Section LockServ.
Variable num_Clients : nat.
Definition Client_index := (fin num_Clients).
Inductive Name :=
| Client : Client_index -> Name
| Server : Name.
Definition list_Clients := map Client (all_fin num_Clients).
Definition Name_eq_dec : forall a b : Name, {a = b} + {a <> b}.
decide equality. apply fin_eq_dec.
Qed.
Inductive Msg :=
| Lock   : Msg
| Unlock : Msg
| Locked : Msg.
Definition Msg_eq_dec : forall a b : Msg, {a = b} + {a <> b}.
decide equality.
Qed.
Definition Input := Msg.
Definition Output := Msg.
Record Data := mkData { queue : list Client_index ; held : bool }.
Definition init_data (n : Name) : Data := mkData [] false.
Inductive Label :=
| InputLock : Client_index -> Label
| InputUnlock : Client_index -> Label
| MsgUnlock : Label
| MsgLock : Client_index -> Label
| MsgLocked : Client_index -> Label
| Nop
| Silent.
Definition Handler (S : Type) := GenHandler (Name * Msg) S Output Label.
Definition ClientNetHandler (i : Client_index) (m : Msg) : Handler Data :=
match m with
| Locked => (put (mkData [] true)) ;; write_output Locked ;; ret (MsgLocked i)
| _ => ret Nop
end.
Definition ClientIOHandler (i : Client_index) (m : Msg) : Handler Data :=
match m with
| Lock => send (Server, Lock) ;; ret (InputLock i)
| Unlock => data <- get ;;
when (held data)
(put (mkData [] false) >>
send (Server, Unlock));;
ret (InputUnlock i)
| _ => ret Nop
end.
Definition ServerNetHandler (src : Name) (m : Msg) : Handler Data :=
st <- get ;;
let q := queue st in
match m with
| Lock =>
match src with
| Server => ret Nop
| Client c =>
when (null q) (send (src, Locked)) >> put (mkData (q++[c]) (held st)) >> ret (MsgLock c)
end
| Unlock => match q with
| _ :: x :: xs => put (mkData (x :: xs) (held st)) >> send (Client x, Locked)
| _ => put (mkData [] (held st))
end ;;
ret MsgUnlock
| _ => ret Nop
end.
Definition ServerIOHandler (m : Msg) : Handler Data := ret Nop.
Definition NetHandler (nm src : Name) (m : Msg) : Handler Data :=
match nm with
| Client c => ClientNetHandler c m
| Server   => ServerNetHandler src m
end.
Definition InputHandler (nm : Name) (m : Msg) : Handler Data :=
match nm with
| Client c => ClientIOHandler c m
| Server   => ServerIOHandler m
end.
Ltac handler_unfold :=
repeat (monad_unfold; unfold NetHandler,
InputHandler,
ServerNetHandler,
ClientNetHandler,
ClientIOHandler,
ServerIOHandler in *).
Definition Nodes := Server :: list_Clients.
Proof using.
intros.
unfold Nodes, list_Clients.
simpl.
destruct n.
- right.
apply in_map.
apply all_fin_all.
- left.
reflexivity.
Qed.
Proof using.
unfold Nodes, list_Clients.
apply NoDup_cons.
- in_crush. discriminate.
- apply NoDup_map_injective.
+ intros. congruence.
+ apply all_fin_NoDup.
Qed.
Global Instance LockServ_BaseParams : BaseParams :=
{
data   := Data ;
input  := Input ;
output := Output
}.
Global Instance LockServ_LabeledParams : LabeledMultiParams LockServ_BaseParams :=
{
lb_name := Name ;
lb_msg  := Msg ;
lb_msg_eq_dec := Msg_eq_dec ;
lb_name_eq_dec := Name_eq_dec ;
lb_nodes := Nodes ;
lb_all_names_nodes := In_n_Nodes ;
lb_no_dup_nodes := nodup ;
label := Label ;
label_silent := Silent;
lb_init_handlers := init_data ;
lb_net_handlers := fun dst src msg s =>
runGenHandler s (NetHandler dst src msg) ;
lb_input_handlers := fun nm msg s =>
runGenHandler s (InputHandler nm msg)
}.
Global Instance LockServ_MultiParams : MultiParams LockServ_BaseParams :=
unlabeled_multi_params.
(* This is the fundamental safety property of the system: No two different clients can (think they) hold the lock at once. *)
Definition mutual_exclusion (sigma : name -> data) : Prop :=
forall m n,
held (sigma (Client m)) = true ->
held (sigma (Client n)) = true ->
m = n.
(* The system enforces mutual exclusion at the server. Whenever a client believs it holds the lock, that client is at the head of the server's queue. *)
Definition locks_correct (sigma : name -> data) : Prop :=
forall n,
held (sigma (Client n)) = true ->
exists t,
queue (sigma Server) = n :: t.
Proof using.
unfold locks_correct, mutual_exclusion.
intros.
repeat find_apply_hyp_hyp.
break_exists.
find_rewrite. find_inversion.
auto.
Qed.
Definition valid_unlock q h c p :=
pSrc p = Client c /\
(exists t, q = c :: t) /\
h = false.
Definition locks_correct_unlock (sigma : name -> data) (p : packet) : Prop :=
pBody p = Unlock ->
exists c, valid_unlock (queue (sigma Server)) (held (sigma (Client c))) c p.
Definition valid_locked q h c p :=
pDst p = Client c /\
(exists t, q = c :: t) /\
h = false.
Definition locks_correct_locked (sigma : name -> data) (p : packet) : Prop :=
pBody p = Locked ->
exists c, valid_locked (queue (sigma Server)) (held (sigma (Client c))) c p.
Definition LockServ_network_invariant (sigma : name -> data) (p : packet) : Prop :=
locks_correct_unlock sigma p /\
locks_correct_locked sigma p.
