Section CommonTheorems.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Proof using.
unfold uniqueIndices.
eauto using NoDup_map_elim.
Qed.
Proof using.
intros.
simpl in *. intuition;
find_apply_hyp_hyp; intuition.
Qed.
Proof using.
induction ys; intros; simpl in *.
- break_match; intuition.
- break_match; intuition.
subst. apply sorted_cons; eauto.
intros. eauto using subseq_In.
Qed.
Proof using.
induction l; intros.
- simpl in *. intuition.
- simpl in *. intuition.
+ subst. auto with *.
+ find_apply_hyp_hyp. lia.
Qed.
Proof using.
induction l; intros.
- simpl in *. intuition.
- simpl in *. intuition.
+ subst. auto with *.
+ find_apply_hyp_hyp. lia.
Qed.
Proof using.
intros. intro.
find_apply_lem_hyp maxIndex_is_max; auto.
subst. lia.
Qed.
Proof using.
destruct l; intros; simpl in *; eauto; congruence.
Qed.
Proof using.
induction l; intros; simpl; auto.
repeat break_match; intuition.
- find_inversion. eauto using subseq_refl.
- right. find_reverse_rewrite. auto.
Qed.
Proof using.
intros. eauto using removeAfterIndex_subseq, sorted_subseq.
Qed.
Proof using.
eauto using removeAfterIndex_subseq, subseq_In.
Qed.
Proof using.
induction l; intros; intro.
- intuition.
- simpl in *. break_match; try discriminate. intuition.
+ subst. rewrite <- beq_nat_refl in *. discriminate.
+ find_copy_apply_hyp_hyp. intuition. break_if; do_bool; eauto. lia.
Qed.
Proof using.
induction l; intros.
- discriminate.
- simpl in *. break_match.
+ find_inversion. auto.
+ break_if; eauto; discriminate.
Qed.
Proof using.
induction l; intros.
- discriminate.
- simpl in *. break_match.
+ find_inversion. apply beq_nat_true in Heqb. auto.
+ break_if; eauto; discriminate.
Qed.
Proof using.
eauto using subseq_NoDup, removeAfterIndex_subseq.
Qed.
Notation disjoint xs ys := (forall x, In x xs -> In x ys -> False).
Proof using.
induction xs; intros.
- intuition.
- simpl in *. break_if; simpl in *; intuition.
subst. do_bool. lia.
Qed.
Proof using.
induction xs; intros.
- simpl in *. intuition.
- simpl in *.
break_if; simpl in *; do_bool; intuition; subst; auto.
find_apply_hyp_hyp. lia.
Qed.
Proof using.
induction xs; intros.
- simpl in *. intuition.
- simpl in *.
break_match; simpl in *; intuition;
subst; do_bool;
match goal with
| e : entry, H : forall _, _ = _ \/ _ -> _ |- _ =>
specialize (H e)
end;
intuition.
+ eauto using removeAfterIndex_le_In.
+ find_apply_hyp_hyp. intuition.
match goal with
| _ : eIndex ?e <= ?li, _ : eIndex ?e > eIndex ?e' |- _ =>
assert (eIndex e' <= li) by lia
end.
eauto using removeAfterIndex_le_In.
Qed.
Proof using.
induction xs; intros.
- reflexivity.
- simpl.
find_copy_apply_hyp_hyp.
repeat (break_if; simpl in *; intuition); try discriminate.
do_bool. lia.
Qed.
Proof using.
induction xs; intros; simpl.
- auto.
- repeat (break_match; simpl); intuition; try discriminate.
+ eauto using subseq_refl.
+ do_bool. assert (j < i) by lia.
rewrite removeAfterIndex_le with (j := i) (i := j) at 1; auto; lia.
+ do_bool. assert (i < j) by lia.
rewrite removeAfterIndex_le with (i := i) (j := j) at 2; auto; lia.
Qed.
Proof using.
auto using subseq_subseq_eq, removeAfterIndex_2_subseq.
Qed.
Proof using.
intros.
pose proof Min.min_spec i j. intuition.
- find_rewrite. rewrite removeAfterIndex_le with (i := i) (j := j) at 2;
eauto using removeAfterIndex_comm; lia.
- find_rewrite.
rewrite <- removeAfterIndex_le with (i := j) (j := i);
auto; lia.
Qed.
Proof using.
induction xs; intros; simpl in *; intuition; break_match; try discriminate.
- subst. do_bool. congruence.
- do_bool. break_if; eauto.
do_bool. find_apply_hyp_hyp. intuition.
Qed.
Proof using.
intros.
eapply NoDup_map_elim with (f := eIndex); eauto using findAtIndex_in, removeAfterIndex_in.
apply findAtIndex_index in H0.
apply findAtIndex_index in H1.
congruence.
Qed.
Proof using.
unfold uniqueIndices.
induction ys; intros.
- simpl in *. break_match; intuition.
- simpl in *. break_match; intuition.
+ simpl. constructor.
+ subst. simpl in *. invc H0. constructor; auto.
intro. apply H3.
eapply subseq_In; eauto.
apply subseq_map. auto.
+ subst. invc H0. eauto.
Qed.
Proof using.
induction xs; intros.
- simpl. auto.
- simpl. repeat break_match; auto.
