelve. all:eauto.
+ unfold prefix_within_term, allEntries_append_entries_prefix_within_term_nw in *.
intros.
find_apply_lem_hyp update_elections_data_appendEntries_allEntries. intuition.
eapply_prop_hyp pBody pBody; eauto. intuition; try lia.
find_apply_lem_hyp allEntries_gt_0_invariant; eauto. lia.
+ unfold prefix_within_term, allEntries_append_entries_prefix_within_term_nw in *.
intros.
find_apply_lem_hyp update_elections_data_appendEntries_allEntries. intuition.
do_in_app. intuition.
* {
copy_eapply_prop_hyp pBody pBody; eauto. intuition.
- subst. apply in_app_iff. right.
apply removeAfterIndex_le_In; auto.
break_exists. intuition.
assert (x = e) by (eapply allEntries_log_matching_invariant; eauto).
subst. auto.
- break_exists. intuition. subst.
(* use the fact that the lists are sorted to prove eTerm e = eTerm x, then use allEntries_log_pwt *)
match goal with
| _ : eIndex ?e < eIndex ?x |- _ =>
destruct (lt_eq_lt_dec (eTerm e) (eTerm x))
end; intuition; try lia.
+ exfalso.
eapply append_entries_request_term_sanity_invariant in H1; eauto.
conclude_using eauto; lia.
+ apply in_app_iff. right.
apply removeAfterIndex_le_In; [lia|].
eapply_prop allEntries_log_prefix_within_term; eauto; lia.
}
* find_copy_apply_lem_hyp removeAfterIndex_in;
find_apply_lem_hyp removeAfterIndex_In_le; [|eapply entries_sorted_invariant; eauto].
apply in_app_iff; right. apply removeAfterIndex_le_In; eauto; try lia.
eapply_prop allEntries_log_prefix_within_term; eauto.
+ unfold prefix_within_term, allEntries_append_entries_prefix_within_term_nw in *.
intros.
eapply_prop_hyp pBody pBody; eauto. intuition; try lia.
find_apply_lem_hyp allEntries_gt_0_invariant; eauto. lia.
+ unfold prefix_within_term, allEntries_append_entries_prefix_within_term_nw in *.
intros.
do_in_app. intuition.
* {
copy_eapply_prop_hyp pBody pBody; eauto. intuition.
- subst. apply in_app_iff. right.
apply removeAfterIndex_le_In; auto.
break_exists. intuition.
assert (x = e) by (eapply allEntries_log_matching_invariant; eauto).
subst. auto.
- break_exists. intuition. subst.
(* use the fact that the lists are sorted to prove eTerm e = eTerm x, then use allEntries_log_pwt *)
match goal with
| _ : eIndex ?e < eIndex ?x |- _ =>
destruct (lt_eq_lt_dec (eTerm e) (eTerm x))
end; intuition; try lia.
+ exfalso.
eapply append_entries_request_term_sanity_invariant in H1; eauto.
conclude_using eauto. lia.
+ apply in_app_iff. right.
apply removeAfterIndex_le_In; [lia|].
eapply_prop allEntries_log_prefix_within_term; eauto; lia.
}
* find_copy_apply_lem_hyp removeAfterIndex_in;
find_apply_lem_hyp removeAfterIndex_In_le; [|eapply entries_sorted_invariant; eauto].
apply in_app_iff; right. apply removeAfterIndex_le_In; eauto; try lia.
eapply_prop allEntries_log_prefix_within_term; eauto.
- (* nw nw invariant *)
unfold allEntries_append_entries_prefix_within_term_nw. intros.
simpl in *. subst. repeat find_higher_order_rewrite.
repeat destruct_update; simpl in *; eauto;
try find_rewrite_lem update_elections_data_appendEntries_leaderLogs; eauto.
+ find_apply_lem_hyp update_elections_data_appendEntries_allEntries.
intuition.
* eapply_prop_hyp allEntries_append_entries_prefix_within_term_nw In;
try eapply H13; eauto; repeat find_rewrite; intuition.
find_apply_hyp_hyp. intuition; [in_crush|].
subst. simpl in *. subst.
unfold handleAppendEntries in *. repeat break_match; find_inversion; congruence.
*
find_eapply_lem_hyp append_entries_append_entries_prefix_within_term_invariant.
conclude_using eauto. find_rewrite.
eapply H0.
5: { eauto. }
all:eauto.
find_apply_hyp_hyp. intuition; [find_rewrite; in_crush|].
repeat (subst; simpl in *).
unfold handleAppendEntries in *; repeat break_match; find_inversion; congruence.
+ find_apply_hyp_hyp. intuition.