Definition LockServ_network_network_invariant (p q : packet) : Prop :=
(pBody p = Unlock -> pBody q = Unlock -> False) /\
(pBody p = Locked -> pBody q = Unlock -> False) /\
(pBody p = Unlock -> pBody q = Locked -> False) /\
(pBody p = Locked -> pBody q = Locked -> False).
Proof using.
unfold LockServ_network_network_invariant.
intuition.
Qed.
Proof using.
unfold locks_correct. simpl. discriminate.
Qed.
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion;
subst; simpl in *; subst; simpl in *.
- left. eexists. intuition.
- left. eexists. intuition.
- left. eexists. intuition.
- auto.
- auto.
Qed.
Proof using.
unfold locks_correct.
intuition.
destruct (Name_eq_dec (Client x) (Client n)).
- find_inversion. exfalso.
rewrite_update.
congruence.
- rewrite_update.
auto.
Qed.
Ltac set_up_input_handlers :=
intros;
find_apply_lem_hyp InputHandler_cases;
intuition idtac; try break_exists; intuition idtac; subst;
repeat find_rewrite;
simpl in *; intuition idtac; repeat find_inversion;
try now rewrite update_nop_ext.
Proof using.
set_up_input_handlers;
auto using locks_correct_update_false.
Qed.
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion; subst; intuition (auto; congruence).
Qed.
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion; subst.
- find_apply_lem_hyp null_sound. find_rewrite. simpl.
intuition. left. eexists. intuition.
- simpl. find_apply_lem_hyp null_false_neq_nil.
intuition. left. eexists. intuition.
- simpl. intuition (auto; congruence).
- simpl. destruct st; simpl in *; subst; intuition (auto; congruence).
- simpl in *. intuition.
- simpl in *. intuition eauto.
- simpl. intuition (auto; congruence).
Qed.
Definition at_head_of_queue sigma c := (exists t, queue (sigma Server) = c :: t).
Proof using.
unfold at_head_of_queue.
firstorder.
Qed.
Proof using.
unfold locks_correct.
intros.
destruct (Name_eq_dec (Client c) (Client n)); rewrite_update; try find_inversion; auto.
Qed.
Proof using.
unfold locks_correct_locked.
firstorder.
repeat find_rewrite. find_inversion.
eauto using at_head_of_queue_intro.
Qed.
Proof using.
unfold locks_correct.
intros.
rewrite_update.
now find_higher_order_rewrite.
Qed.
Proof using.
unfold locks_correct.
intros.
find_apply_hyp_hyp. break_exists.
eauto using at_head_of_queue_intro.
Qed.
Proof using.
unfold at_head_of_queue.
firstorder.
congruence.
Qed.
Proof using.
intuition.
destruct (held (sigma (Client c))) eqn:?; auto.
exfalso. eauto using at_head_of_nil, locks_correct_true_at_head_of_queue.
Qed.
Proof using.
intros.
destruct (held (sigma (Client c))) eqn:?; auto.
firstorder.
find_copy_apply_lem_hyp locks_correct_true_at_head_of_queue; auto.
unfold at_head_of_queue in *. break_exists.
congruence.
Qed.
Proof using.
unfold locks_correct, at_head_of_queue.
firstorder.
rewrite_update.
eauto.
Qed.
Proof using.
unfold at_head_of_queue.
intuition. break_exists.
rewrite_update.
find_rewrite.
eauto.
Qed.
Ltac set_up_net_handlers :=
intros;
match goal with
| [ H : context [ NetHandler (pDst ?p) _ _ _ ] |- _ ] =>
destruct (pDst p) eqn:?
end; simpl in *;
[find_apply_lem_hyp ClientNetHandler_cases |
find_apply_lem_hyp ServerNetHandler_cases; intuition; try break_exists ];
intuition; subst;
simpl in *; intuition;
repeat find_rewrite;
repeat find_inversion;
simpl in *;
try now rewrite update_nop_ext.
Proof using.
set_up_net_handlers;
eauto using
locks_correct_update_true, locks_correct_locked_at_head,
all_clients_false_locks_correct_server_update, empty_queue_all_clients_false,
locks_correct_at_head_preserved, snoc_at_head_of_queue_preserved,
all_clients_false_locks_correct_server_update, unlock_in_flight_all_clients_false.
Qed.
Proof using.
unfold locks_correct_unlock.
intuition. congruence.
Qed.
Proof using.
unfold locks_correct_unlock.
intuition. congruence.
Qed.
Proof using.
set_up_input_handlers.
destruct (pBody p) eqn:?.
- auto using locks_correct_unlock_sent_lock.
- now erewrite unlock_in_flight_all_clients_false in * by eauto.
- auto using locks_correct_unlock_sent_locked.
Qed.
Proof using.
intros.
destruct (held (sigma (Client c))) eqn:?; auto.
firstorder.
find_copy_apply_lem_hyp locks_correct_true_at_head_of_queue; auto.
unfold at_head_of_queue in *. break_exists.
congruence.
Qed.
Proof using.
unfold locks_correct_locked.
intuition. congruence.
Qed.
Proof using.
unfold locks_correct_locked.
intuition. congruence.
Qed.
Proof using.
set_up_input_handlers.
destruct (pBody p) eqn:?.
- auto using locks_correct_locked_sent_lock.
- auto using locks_correct_locked_sent_unlock.
- now erewrite locked_in_flight_all_clients_false in * by eauto.
Qed.
Proof using.
unfold locks_correct_unlock, valid_unlock.
intros.
exists x.
intuition; now rewrite_update.
Qed.
Proof using.
set_up_input_handlers;
auto using locks_correct_unlock_sent_lock,
locks_correct_unlock_true_to_false,
locks_correct_true_at_head_of_queue.