+ find_inversion. eauto.
+ congruence.
Qed.
Proof using.
eauto using subseq_In, subseq_findGtIndex.
Qed.
Proof using.
induction entries; intros.
- simpl in *. intuition.
- simpl in *. break_if; intuition.
+ subst. in_crush.
+ subst. do_bool. lia.
+ do_bool. find_apply_hyp_hyp. lia.
Qed.
Definition contiguous_range_exact_lo xs lo :=
(forall i,
lo < i <= maxIndex xs ->
exists e,
eIndex e = i /\
In e xs) /\
(forall e,
In e xs ->
lo < eIndex e).
Proof with eauto using subseq_uniqueIndices, removeAfterIndex_subseq.
intros...
Qed.
Proof using.
destruct xs; intros.
- simpl. lia.
- destruct ys; simpl in *.
+ match goal with
| [ H : forall _, _ = _ \/ _ -> _, a : entry |- _ ] =>
solve [ specialize (H a); intuition ]
end.
+ match goal with
| [ H : forall _, _ = _ \/ _ -> _ |- eIndex ?a <= _ ] =>
specialize (H a); intuition
end; subst; auto.
find_apply_hyp_hyp. lia.
Qed.
Proof using.
destruct xs; intros.
- simpl in *. lia.
- simpl in *. eauto.
Qed.
Proof using.
induction l; intuition.
Qed.
Proof using.
intros.
apply maxIndex_subset; eauto using removeAfterIndex_sorted.
intros. eauto using removeAfterIndex_in.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
- subst. break_if; do_bool; try lia.
reflexivity.
- break_if; simpl in *.
+ do_bool.
match goal with
| H : forall _, In _ _ -> _ |- _ =>
specialize (H2 e)
end. intuition. lia.
+ eauto.
Qed.
Proof using.
intros.
destruct es; simpl in *.
- lia.
- congruence.
Qed.
Proof using one_node_params.
intros new current prev e Hsorted Huniq Hinv. intros. red. intros.
intuition.
- destruct (le_lt_dec i prev).
+ unfold contiguous_range_exact_lo in *. intuition.
match goal with
| H: forall _, _ < _ <= _ current -> _, H' : In _ current |- _ =>
specialize (H i); apply maxIndex_is_max in H'; auto; forward H; intuition
end.
break_exists. exists x. intuition.
apply in_or_app. right. subst.
eapply removeAfterIndex_le_In; eauto.
+ pose proof maxIndex_app new (removeAfterIndex current prev). intuition.
* find_rewrite.
unfold contiguous_range_exact_lo in *. intuition.
match goal with
| H: forall _, _ < _ <= _ new -> _ |- _ =>
specialize (H i); auto; forward H; intuition
end. break_exists. exists x. intuition.
* subst. simpl in *. clean.
exfalso.
pose proof maxIndex_removeAfterIndex current (eIndex e) e.
intuition.
- unfold contiguous_range_exact_lo in *.
do_in_app. intuition.
+ firstorder auto with zarith.
+ firstorder using removeAfterIndex_in.
Qed.
Proof using.
intros.
exists x. intuition.
Qed.
Proof using.
induction entries; intros; simpl in *; intuition.
- break_if; simpl in *; intuition; right; eapply IHentries; eauto.
- break_if;
simpl in *; intuition.
+ do_bool. subst. lia.
+ simpl in *; intuition; eapply IHentries; eauto.
Qed.
Proof using.
intros entries x Hsorted; intros. specialize (H i).
conclude H ltac:(lia).
break_exists. exists x0. intuition.
apply findGtIndex_sufficient; auto; lia.
Qed.
Proof using.
intros. destruct entries; simpl; auto.
break_if; simpl; intuition.
Qed.
Proof using.
intros.
pose proof findAtIndex_in _ _ _ H.
pose proof findAtIndex_index _ _ _ H.
eapply uniqueIndices_elim_eq; eauto.
Qed.
Definition entries_match' entries entries' :=
forall e e' e'',
eIndex e = eIndex e' ->
eTerm e = eTerm e' ->
In e entries ->
In e' entries' ->
eIndex e'' <= eIndex e ->
(In e'' entries -> In e'' entries').
Proof using.
unfold entries_match, entries_match'.
intros. intuition.
- eapply H; eauto.
- eapply (H e' e); eauto with *.
Qed.
Definition contiguous
(prevLogIndex : logIndex)
(prevLogTerm : term)
(leaderLog entries : list entry) : Prop :=
(prevLogIndex = 0 \/
exists e, findAtIndex leaderLog prevLogIndex = Some e /\
eTerm e = prevLogTerm) /\
(forall e,
In e leaderLog ->
eIndex e > prevLogIndex ->
eIndex e <= maxIndex entries ->
In e entries) /\
forall e e',
eIndex e = eIndex e' ->
eTerm e = eTerm e' ->
In e entries ->
In e' leaderLog ->
e = e'.
Proof using.
unfold entries_match. intuition.
Qed.
Proof using.
intros.
unfold entries_match in *.
intros. intuition.
- apply H with (e:=e')(e':=e); auto.
repeat find_rewrite. auto.
- apply H with (e:=e')(e':=e); auto.
repeat find_rewrite. auto.