* copy_eapply_prop_hyp allEntries_append_entries_prefix_within_term_nw pBody;
[|repeat find_rewrite; in_crush].
repeat conclude_using eauto. repeat find_rewrite. intuition.
* exfalso.
subst. simpl in *. subst.
unfold handleAppendEntries in *; repeat break_match; find_inversion; congruence.
- (* trivial *)
unfold append_entries_leaderLogs_prefix_within_term. intros.
simpl in *. find_apply_hyp_hyp. intuition;
[|subst; simpl in *; subst;
unfold handleAppendEntries in *; repeat break_match; find_inversion; congruence].
repeat find_higher_order_rewrite.
repeat destruct_update; simpl in *; eauto;
try find_rewrite_lem update_elections_data_appendEntries_leaderLogs; eauto.
- unfold append_entries_log_prefix_within_term. intros. simpl in *. subst.
find_apply_hyp_hyp.
intuition; [| subst; simpl in *; subst;
unfold handleAppendEntries in *; repeat break_match;
find_inversion; congruence].
find_higher_order_rewrite.
destruct_update; simpl in *; eauto.
find_apply_lem_hyp handleAppendEntries_log. intuition; subst; repeat find_rewrite; eauto.
+ unfold prefix_within_term. intros.
find_copy_apply_lem_hyp append_entries_append_entries_prefix_within_term_invariant.
match goal with
| H : append_entries_append_entries_prefix_within_term_nw _,
_ : pBody ?p = AppendEntries ?t ?n ?pli ?plt ?es ?ci,
_ : pBody ?p' = AppendEntries ?t' ?n' ?pli' ?plt' ?es' ?ci',
_ : In ?e' ?es', _ : In ?e ?es |-
In ?e ?es' =>
specialize (H p t n pli plt es ci p' t' n' pli' plt' es' ci' e e')
end.
conclude_using ltac:(repeat find_rewrite; in_crush).
concludes.
conclude_using ltac:(repeat find_rewrite; in_crush).
repeat concludes. intuition.
* exfalso.
match goal with
| _ : eIndex ?e = 0 |- _ =>
cut (eIndex e > 0); [intuition|]
end.
eapply entries_gt_0_nw_invariant; [|idtac|idtac|eauto]; [|idtac|eauto]; eauto.
* lia.
+ { unfold prefix_within_term. intros.
do_in_app. intuition.
- find_copy_apply_lem_hyp append_entries_append_entries_prefix_within_term_invariant.
match goal with
| H : append_entries_append_entries_prefix_within_term_nw _,
_ : pBody ?p = AppendEntries ?t ?n ?pli ?plt ?es ?ci,
_ : pBody ?p' = AppendEntries ?t' ?n' ?pli' ?plt' ?es' ?ci',
_ : In ?e' ?es', _ : In ?e ?es |-
In ?e (?es' ++ _) =>
specialize (H p t n pli plt es ci p' t' n' pli' plt' es' ci' e e')
end.
conclude_using ltac:(repeat find_rewrite; in_crush).
concludes.
conclude_using ltac:(repeat find_rewrite; in_crush).
repeat concludes. intuition.
+ (* use  log matching *)
break_exists. intuition.
subst.
apply in_app_iff. right.
apply removeAfterIndex_le_In; auto.
eapply entries_match_nw_host_invariant.
8: { eauto. }
3: { eauto. }
all:eauto.
+ (* contra terms_ge_prevTerm *)
find_eapply_lem_hyp append_entries_request_term_sanity_invariant; eauto.
repeat find_rewrite.
match goal with
| _ : ?x >= ?y, _ : ?x <= ?y |- _ =>
assert (x = y) by eauto using le_antisym
end. subst.
break_exists. intuition.
subst.
apply in_app_iff. right.
apply removeAfterIndex_le_In; auto; [lia|].
eapply_prop append_entries_log_prefix_within_term.
6: { eauto. }
5: { eauto. }
2: { eauto. }
all:eauto; repeat find_rewrite; intuition.
- apply in_app_iff. right.
find_copy_apply_lem_hyp removeAfterIndex_in.
find_apply_lem_hyp removeAfterIndex_In_le; [|eapply entries_sorted_invariant; eauto].
eapply removeAfterIndex_le_In; [lia|].
eapply_prop append_entries_log_prefix_within_term.
6: { eauto. }
5: { eauto. }
2: { eauto. }
all:eauto.
}
Qed.
Proof using.
intros. find_apply_lem_hyp handleClientRequest_log.
intuition; repeat find_rewrite; intuition.
break_exists; intuition; repeat find_rewrite; simpl in *; intuition.
subst. auto.
Qed.

Lemma prefix_within_term_inductive_client_request :
refined_raft_net_invariant_client_request prefix_within_term_inductive.