Qed.
Proof using.
set_up_input_handlers;
auto using locks_correct_locked_sent_lock, locks_correct_locked_sent_unlock.
Qed.
Proof using.
unfold LockServ_network_network_invariant.
intros.
destruct (pBody q); intuition; try discriminate.
Qed.
Proof using.
unfold LockServ_network_network_invariant.
intros.
destruct (pBody q); intuition; try discriminate.
Qed.
Proof using.
unfold locks_correct_unlock.
intros.
find_apply_hyp_hyp. clear H1.
break_exists.
unfold valid_unlock in *. intuition.
break_exists.
repeat find_rewrite. repeat find_inversion.
eauto using at_head_of_queue_intro.
Qed.
Proof using.
unfold locks_correct_unlock, valid_unlock.
intuition.
break_exists.
exists x.
intuition.
- firstorder.
- now rewrite_update.
Qed.
Proof using.
unfold at_head_of_queue.
firstorder.
congruence.
Qed.
Proof using.
set_up_net_handlers;
eauto using locks_correct_unlock_sent_lock, nwnw_locked_lock,
locks_correct_unlock_at_head_preserved, snoc_at_head_of_queue_preserved,
nwnw_unlock_lock, nil_at_head_of_queue_preserved.
Qed.
Proof using.
unfold locks_correct_locked, valid_locked.
intuition.
break_exists.
exists x.
intuition.
- firstorder.
- now rewrite_update.
Qed.
Proof using.
set_up_net_handlers;
eauto using locks_correct_locked_sent_lock, nwnw_locked_lock,
locks_correct_locked_at_head_preserved, snoc_at_head_of_queue_preserved,
nwnw_unlock_lock, nil_at_head_of_queue_preserved.
Qed.
Proof using.
set_up_net_handlers;
auto using locks_correct_unlock_sent_locked.
Qed.
Proof using.
unfold locks_correct_locked, valid_locked.
intros.
exists c.
intuition.
- exists t. now rewrite_update.
- now rewrite_update.
Qed.
Proof using.
set_up_net_handlers;
eauto using locks_correct_locked_intro,
empty_queue_all_clients_false,
unlock_in_flight_all_clients_false.
Qed.
Proof using.
unfold LockServ_network_network_invariant.
intuition; simpl in *; congruence.
Qed.
Proof using.
unfold LockServ_network_invariant.
set_up_input_handlers.
- auto using nwnw_sym, nwnw_lock.
- destruct (pBody p) eqn:?.
+ auto using nwnw_lock.
+ now erewrite unlock_in_flight_all_clients_false in * by eauto.
+ now erewrite locked_in_flight_all_clients_false in * by eauto.
Qed.
Proof using.
set_up_input_handlers.
Qed.
Proof using.
unfold LockServ_network_invariant,
locks_correct_unlock, locks_correct_locked,
valid_unlock, valid_locked.
intuition.
destruct (pBody p) eqn:?; intuition; break_exists; intuition; break_exists;
congruence.
Qed.
Proof using.
set_up_net_handlers;
eauto using nwnw_sym, nwnw_lock, nw_empty_queue_lock, nwnw_unlock_lock.
Qed.
Proof using.
set_up_net_handlers.
Qed.
Ltac unlabeled_unfold :=
unfold unlabeled_net_handlers, unlabeled_input_handlers in *.
Instance LockServ_Decompositition : Decomposition _ LockServ_MultiParams.
apply Build_Decomposition with (state_invariant := locks_correct)
(network_invariant := LockServ_network_invariant)
(network_network_invariant := LockServ_network_network_invariant);
simpl; intros; monad_unfold; unlabeled_unfold; repeat break_let; repeat find_inversion.
- auto using nwnw_sym.
- auto using locks_correct_init.
- eauto using locks_correct_input_handlers.
- unfold LockServ_network_invariant in *. intuition.
eauto using locks_correct_net_handlers.
- unfold LockServ_network_invariant in *.
intuition eauto using locks_correct_unlock_input_handlers_old,
locks_correct_locked_input_handlers_old.
- unfold LockServ_network_invariant in *.
intuition eauto using locks_correct_unlock_input_handlers_new,
locks_correct_locked_input_handlers_new.
- unfold LockServ_network_invariant in *.
intuition eauto using locks_correct_unlock_net_handlers_old,
locks_correct_locked_net_handlers_old.
- unfold LockServ_network_invariant in *.
intuition eauto using locks_correct_unlock_net_handlers_new,
locks_correct_locked_net_handlers_new.
- eauto using LockServ_nwnw_input_handlers_old_new.
- eauto using LockServ_nwnw_input_handlers_new_new.
- eauto using LockServ_nwnw_net_handlers_old_new.
- eauto using LockServ_nwnw_net_handlers_new_new.
Defined.
Proof using.
pose proof decomposition_invariant.
find_apply_lem_hyp inductive_invariant_true_in_reachable.
unfold true_in_reachable in *.
intros.
apply locks_correct_implies_mutex.
match goal with
| [ H : _ |- _ ] => apply H
end.
auto.
Qed.
Fixpoint last_holder' (holder : option Client_index) (trace : list (name * (input + list output))) : option Client_index :=
match trace with
| [] => holder
| (Client n, inl Unlock) :: tr => match holder with
| None => last_holder' holder tr
| Some m => if fin_eq_dec _ n m
then last_holder' None tr
else last_holder' holder tr
end
| (Client n, inr [Locked]) :: tr => last_holder' (Some n) tr
| (n, _) :: tr => last_holder' holder tr
end.