Qed.
Proof using.
unfold advanceCurrentTerm. intros.
break_if; auto.
Qed.
Proof using.
unfold tryToBecomeLeader.
intros. find_inversion. auto.
Qed.
Proof using.
unfold handleRequestVote.
intros.
repeat (break_match; try discriminate; repeat (find_inversion; simpl in *));
auto using advanceCurrentTerm_same_log.
Qed.
Proof using.
unfold handleRequestVoteReply.
intros. repeat break_match; simpl; auto using advanceCurrentTerm_same_log.
Qed.
Proof using.
unfold advanceCurrentTerm. intros.
break_if; auto.
Qed.
Proof using.
intros. unfold handleTimeout, tryToBecomeLeader in *.
break_match; find_inversion; subst; auto.
Qed.
Proof using.
intros. unfold handleClientRequest in *.
break_match; find_inversion; subst; auto.
Qed.
Proof using.
unfold tryToBecomeLeader.
intros. find_inversion. auto.
Qed.
Proof using.
unfold handleRequestVote.
intros.
repeat (break_match; try discriminate; repeat (find_inversion; simpl in *));
auto using advanceCurrentTerm_same_lastApplied.
Qed.
Proof using.
unfold handleRequestVoteReply.
intros. repeat break_match; simpl; auto using advanceCurrentTerm_same_lastApplied.
Qed.
Proof using.
eauto using findAtIndex_in, findAtIndex_index.
Qed.
Proof using.
intros. split.
- lia.
- etransitivity; eauto. apply maxIndex_is_max; auto.
Qed.
Proof using.
intros.
assert (x = x').
- eapply uniqueIndices_elim_eq; eauto.
- subst. auto.
Qed.
Proof using.
induction l; intros.
- simpl in *. intuition.
- simpl in *. intuition; break_if; subst; do_bool.
+ auto.
+ congruence.
+ f_equal. eauto using uniqueIndices_elim_eq with *.
+ break_if; eauto.
* do_bool. find_apply_hyp_hyp. lia.
* eapply IHl; auto. unfold uniqueIndices in *.
simpl in *. solve_by_inversion.
Qed.
Proof using.
intros; induction l; simpl; auto.
- unfold uniqueIndices. simpl. constructor.
- unfold uniqueIndices in *. simpl in *. intuition. constructor; eauto.
intuition. do_in_map. find_apply_hyp_hyp. lia.
Qed.
Proof using.
intros.
apply findAtIndex_intro; auto using sorted_uniqueIndices.
Qed.
Proof using.
unfold doLeader.
intros.
repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
intros.
unfold handleAppendEntriesReply in *.
repeat (break_match; repeat (find_inversion; simpl in *)); auto using advanceCurrentTerm_same_log.
Qed.
Proof using.
intros.
unfold handleAppendEntriesReply in *.
repeat (break_match; repeat (find_inversion; simpl in *)); auto using advanceCurrentTerm_same_lastApplied.
Qed.
Proof using.
intros.
unfold handleAppendEntries in *.
repeat (break_match; repeat (find_inversion; simpl in *)); auto using advanceCurrentTerm_same_lastApplied.
Qed.
Definition term_of (m : msg) :=
match m with
| RequestVote t _ _ _ => Some t
| RequestVoteReply t _ => Some t
| AppendEntries t _ _ _ _ _ => Some t
| AppendEntriesReply t _ _ => Some t
end.
Proof using.
intros.
unfold wonElection in *. do_bool.
lia.
Qed.
Proof using.
intros.
find_eapply_lem_hyp subset_length;
eauto using name_eq_dec, wonElection_length.
Qed.
Proof using.
unfold wonElection.
intros.
destruct l; try discriminate.
simpl. eauto.
Qed.
Proof using.
intros. induction l; simpl in *; intuition.
find_rewrite. break_match; intuition.
repeat find_higher_order_rewrite. auto.
Qed.
Proof using.
intros. destruct l; simpl in *; intuition.
repeat break_match; congruence.
Qed.
Proof using.
induction l; intros; simpl in *; [congruence|].
repeat break_match; find_inversion.
- do_bool.
match goal with
| H : forall _, Some ?a = Some _ -> _ |- _ =>
specialize (H a)
end. intuition; subst; auto.
find_apply_hyp_hyp. lia.
- do_bool.
match goal with
| H : forall _, Some ?a = Some _ -> _ |- _ =>
specialize (H a)
end. intuition; subst; auto.
find_apply_hyp_hyp. lia.
- intuition; subst; auto.
find_apply_lem_hyp argmax_None.
subst. solve_by_inversion.
Qed.
Proof using.
intros. find_apply_lem_hyp argmax_elim. intuition.
Qed.
Proof using.
intros. induction l; simpl in *; intuition.
conclude IHl intuition.
conclude IHl intuition. intuition.
- find_rewrite. break_match; intuition.
repeat break_if; intuition.
+ do_bool. right.
find_apply_lem_hyp argmax_in; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition;
specialize (H0 a1); specialize (H a0); intuition; repeat find_rewrite; lia.
+ do_bool. right.
find_apply_lem_hyp argmax_in; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition.
* specialize (H a1); specialize (H0 a0); intuition. repeat find_rewrite. lia.
* specialize (H a1); specialize (H0 a0); intuition. repeat find_rewrite. lia.
- find_rewrite. repeat break_match; subst; intuition.
do_bool.
repeat find_apply_lem_hyp argmax_elim; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition.
+ specialize (H a0); specialize (H0 a1); intuition. repeat find_rewrite. lia.
+ pose proof H a0; pose proof H a1; intuition. repeat find_rewrite.
specialize (H3 a1). intuition. lia.
Qed.
Proof using.
intros. induction l; simpl in *; intuition.
find_rewrite. break_match; intuition.
repeat find_higher_order_rewrite. auto.
Qed.
Proof using.
intros. destruct l; simpl in *; intuition.
repeat break_match; congruence.
Qed.
Proof using.
induction l; intros; simpl in *; [congruence|].
repeat break_match; find_inversion.
- do_bool.
match goal with
| H : forall _, Some ?a = Some _ -> _ |- _ =>
specialize (H a)
end. intuition; subst; auto.
find_apply_hyp_hyp. lia.
- do_bool.
match goal with
| H : forall _, Some ?a = Some _ -> _ |- _ =>
specialize (H a)
end. intuition; subst; auto.
find_apply_hyp_hyp. lia.
- intuition; subst; auto.
find_apply_lem_hyp argmin_None.
subst. solve_by_inversion.
Qed.
Proof using.
intros. find_apply_lem_hyp argmin_elim. intuition.
Qed.
Proof using.
intros. induction l; simpl in *; intuition.
conclude IHl intuition.
conclude IHl intuition. intuition.
- find_rewrite. break_match; intuition.
repeat break_if; intuition.
+ do_bool. right.
find_apply_lem_hyp argmin_in; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition;
specialize (H0 a1); specialize (H a0); intuition; repeat find_rewrite; lia.
+ do_bool. right.
find_apply_lem_hyp argmin_in; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition.
* specialize (H a1); specialize (H0 a0); intuition. repeat find_rewrite. lia.
* specialize (H a1); specialize (H0 a0); intuition. repeat find_rewrite. lia.
- find_rewrite. repeat break_match; subst; intuition.
do_bool.
repeat find_apply_lem_hyp argmin_elim; intuition.
destruct (A_eq_dec a a1); destruct (A_eq_dec a a0); repeat subst; intuition.
+ specialize (H a0); specialize (H0 a1); intuition. repeat find_rewrite. lia.
+ pose proof H a0; pose proof H a1; intuition. repeat find_rewrite.
specialize (H3 a1). intuition. lia.
Qed.
Proof using.
intros.
unfold applied_entries in *.
repeat break_match; intuition;
try solve [find_apply_lem_hyp argmax_None;
exfalso;
pose proof (all_fin_all _ h); find_rewrite; intuition].
match goal with
| _ : argmax ?f ?l = _, _ : argmax ?g ?l = _ |- _ =>
pose proof argmax_one_different name name_eq_dec g f l h as Hproof
end.
forward Hproof; [intros; rewrite_update; intuition|]; concludes.
forward Hproof; [intros; update_destruct; subst; rewrite_update; intuition|]; concludes.
intuition.
- repeat find_rewrite. find_inversion.
update_destruct; subst; rewrite_update; intuition. left.
intuition. eexists; intuition eauto. repeat find_apply_lem_hyp argmax_elim; intuition eauto.
match goal with
H : _ |- _ =>
solve [specialize (H h); rewrite_update; eauto using all_fin_all]
end.
- repeat find_rewrite. find_inversion. rewrite_update. intuition.
Qed.
Proof using.
intros. unfold applied_entries in *.
repeat break_match; repeat find_rewrite; intuition;
match goal with
| _ : argmax ?f ?l = _, _ : argmax ?g ?l = _ |- _ =>
assert (argmax f l = argmax g l) by
(apply argmax_fun_ext; intros; update_destruct; subst; rewrite_update; auto)
end; repeat find_rewrite; try congruence.
match goal with | H : Some _ = Some _ |- _ => inversion H end.
subst. clean.
f_equal. update_destruct; subst; rewrite_update; repeat find_rewrite; auto.
Qed.
Proof using.
intros.
unfold applied_entries in *.
rewrite argmax_fun_ext with (g := fun h : name => lastApplied (sigma h)); intuition.
break_match; auto.
repeat find_higher_order_rewrite. auto.
Qed.
Proof using.
intros.
apply applied_entries_log_lastApplied_same;
intros; update_destruct; subst; rewrite_update; auto.
Qed.
Proof using.
intros.
unfold applied_entries in *.
break_match; simpl in *; intuition eauto.
Qed.
Proof using.
intros; induction l; simpl in *; intuition eauto using app_nil_r.
break_exists. break_if; [exists nil; eauto|].
do_bool.
match goal with
| l : list entry, e : entry |- _ =>
solve [exists (e :: l); simpl in *; f_equal; auto]
end.
Qed.
Proof using.
intros.
unfold entries_match. intuition.
- match goal with
| [ H : _ |- _ ] => solve [eapply H; eauto]
end.