Fixpoint trace_mutual_exclusion' (holder : option Client_index) (trace : list (name * (input + list output))) : Prop :=
match trace with
| [] => True
| (Client n, (inl Unlock)) :: tr' => match holder with
| Some m => if fin_eq_dec _ n m
then trace_mutual_exclusion' None tr'
else trace_mutual_exclusion' holder tr'
| _ => trace_mutual_exclusion' holder tr'
end
| (n, (inl _)) :: tr' => trace_mutual_exclusion' holder tr'
| (Client n, (inr [Locked])) :: tr' => match holder with
| None => trace_mutual_exclusion' (Some n) tr'
| Some _ => False
end
| (_, (inr [])) :: tr' => trace_mutual_exclusion' holder tr'
| (_, (inr _)) :: tr' => False
end.
Definition trace_mutual_exclusion (trace : list (name * (input + list output))) : Prop :=
trace_mutual_exclusion' None trace.
Definition last_holder (trace : list (name * (input + list output))) : option Client_index :=
last_holder' None trace.
Proof using.
intros.
find_apply_lem_hyp refl_trans_1n_n1_trace.
prep_induction H1.
induction H1; intros; subst; eauto.
eapply H3; eauto.
- apply refl_trans_n1_1n_trace. auto.
- apply IHrefl_trans_n1_trace; auto.
Qed.
Proof using.
induction tr; intuition; unfold trace_mutual_exclusion in *; simpl in *;
repeat break_match; subst; intuition.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; subst; intuition; eauto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; subst; intuition.
Qed.
Proof using.
pose proof decomposition_invariant.
find_apply_lem_hyp inductive_invariant_true_in_reachable.
unfold true_in_reachable, reachable in *.
intuition.
unfold composed_invariant in *.
apply H; eauto.
Qed.
Proof using.
induction tr; intros; simpl in *.
- subst. auto.
- simpl in *. repeat break_match; subst; intuition.
Qed.
Proof using.
unfold reachable.
intros. eauto.
Qed.
Proof using.
intros.
pose proof decomposition_invariant.
find_apply_lem_hyp inductive_invariant_true_in_reachable.
unfold true_in_reachable in *. apply H1; auto.
Qed.
Proof using.
intros.
pose proof decomposition_invariant.
find_apply_lem_hyp inductive_invariant_true_in_reachable.
unfold true_in_reachable in *. apply H0; auto.
Qed.
Proof using.
intros.
apply locks_correct_implies_mutex.
auto using locks_correct_invariant.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; subst; eauto.
congruence.
Qed.
Ltac my_update_destruct :=
match goal with
| [H : context [ update _ _ ?x _ ?y ] |- _ ] => destruct (Name_eq_dec x y)
| [ |- context [ update _ _ ?x _ ?y ] ] => destruct (Name_eq_dec x y)
end.
Proof using.
induction tr; intros; simpl in *; repeat break_match; auto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; auto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; intuition.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; intuition.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; auto; try congruence; subst; eauto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; auto; try congruence; subst; eauto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; auto.
congruence.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; intuition (auto; try congruence).
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; intuition.
congruence.
Qed.
Proof using.
intros.
apply last_holder'_unlock_none. auto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; subst;
intuition; try congruence; eauto.
Qed.
Proof using.
induction tr; intros; simpl in *; repeat break_match; subst; try congruence; intuition.
Qed.
Proof using.
apply cross_relation; intros.
- intuition.
+ red. red. auto.
+ unfold last_holder in *. simpl in *. discriminate.
+ unfold last_holder in *. simpl in *. discriminate.
- match goal with
| [ H : step_async _ _ _ |- _ ] => invcs H
end; monad_unfold; unlabeled_unfold;
unfold lb_net_handlers, lb_input_handlers in *; simpl in *;
repeat break_let; repeat find_inversion.
+ unfold NetHandler in *. break_match.
* find_apply_lem_hyp ClientNetHandler_cases; eauto.
break_and.
{ break_or_hyp.
- intuition; subst.
+ apply trace_mutex'_no_out_extend; auto.
+ rewrite update_nop_ext.
find_apply_lem_hyp last_holder'_no_out_inv.
auto.
+ match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
find_apply_hyp_hyp.
apply last_holder'_no_out_extend. auto.
- intuition; subst.
+ apply trace_mutex'_locked_extend. auto.
destruct (last_holder' None tr) eqn:?; auto.
find_apply_hyp_hyp.
erewrite locked_in_flight_all_clients_false in * by
eauto using locks_correct_locked_invariant, reachable_intro,
locks_correct_invariant.
discriminate.
+ my_update_destruct; try find_inversion; rewrite_update; auto.
find_apply_lem_hyp last_holder'_locked_some_eq. congruence.
+ my_update_destruct; try find_inversion; rewrite_update.
* apply last_holder'_locked_extend.
* erewrite locked_in_flight_all_clients_false in * by
eauto using locks_correct_locked_invariant, reachable_intro,
locks_correct_invariant.
discriminate.
}
* { find_apply_lem_hyp ServerNetHandler_cases. break_and. subst.
repeat split.
- apply trace_mutex'_no_out_extend. auto.
- intros. my_update_destruct; try discriminate.
rewrite_update.
find_apply_lem_hyp last_holder'_no_out_inv.
auto.
- intros. my_update_destruct; try discriminate; rewrite_update.
apply last_holder'_no_out_extend. auto.
}
+ unfold InputHandler in *. break_match.
* unfold ClientIOHandler in *.
{ monad_unfold.
repeat break_match; repeat find_inversion; intuition;
repeat rewrite snoc_assoc in *;
try apply trace_mutex'_no_out_extend;
try find_apply_lem_hyp last_holder'_no_out_inv;
try (apply last_holder'_no_out_extend; auto).
- apply trace_mutual_exclusion'_extend_input; auto. congruence.
- rewrite update_nop_ext.
find_apply_lem_hyp last_holder'_input_inv; try congruence.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
apply last_holder'_input_extend; auto. congruence.
- apply trace_mutex'_unlock_extend; auto.
- rewrite last_holder_unlock_none in *; auto. discriminate.
- my_update_destruct; try find_inversion; rewrite_update.
+ discriminate.
+ assert (mutual_exclusion (nwState st))
by eauto using mutual_exclusion_invariant, reachable_intro.
unfold mutual_exclusion in *.
assert (c = n) by eauto. congruence.
- apply trace_mutex'_unlock_extend. auto.
- rewrite update_nop.
find_apply_lem_hyp last_holder_some_unlock_inv.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
assert (n <> c) by congruence.
find_apply_hyp_hyp.
apply last_holder'_neq_unlock_extend; auto.
- apply trace_mutual_exclusion'_extend_input; auto. congruence.
- rewrite update_nop_ext. find_apply_lem_hyp last_holder'_input_inv; try congruence.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
apply last_holder'_input_extend; auto. congruence.
}
* unfold ServerIOHandler in *.
monad_unfold. find_inversion.
{ intuition;
repeat rewrite snoc_assoc in *.
- apply trace_mutex'_no_out_extend.
apply trace_mutual_exclusion'_extend_input_server. auto.
- find_apply_lem_hyp last_holder'_no_out_inv.
rewrite update_nop. find_apply_lem_hyp last_holder'_input_inv_server. auto.
- apply last_holder'_no_out_extend; auto.
rewrite_update. unfold last_holder. rewrite last_holder'_server_extend.
auto.
}
Qed.
Proof using.
intros.
find_apply_lem_hyp refl_trans_1n_n1_trace.
prep_induction H.
induction H; intros; subst.
- discriminate.
- match goal with
| [ H : step_async _ _ _ |- _ ] => invcs H
end; unlabeled_unfold;
unfold lb_net_handlers, lb_input_handlers in *; simpl in *;
monad_unfold;
repeat break_let; repeat find_inversion.
+ unfold NetHandler in *.
break_match; rewrite_update.
* find_apply_lem_hyp ClientNetHandler_cases.
intuition.
-- subst. rewrite update_nop_ext.
find_apply_lem_hyp IHrefl_trans_n1_trace; auto; [idtac].
repeat find_rewrite.
simpl.
in_crush.
discriminate.
-- subst.
find_apply_lem_hyp refl_trans_n1_1n_trace.
find_apply_lem_hyp reachable_intro.
match goal with
| [ H : reachable _ _ _ |- _ ] =>
let H' := fresh H in
pose H as H';
eapply locks_correct_locked_invariant with (p := p) in H';
[| now repeat find_rewrite; in_crush];
eapply locks_correct_locked_at_head in H'; eauto
end.
unfold at_head_of_queue in *. break_exists. find_rewrite. find_inversion.
rewrite_update. auto.
* find_apply_lem_hyp ServerNetHandler_cases. intuition.
-- break_exists. intuition.
++ subst. repeat find_rewrite. simpl in *.
find_inversion. auto.
++ subst. repeat find_rewrite.
destruct (queue (nwState x' Server)); try congruence.
simpl in *. find_inversion.
do 2 insterU IHrefl_trans_n1_trace.
repeat conclude_using eauto.
intuition.
** in_crush. discriminate.
** in_crush. discriminate.
-- congruence.
-- break_exists. intuition. subst. simpl.
repeat find_rewrite. find_inversion. auto.
-- subst. simpl.
find_apply_hyp_hyp. intuition.
++ repeat find_rewrite. in_crush. discriminate.
++ repeat find_rewrite. in_crush.
+ find_apply_lem_hyp InputHandler_cases.
intuition.
* break_exists. break_and. subst.
rewrite_update.
find_apply_hyp_hyp.
intuition.
-- subst. rewrite update_nop_ext. auto.
-- find_apply_lem_hyp refl_trans_n1_1n_trace.
find_apply_lem_hyp reachable_intro.
match goal with
| [ H : reachable _ _ _ |- _ ] =>
pose H as Hmutex;
eapply mutual_exclusion_invariant in Hmutex
end.
unfold mutual_exclusion in *.
assert (c = x) by auto. clear Hmutex.
subst. simpl. auto.
-- subst. rewrite_update. auto.
* subst. simpl. rewrite update_nop_ext in *.
match goal with
| [ H : _ |- _ ] => apply IHrefl_trans_n1_trace in H; auto
end.
Qed.
(* LIVENESS *)
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion;
subst; simpl in *; subst; simpl in *.
- left. eexists. intuition.
- left. eexists. intuition.
- left. eexists. intuition.
- auto.
- auto.
Qed.
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion; subst; intuition.
Qed.
Proof using.
handler_unfold.
intros.
repeat break_match; repeat tuple_inversion; subst.
- find_apply_lem_hyp null_sound. find_rewrite. simpl.
intuition. left. eexists. intuition.
- simpl. find_apply_lem_hyp null_false_neq_nil.
intuition. left. eexists. intuition.
- simpl. intuition.
- simpl. destruct st; simpl in *; subst; intuition.
- simpl in *. intuition.
- simpl in *. intuition eauto.
- simpl. intuition.
Qed.
Definition message_enables_label p l :=
forall net,
In p (nwPackets net) ->
lb_step_ex lb_step_async l net.
Proof using.
unfold message_enables_label.
intros.
find_apply_lem_hyp in_split.
break_exists_name xs. break_exists_name ys.
unfold enabled.
destruct (ServerNetHandler (Client i) Lock (nwState net Server)) eqn:?.
destruct p. destruct p.
cut (l0 = MsgLock i); intros.
subst.
- repeat eexists. econstructor; eauto.