- match goal with
| [ H : forall _ _, _, H' : eIndex ?e1 = eIndex ?e2 |- _ ] =>
specialize (H e1 e2); do 4 concludes
end. intuition.
match goal with
| [ H : forall _, _ < _ <= _ -> _,
_ : eIndex ?e3 <= eIndex _
|- _ ] =>
specialize (H (eIndex e3));
conclude H
ltac:(split; [eauto|
eapply le_trans; eauto; apply maxIndex_is_max; eauto])
end.
break_exists. intuition.
match goal with
| [ _ : In ?x _,
_ : eIndex ?x = eIndex ?e3,
_ : eIndex ?e3 <= eIndex _ |- _ ] =>
eapply rachet with (x' := x); eauto using sorted_uniqueIndices
end.
match goal with
| [ H : _ |- _ ] => solve [ eapply H; eauto; congruence ]
end.
Qed.
Ltac use_entries_match :=
match goal with
| [ _ : eIndex ?e1 = eIndex ?e2,
H : context [entries_match]
|- _ ] =>
first [ solve [eapply H with (e:=e2)(e':=e1); eauto; congruence] |
solve [eapply H with (e:=e1)(e':=e2); eauto; congruence]]
end.
Proof using.
intros.
unfold entries_match. intros. split; intros.
- in_crush_start.
+ match goal with
| [ H : _ |- _ ] => solve [eapply H; eauto]
end.
+ exfalso.
find_apply_lem_hyp removeAfterIndex_In_le; intuition.
find_apply_hyp_hyp. lia.
+ find_apply_lem_hyp findAtIndex_elim.
intuition subst.
match goal with
| [ H : forall _ _, _, H' : eIndex ?e1 = eIndex ?e2 |- _ ] =>
specialize (H e1 e2); do 4 concludes
end.
intuition. break_exists.
intuition.
find_copy_apply_lem_hyp removeAfterIndex_In_le; intuition.
find_apply_lem_hyp removeAfterIndex_in.
use_entries_match.
+ repeat find_apply_lem_hyp removeAfterIndex_in. use_entries_match.
- in_crush_start.
+ match goal with
| [ H : forall _ _, _, H' : eIndex ?e1 = eIndex ?e2 |- _ ] =>
specialize (H e1 e2); do 4 concludes
end.
intuition. break_exists. intuition.
destruct (le_lt_dec (eIndex e'') pli).
* apply in_or_app. right.
apply removeAfterIndex_le_In; auto.
find_apply_lem_hyp findAtIndex_elim. intuition.
subst. use_entries_match.
* apply in_or_app. left.
match goal with
| H : forall _, _ < _ <= _ -> _ |- In ?e _ =>
specialize (H (eIndex e))
end.
intuition.
conclude_using ltac:(eapply le_trans; eauto; apply maxIndex_is_max; eauto).
break_exists. intuition.
match goal with
| _: eIndex ?e1 = eIndex ?e2 |- context [ ?e2 ] =>
eapply rachet with (x' := e1); eauto using sorted_uniqueIndices with *
end.
+ apply in_or_app. right.
find_copy_apply_lem_hyp removeAfterIndex_In_le; eauto.
apply removeAfterIndex_le_In; [lia|].
find_apply_lem_hyp removeAfterIndex_in.
use_entries_match.
Qed.
Proof using.
intros.
apply applied_entries_log_lastApplied_same.
- intros. update_destruct; subst; rewrite_update; auto.
eapply doLeader_same_log; eauto.
- intros. update_destruct; subst; rewrite_update; auto.
unfold doLeader in *. repeat break_match; find_inversion; auto.
Qed.
Proof using.
induction es; intros; simpl in *; intuition.
- find_inversion. destruct st'; repeat eexists; eauto.
- unfold cacheApplyEntry, applyEntry in *.
repeat break_match; repeat find_inversion;
find_apply_hyp_hyp; break_exists; repeat eexists; eauto.
Qed.
Proof using.
intros.
find_apply_lem_hyp applyEntries_spec.
break_exists. subst. eauto.
Qed.
Proof using.
unfold handleClientRequest.
intros.
repeat break_match; find_inversion; auto.
Qed.
Proof using.
unfold handleTimeout, tryToBecomeLeader; intros; repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
unfold handleAppendEntriesReply, advanceCurrentTerm.
intros.
repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
unfold handleRequestVote, advanceCurrentTerm.
intros.
repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
unfold handleRequestVoteReply, advanceCurrentTerm.
intros. repeat break_match; simpl; auto.
Qed.
Proof using.
unfold doGenericServer.
intros.
repeat break_match; repeat find_inversion; simpl;
eapply applyEntries_spec_ind; eauto.
Qed.
Functional Scheme div2_ind := Induction for div2 Sort Prop.
Proof using.
intro n. functional induction (div2 n); lia.
Qed.
Proof using.
intros n. functional induction (div2 n); intros; try lia.
specialize (IHn0 (pred a) (pred b)). lia.
Qed.
Proof using.
intros. unfold wonElection in *. do_bool.
cut (exists h, In h (dedup name_eq_dec l) /\ In h (dedup name_eq_dec l'));
[intros; break_exists; exists x; intuition eauto using in_dedup_was_in|].
eapply pigeon with (l := nodes); eauto using all_fin_all, all_fin_NoDup, NoDup_dedup, name_eq_dec, div2_correct.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl in *. repeat break_let.
rewrite IHxs. break_let. find_inversion. auto.
Qed.
Proof using.
induction log; intros.
- simpl. rewrite app_nil_r. auto.
- simpl. break_let. rewrite IHlog. rewrite app_ass. simpl.
rewrite IHlog with (tr := [(eInput a, o)]).
auto.
Qed.
Proof using.
induction log; intros.
- auto.
- simpl. break_let. rewrite IHlog.
rewrite IHlog with (tr := [(eInput a, o)]).
auto.
Qed.
Proof using.
induction log; intros.
- simpl. constructor.
- simpl. break_let.
rewrite fst_execute_log'.
rewrite snd_execute_log'.
unfold step_1_star in *.
econstructor.
+ constructor. eauto.
+ auto.
Qed.
Proof using.
intros. apply execute_log_correct'.
Qed.
Proof using.
unfold contiguous_range_exact_lo. intuition.
- simpl in *. lia.
- contradiction.
Qed.
Proof using.
intros. unfold contiguous_range_exact_lo in *. intuition.
specialize (H1 a).
specialize (H0 (S i)).
assert (H_a: In a [a]) by auto with datatypes.
concludes.
assert (H_s: i < S i) by auto.
concludes.
break_exists.
break_and.
inversion H0; subst; auto.
inversion H2.
Qed.
Proof using.
intros. unfold contiguous_range_exact_lo in *. intuition.
assert (i < S (eIndex b) <= eIndex a).
simpl in *. intuition. specialize (H0 b). concludes. intuition.
specialize (H1 (S (eIndex b))). concludes.
break_exists. simpl In in *. intuition; subst.
- auto.
- lia.
- simpl in *. intuition. specialize (H x). concludes. lia.
Qed.
Proof using.
induction l; intros.
- apply contiguous_nil.
- eapply contiguous_index_adjacent in H; eauto.
unfold contiguous_range_exact_lo in *. break_and.
intuition. simpl maxIndex in *. specialize (H0 i0).
assert (i < i0 <= eIndex a0) by lia.
concludes. break_exists. intuition. simpl in *. intuition; subst.
+ lia.
+ exists x. intuition.
+ exists x. intuition.
Qed.
Proof using.
induction l1; intros.
- auto.
- simpl ((a :: l1) ++ l2) in *.
find_apply_lem_hyp cons_contiguous_sorted; auto.
simpl in *. intuition.
Qed.
Proof using.
induction l; intros.
- find_apply_lem_hyp Prefix_nil. subst. auto.
- destruct l'.
+ auto.
+ simpl. split.
* intros. simpl in *. break_and. find_eapply_lem_hyp Prefix_in; eauto.
find_insterU. econcludes. subst. intuition.
* apply IHl; simpl in *; intuition.
Qed.
Proof using.
induction l; intros.
- contradiction.
- destruct l'; try congruence.
find_copy_apply_lem_hyp prefix_sorted; auto. simpl in *. intuition.
+ left. subst. reflexivity.
+ right. subst. destruct l'.
* find_apply_lem_hyp contiguous_index_singleton.
specialize (H0 e). concludes. lia.
* eapply IHl; try discriminate; eauto. eapply cons_contiguous_sorted; eauto.
simpl in *. intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
break_if; auto.
do_bool.
eapply IHl; eauto.
eapply cons_contiguous_sorted; eauto.
simpl; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; auto.
- constructor.
- constructor; intuition.
match goal with
| H : forall _, _ |- _ => specialize (H a)
end. intuition.
Qed.
Proof using.
induction l; intros.
- symmetry. apply Permutation_nil. assumption.
- destruct l'.
+ apply Permutation_nil. apply Permutation_sym. assumption.
+ simpl in *. intuition.
find_copy_eapply_lem_hyp Permutation_in; intuition.
find_copy_apply_lem_hyp Permutation_sym.
find_copy_eapply_lem_hyp Permutation_in; intuition.
simpl in *. intuition;
try (subst a; f_equal; eauto using Permutation_cons_inv).
repeat find_apply_hyp_hyp. intuition.
lia.
Qed.
Proof using.
intros. apply sorted_Permutation_eq;
try (apply removeAfterIndex_sorted; assumption).
apply NoDup_Permutation;
try (apply NoDup_removeAfterIndex; apply sorted_NoDup; assumption).
split; intros; apply removeAfterIndex_le_In;
eauto using removeAfterIndex_In_le, removeAfterIndex_in.
Qed.
Proof using.
intros.
eapply removeAfterIndex_same_sufficient; eauto.
intros.
unfold contiguous_range_exact_lo in *. intuition.
specialize (H7 (eIndex e)).
intuition. find_copy_apply_hyp_hyp.
repeat conclude_using lia.
break_exists. intuition.
symmetry in H9. copy_apply H4 H10; try lia.
eapply rachet with (xs := l'); eauto using sorted_uniqueIndices.
Qed.
Proof using one_node_params.
induction l; try congruence; intros.
destruct l'; simpl.
- contradiction.
- simpl Prefix in *. intuition. subst a. f_equal. break_if.