- handler_unfold. repeat break_match; repeat find_inversion; auto.
Qed.
Definition message_delivered_label p l :=
forall l' net net' tr,
lb_step_async net l' net' tr ->
In p (nwPackets net) ->
~ In p (nwPackets net') ->
l = l'.
Proof using.
intros.
find_apply_lem_hyp in_app_iff.
simpl in *; intuition;
find_false; apply in_app_iff; right; apply in_app_iff; auto.
Qed.
Proof using.
unfold message_delivered_label.
intros.
invcs H.
- repeat find_rewrite.
find_eapply_lem_hyp In_split_not_In; eauto. subst.
monad_unfold. simpl in *.
handler_unfold. repeat break_match; repeat find_inversion; auto.
- unfold not in *. find_false.
apply in_app_iff; auto.
- intuition.
Qed.
Definition label_eq_dec :
forall x y : label,
{x = y} + {x <> y}.
Proof using.
decide equality; apply fin_eq_dec.
Qed.
Proof using.
intros l p Henabled Hdelivered.
cofix c.
destruct s. destruct e.
simpl.
intros Hexec Hin.
invcs Hexec.
destruct (label_eq_dec l evt_l).
- subst evt_l.
apply W0. simpl. reflexivity.
- apply W_tl.
+ simpl.
unfold message_enables_label in *.
unfold enabled. simpl. now auto.
+ simpl.
apply c; auto.
simpl.
match goal with
| |- In ?p ?ps =>
destruct (In_dec packet_eq_dec p ps)
end; auto.
unfold message_delivered_label in *.
now find_apply_hyp_hyp.
Qed.
Proof using.
intros.
find_eapply_lem_hyp messages_trigger_labels; eauto.
find_apply_lem_hyp weak_until_until_or_always.
intuition.
- now eauto using until_eventually.
- find_apply_lem_hyp always_continuously.
eapply_prop_hyp weak_fairness continuously; auto.
destruct s.
now find_apply_lem_hyp always_now.
Qed.
Ltac coinductive_case CIH :=
apply W_tl; simpl in *; auto;
apply CIH; simpl in *; auto.
Proof using.
induction l; intros; simpl in *; try solve_by_inversion.
invcs H.
- constructor.
- constructor. auto.
Qed.
Proof using.
induction l; intros; solve_by_inversion.
Qed.
Proof using.
intros n c.
cofix CIH.
destruct s.
destruct e.
intros Hexec HNth.
invcs Hexec.
invcs H1.
- unfold runGenHandler, NetHandler in *.
break_match.
+ coinductive_case CIH.
find_rewrite. simpl.
now rewrite_update.
+ find_apply_lem_hyp ServerNetHandler_lbcases.
intuition.
* coinductive_case CIH.
repeat find_rewrite. simpl.
rewrite_update.
break_exists.
intuition; repeat find_rewrite; try solve_by_inversion.
now eauto using Nth_app.
* exfalso. clear CIH.
subst.
invcs HNth.
repeat find_reverse_rewrite. simpl in *.
repeat find_rewrite. now solve_by_inversion.
* clear CIH.
apply W0. simpl.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite. simpl.
rewrite_update. repeat find_rewrite.
intuition eauto using Nth_tl.
break_exists.
intuition. subst.
find_apply_lem_hyp Nth_tl.
repeat find_rewrite.
invcs HNth. auto.
* coinductive_case CIH.
repeat find_rewrite. simpl.
now rewrite_update.
- unfold runGenHandler in *.
find_apply_lem_hyp InputHandler_cases.
intuition.
+ break_exists. break_and. subst.
coinductive_case CIH.
repeat find_rewrite. simpl.
now rewrite_update.
+ coinductive_case CIH.
repeat find_rewrite. simpl.
update_destruct; subst; now rewrite_update.
- coinductive_case CIH.
Qed.
Proof using.
intros n c s Hexec HNth Hlabel.
destruct s. simpl.
invcs Hexec.
match goal with
| H : lb_step_async _ _ _ _ |- _ => invcs H
end.
- unfold occurred in *.
match goal with
| H : MsgUnlock = _ |- _ => symmetry in H; repeat find_rewrite; clear H
end.
monad_unfold. unfold NetHandler in *.
break_match.
+ find_apply_lem_hyp ClientNetHandler_lbcases.
intuition; congruence.
+ find_apply_lem_hyp ServerNetHandler_lbcases.
(* not using intuition because i don't want to break or in the goal *)
repeat (break_and; try break_or_hyp);
break_exists;
repeat (break_and; try break_or_hyp);
try congruence.
* exfalso.
repeat find_rewrite.
invcs HNth.
repeat find_reverse_rewrite.
simpl in *. subst. solve_by_inversion.
* fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite.
simpl in *.
find_apply_lem_hyp Nth_tl.
repeat find_rewrite.
intuition; [|intros; subst; solve_by_inversion].
rewrite_update. congruence.
- unfold occurred in *.
match goal with
| H : MsgUnlock = _ |- _ => symmetry in H; repeat find_rewrite; clear H
end.
monad_unfold. find_apply_lem_hyp InputHandler_lbcases.
intuition; break_exists; intuition; congruence.
- unfold occurred in *. congruence.
Qed.
Proof using.
unfold message_enables_label.
intros.
find_apply_lem_hyp in_split.
break_exists_name xs. break_exists_name ys.
unfold enabled.
destruct (ServerNetHandler n Unlock (nwState net Server)) eqn:?.
destruct p. destruct p.
cut (l0 = MsgUnlock); intros.
subst.
- repeat eexists. econstructor; eauto.
- handler_unfold. repeat break_match; repeat find_inversion; auto.
Qed.
Proof using.
unfold message_delivered_label.
intros.
invcs H.
- repeat find_rewrite.
find_eapply_lem_hyp In_split_not_In; eauto. subst.
monad_unfold. simpl in *.
handler_unfold. repeat break_match; repeat find_inversion; auto.
- unfold not in *. find_false.
apply in_app_iff; auto.
- intuition.
Qed.
Proof using.
intros.
eapply message_labels_eventually_occur;
eauto using Unlock_enables_MsgUnlock, Unlock_delivered_MsgUnlock.
unfold label_silent. simpl. congruence.
Qed.
Proof using.
intros.
solve_by_inversion' eauto.
Qed.
Proof using.
intros.
invcs H.
invcs H2.
- monad_unfold.
unfold NetHandler in *.
break_match_hyp.
+ unfold occurred in *.
find_apply_lem_hyp ClientNetHandler_lbcases; intuition; congruence.
+ unfold occurred in *.
find_apply_lem_hyp ServerNetHandler_lbcases; intuition;
break_exists; intuition; congruence.
- monad_unfold.
find_apply_lem_hyp InputHandler_lbcases.
intuition; try congruence.
break_exists. intuition; try congruence.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite.
simpl. left. unfold occurred in *.
congruence.
- unfold occurred in *. congruence.
Qed.
Proof using.
intros.
unfold InputHandler. unfold ClientIOHandler.
monad_unfold. repeat break_let.
find_inversion. eauto.
Qed.
Proof using.
intros.
destruct s. simpl.
unfold enabled, enabled.
pose proof (InputHandler_Client_Unlock c (nwState (evt_a e) (Client c))).
break_exists.
repeat eexists.
unfold InputHandler in *. unfold ClientIOHandler in *.
eapply LabeledStepAsync_input with (h := (Client c)) (inp := Unlock); eauto.
Qed.
Proof using.
unfold cont_enabled.
intros.
apply always_continuously.
eapply always_monotonic;
[|eapply always_inv; eauto; eauto using lb_step_execution_invar];
eauto using InputUnlock_enabled.
Qed.
Proof using.
intros c.
cofix CIH.
destruct s. simpl.
intros.
invcs H.
invcs H3.
- coinductive_case CIH.
monad_unfold.
unfold NetHandler in *.
break_match_hyp.
+ find_apply_lem_hyp ClientNetHandler_cases.
repeat find_rewrite. simpl.
intuition;
update_destruct_max_simplify; repeat find_rewrite; auto.
+ find_apply_lem_hyp ServerNetHandler_cases.
repeat find_rewrite. simpl.
intuition; break_exists; intuition;
rewrite_update; repeat find_rewrite; auto.
- monad_unfold.
find_apply_lem_hyp InputHandler_lbcases.
intuition; break_exists; intuition.
+ coinductive_case CIH.
repeat find_rewrite.
simpl.
update_destruct_max_simplify; repeat find_rewrite; auto.
+ subst.
destruct (fin_eq_dec _ c x).
* clear CIH. subst.
apply W0; simpl.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite.
simpl. auto.
* coinductive_case CIH.
repeat find_rewrite.
simpl. now rewrite_update.
+ coinductive_case CIH.
clear CIH.
repeat find_rewrite.
simpl.
update_destruct_max_simplify; repeat find_rewrite; auto.
+ coinductive_case CIH. clear CIH.
repeat find_rewrite.
simpl.
update_destruct_max_simplify; repeat find_rewrite; auto.
- coinductive_case CIH.
congruence.
Qed.
Proof using.
intros.
pose proof (@InputUnlock_continuously_enabled s c). intuition.
eapply_prop_hyp weak_fairness cont_enabled; [|now unfold label_silent].
solve_by_inversion.
Qed.
Proof using.
intros. apply eventually_next.
match goal with
| H : context [held] |- _ =>
pattern s in H
end.
match goal with
| H1 : ?J1 s, H2 : ?J2 s, H3 : ?J3 s |- _ =>
assert ((J1 /\_ J2 /\_ J3) s) by (now unfold and_tl);
eapply weak_until_eventually with (J := (and_tl J1 (and_tl J2 J3)))
end; simpl in *.
2:now unfold and_tl.
3:eauto using held_eventually_InputUnlock.
- intros. unfold and_tl in *. intuition.
eapply InputUnlock_held; eauto.
- apply weak_until_always; eauto using lb_step_execution_invar, always_inv.
apply weak_until_always; eauto using weak_fairness_invar, always_inv.
eauto using held_until_Unlock.
Qed.
Proof using.
unfold message_enables_label, enabled.
intros.
find_apply_lem_hyp in_split.
break_exists_name xs. break_exists_name ys.
do 2 eexists.
eapply LabeledStepAsync_deliver; eauto.
simpl. monad_unfold. simpl. eauto.
Qed.
Proof using.
unfold message_delivered_label.
intros.
invcs H.
- repeat find_rewrite.
find_eapply_lem_hyp In_split_not_In; eauto. subst.
monad_unfold. simpl in *.
handler_unfold. repeat break_match; repeat find_inversion; auto.
- unfold not in *. find_false.
apply in_app_iff; auto.
- intuition.
Qed.
Proof using.
intros.
eapply message_labels_eventually_occur;
eauto using Locked_enables_MsgLocked, Locked_delivered_MsgLocked.
unfold label_silent. simpl. congruence.
Qed.
Proof using.
intros.
invcs H.
invcs H1.
- monad_unfold.
unfold NetHandler in *.
break_match_hyp.
+ unfold occurred in *.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite. simpl.
find_apply_lem_hyp ClientNetHandler_lbcases; intuition; subst;
update_destruct_max_simplify; congruence.
+ unfold occurred in *.
find_apply_lem_hyp ServerNetHandler_lbcases; intuition;
break_exists; intuition; congruence.