+ do_bool. unfold contiguous_range_exact_lo in *.
intuition.
simpl maxIndex in *.
specialize (H6 e).
specialize (H4 e).
assert (H_in': In e (e :: l')).
left. reflexivity.
concludes.
assert (H_in: In e (e :: l)).
left. reflexivity.
concludes.
lia.
+ destruct l.
{ do_bool. simpl in *. find_apply_lem_hyp contiguous_index_singleton. destruct l'.
- reflexivity.
- simpl. intuition. specialize (H0 e0).
assert (H_in: In e0 (e0 :: l')).
left.
reflexivity.
concludes. intuition. find_rewrite. break_if.
+ reflexivity.
+ do_bool. lia. }
{ apply IHl; try discriminate; auto.
- find_apply_lem_hyp cons_contiguous_sorted.
+ firstorder.
+ eauto using Prefix_cons, prefix_sorted.
- find_apply_lem_hyp cons_contiguous_sorted.
+ unfold contiguous_range_exact_lo in *.
inversion H1.
split; intros.
* do_bool.
simpl maxIndex in *.
destruct H2.
destruct H3.
destruct l'; simpl in *; intuition.
subst_max.
specialize (H2 i0).
specialize (H0 e1).
conclude_using eauto.
break_and.
assert (i < i0 <= eIndex e) by lia.
concludes.
break_exists.
break_and.
subst.
exists x.
split; auto.
break_or_hyp; auto.
lia.
* apply H2.
right; auto.
+ eauto using Prefix_cons, prefix_sorted.
- apply cons_contiguous_sorted in H3; auto.
}
Qed.
Proof using one_node_params.
intros.
rewrite removeAfterIndex_same_sufficient with (l := l'); auto.
- apply thing2; auto.
- unfold entries_match in *. intros. eapply H2; eauto.
- unfold entries_match in *. intros. eapply H2; eauto.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
break_if; intuition.
- f_equal. auto.
- do_bool. specialize (H a); intuition; lia.
Qed.
Proof using.
induction l1; intros; simpl in *; intuition.
- subst. break_if; do_bool; try lia.
eexists; repeat (simpl in *; intuition).
- specialize (H1 e); intuition. conclude H1 ltac:(apply in_app_iff; intuition).
break_if; do_bool; try lia. eexists; intuition; eauto.
simpl in *. intuition.
eapply_prop_hyp sorted sorted; eauto. break_exists; intuition.
find_rewrite. eauto.
Qed.
Proof using.
induction l1; intros; simpl in *; intuition.
subst. destruct l2; simpl in *; auto.
specialize (H2 e0); conclude_using eauto; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
break_if; simpl in *; intuition.
Qed.
Proof using.
induction l1; intros; simpl in *; intuition.
- eexists; intuition eauto using findGtIndex_Prefix.
- break_if; simpl in *; intuition.
+ eapply_prop_hyp sorted sorted; eauto.
break_exists; intuition; find_rewrite; eauto.
+ do_bool. specialize (H1 e); conclude H1 ltac:(apply in_app_iff; intuition).
lia.
Qed.
Proof using.
intros; induction l; simpl in *; intuition.
break_if; do_bool; intuition.
specialize (H a); intuition. lia.
Qed.
Proof using.
intros. induction l; simpl in *; auto.
repeat (break_if; simpl; intuition); do_bool;
try congruence.
symmetry. apply findGtIndex_nil.
intros. find_apply_lem_hyp removeAfterIndex_in.
find_apply_hyp_hyp. intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
- destruct l'; simpl in *; intuition.
break_if; do_bool; auto; lia.
- break_if; do_bool; auto.
f_equal. eauto.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
break_if; do_bool; auto.
- f_equal. eauto.
- exfalso.
destruct l'; simpl in *; intuition.
specialize (H1 e); conclude_using intuition; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
subst. destruct l'; simpl in *; intuition.
- exfalso. specialize (H0 e). intuition.
- exfalso. specialize (H3 e0). conclude_using intuition.
intuition.
Qed.
Proof using.
intros. induction l; simpl in *; intuition.
break_if; do_bool; simpl in *; intuition.
congruence.
Qed.
Proof using.
induction l'; intros; simpl in *; intuition.
- destruct l; simpl in *; auto.
specialize (H1 e); intuition.
- break_match; intuition. subst.
simpl in *. intuition. f_equal.
eapply IHl'; eauto.
intros.
specialize (H1 e0); intuition.
subst. specialize (H0 e0); intuition. lia.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
break_if; intuition.
do_bool. specialize (H a). intuition. lia.
Qed.
Proof using.
induction l; intros; simpl in *; intuition;
subst; break_if; do_bool; eauto using app_ass.
lia.
Qed.
Proof using.
induction l; intros; simpl in *; intuition;
subst; break_if; do_bool; eauto using app_ass.
specialize (H a). intuition. lia.
Qed.
Proof using.
intros; induction l; simpl in *; intuition.
break_if; auto. do_bool. lia.
Qed.
Proof using.
red. intuition.
- exists x. intuition. simpl in *. inv H2; [reflexivity | lia].
- simpl in *. intuition. subst. lia.
Qed.
Proof using.
intros. unfold contiguous_range_exact_lo in *. intuition.
- invc H4.
+ eexists; intuition.