- monad_unfold.
find_apply_lem_hyp InputHandler_lbcases.
intuition; break_exists; intuition; congruence.
- congruence.
Qed.
Proof using.
intros.
find_apply_lem_hyp Nth_something_at_head.
break_exists_name holder. break_exists.
exists (Client holder).
remember H0 as Hlbs; clear HeqHlbs.
invcs H0.
find_eapply_lem_hyp head_grant_state_unlock; eauto.
intuition.
- (* eventually the Locked message is delivered, after which this is
the same as the next case. *)
eapply eventually_trans
with (inv := lb_step_execution lb_step_async /\_
weak_fairness
(lb_step_async(labeled_multi_params := LockServ_LabeledParams))
Silent)
(P := now (occurred (MsgLocked holder))).
all:unfold and_tl in *; intuition.
+ eauto using lb_step_execution_invar.
+ eauto using weak_fairness_invar.
+ (* need `now (MsgLocked i) -> held i = true`, then identical to next case below. *)
find_apply_lem_hyp MsgLocked_held; eauto.
destruct s.
simpl in *.
eauto using lb_step_execution_invar, weak_fairness_invar,
E_next, held_eventually_Unlock.
+ apply Locked_in_network_eventually_MsgLocked; auto.
- eauto using held_eventually_Unlock.
- eauto using E0.
Qed.
Proof using.
intros n c s Hstar Hexec Hfair HNth.
pattern s in Hexec. pattern s in Hfair.
find_copy_eapply_lem_hyp eventually_Unlock; eauto.
break_exists.
match goal with
| H1 : (fun x => ?J1) s, H2 : (fun x => ?J2) s |- _ =>
assert (and_tl (fun x => J1) (fun x => J2) s) as Hand by (now unfold and_tl);
clear H1; clear H2
end; simpl in *.
eapply eventually_trans.
4:eauto. 3:apply Hand.
2:intros; eapply Unlock_in_network_eventually_MsgUnlock.
all:unfold and_tl in *; intuition eauto.
- eauto using lb_step_execution_invar.
- simpl. eauto using weak_fairness_invar.
Qed.
Proof using.
intros n c s Hstar Hexec Hfair HNth.
apply eventually_next.
pattern s in HNth.
match goal with
| H1 : ?J1 s, H2 : ?J2 s, H3 : ?J3 s |- _ =>
assert ((J1 /\_ J2 /\_ J3) s) by (now unfold and_tl);
eapply weak_until_eventually with (J := (and_tl J1 (and_tl J2 J3)))
end; simpl in *.
2:now unfold and_tl.
3:eauto using eventually_MsgUnlock.
- intros. unfold and_tl in *. intuition.
eapply MsgUnlock_moves_client; eauto.
- apply weak_until_always; eauto using lb_step_execution_invar, always_inv.
apply weak_until_always; eauto using weak_fairness_invar, always_inv.
eauto using clients_only_move_up_in_queue.
Qed.
Proof using.
induction n; intros; simpl in *; auto.
- intuition.
assert (n' = 0) by lia. subst.
eauto using clients_move_up_in_queue.
- match goal with
| H : _ (hd s) |- _ =>
pattern s in H
end.
match goal with
| H1 : ?J1 s, H2 : ?J2 s, H3 : ?J3 s |- _ =>
assert ((J1 /\_ J2 /\_ J3) s) as Hand by (now unfold and_tl);
clear H1; clear H2; clear H3
end; simpl in *.
find_apply_lem_hyp le_lt_eq_dec. intuition.
+ assert (n' <= n) by lia.
find_eapply_lem_hyp clients_move_up_in_queue; eauto;
try solve [unfold and_tl in *; intuition]; [idtac].
eapply eventually_trans. 4:eauto.
3:apply Hand. all:unfold and_tl in *.
all:intuition eauto using lb_step_execution_invar, weak_fairness_invar.
find_apply_lem_hyp step_async_star_lb_step_execution; auto.
destruct s0. simpl in *.
find_apply_lem_hyp always_Cons.
intuition.
find_apply_lem_hyp always_Cons.
intuition.
+ subst. unfold and_tl in *. intuition.
eauto using clients_move_up_in_queue.
Qed.
Proof using.
intros.
pose proof (@clients_move_way_up_in_queue n 0 c s).
pose proof (Nat.le_0_l n).
repeat concludes. conclude_using ltac:(intuition; congruence).
eapply eventually_monotonic_simple; [|eauto].
intros. simpl in *. intuition.
Qed.
Proof using.
intros.
invcs H.
invcs H1.
- monad_unfold.
unfold NetHandler in *.
break_match_hyp.
+ unfold occurred in *.
find_apply_lem_hyp ClientNetHandler_lbcases; intuition; congruence.
+ unfold occurred in *.
find_apply_lem_hyp ServerNetHandler_lbcases; intuition;
break_exists; intuition; congruence.
- monad_unfold.
find_apply_lem_hyp InputHandler_lbcases.
intuition; try congruence.
break_exists. intuition; try congruence.
fold LockServ_MultiParams in *. (* typeclass stuff *)
repeat find_rewrite.
simpl. left. unfold occurred in *.
congruence.
- unfold occurred in *. congruence.
Qed.
Proof using.
intros.
eapply message_labels_eventually_occur;
eauto using Lock_enables_MsgLock, Lock_delivered_MsgLock.
unfold label_silent. simpl. congruence.
Qed.
Proof using.
intros.
find_apply_lem_hyp InputLock_Lock; auto.
destruct s.
simpl in *.
eauto using E_next, Lock_in_network_eventually_MsgLock,
lb_step_execution_invar, weak_fairness_invar.
Qed.

Lemma Nth_snoc :
forall A (l : list A) x,
Nth (l ++ [x]) (length l) x.