+ find_rewrite. find_apply_lem_hyp Nat.succ_inj. subst.
assert (i < i0 <= maxIndex (y :: l)). simpl. lia.
find_apply_hyp_hyp. break_exists. simpl in *.
intuition; subst; eexists; intuition.
- simpl in *. intuition; subst; auto. specialize (H2 y). concludes. lia.
Qed.
Proof using.
Opaque sorted.
induction l1; intros.
- apply contiguous_nil.
- destruct l1; simpl in *; find_copy_apply_lem_hyp contiguous_index_adjacent; auto.
+ apply contiguous_singleton_sufficient. intuition.
+ intuition. eapply contiguous_adjacent_sufficient; auto.
eauto using contiguous_singleton_sufficient. eapply IHl1.
* eauto using sorted_subseq, subseq_skip, subseq_refl.
* eauto using cons_contiguous_sorted.
Transparent sorted.
Qed.
Proof using.
intros.
break_exists.
exists (rev x). subst. eauto using rev_app_distr.
Qed.
Proof using.
intros. subst. intuition.
Qed.
Proof using.
intros. subst.
destruct l2; eauto using app_nil_r.
simpl in *.
break_match; intuition. subst. simpl in *.
unfold contiguous_range_exact_lo in *.
intuition. specialize (H0 e0). conclude_using intuition.
lia.
Qed.
Proof using.
intros. induction l1; simpl in *; intuition;
eapply H0; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition;
break_match; intuition; repeat subst; simpl in *; auto.
intuition.
eapply_prop_hyp sorted sorted; eauto.
match goal with
| _ : eIndex _ <= maxIndex ?l |- _ =>
destruct l
end.
- simpl in *.
find_apply_lem_hyp Prefix_nil. subst. simpl in *. intuition.
- simpl in *.
match goal with
| [ H : forall _, _ = _ \/ In _ _ -> _, _ : eIndex _ <= eIndex ?e |- _ ] =>
specialize (H e)
end; intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
- destruct l'; simpl in *; intuition; subst; intuition.
find_apply_hyp_hyp. intuition.
- do_in_app. intuition. right. eapply IHl; eauto.
intuition.
Qed.
Proof using.
intros.
destruct l2.
- intuition. subst. rewrite app_nil_r in *. auto.
- match goal with H : _ \/ _ |- _ => clear H end.
simpl in *. break_match; intuition. subst. simpl.
eauto using contiguous_partition.
Qed.
Proof using.
intros.
induction l; simpl in *; intuition; repeat subst; auto;
find_apply_hyp_hyp; intuition.
Qed.
Proof using.
destruct l'.
- intros. simpl in *. rewrite app_nil_r in *.
eapply_prop (contiguous_range_exact_lo l). lia.
- intros. find_eapply_lem_hyp contiguous_app_prefix_contiguous; eauto.
left. intuition. congruence.
Qed.
Proof using.
induction l1; intros.
- simpl in *. intuition.
- rewrite <- app_comm_cons in *.
match goal with
| H : In _ _ |- _ => simpl in H
end. intuition.
+ subst. simpl in *. intuition.
destruct l2; simpl in *.
* unfold contiguous_range_exact_lo in *. intuition.
* match goal with
| H : _ |- eIndex _ > eIndex ?e =>
specialize (H e)
end. conclude_using intuition. intuition.
+ find_apply_lem_hyp cons_contiguous_sorted; eauto.
simpl in *. intuition.
Qed.
Proof using.
induction l; intros; simpl in *; intuition.
repeat break_match; simpl in *; intuition; find_apply_hyp_hyp; auto.
Qed.
Proof using.
induction l; intros; intuition.
simpl in *.
repeat break_if; simpl in *; repeat break_if;
repeat (do_bool; intuition); try lia.
simpl. f_equal.
rewrite IHl; eauto.
apply removeAfterIndex_eq.
intros.
find_apply_hyp_hyp. intuition.
Qed.
Proof using.
induction l; intros; intuition.
simpl in *.
repeat break_if; simpl in *; repeat break_if;
repeat (do_bool; intuition); lia.
Qed.
Proof using.
eauto using sorted_subseq, subseq_skip, subseq_refl.
Qed.
Proof using.
Opaque sorted.
induction l; intros.
- split.
+ simpl in *. find_apply_lem_hyp contiguous_index_singleton. assumption.
+ apply contiguous_nil.
- destruct l.
+ simpl in *. find_copy_apply_lem_hyp contiguous_index_adjacent; auto.
find_apply_lem_hyp cons_contiguous_sorted; auto.
find_copy_apply_lem_hyp contiguous_index_singleton.
intuition. eapply contiguous_singleton_sufficient. lia.
+ simpl in *. find_copy_apply_lem_hyp contiguous_index_adjacent; auto.
find_apply_lem_hyp cons_contiguous_sorted; auto.
find_apply_lem_hyp sorted_cons_elim. split.
* eapply IHl; eauto.
* eapply contiguous_adjacent_sufficient; intuition.
eapply IHl; eauto.
Transparent sorted.
Qed.

Lemma sorted_app_gt :
forall l1 l2 e1 e2,
sorted (l1 ++ l2) ->
In e1 l1 ->
In e2 l2 ->
eIndex e1 > eIndex e2.