Section TwoPhaseInductiveInv.
Variable l : Label.
Variables (cn : nid) (pts : seq nid) (others : seq nid).
Hypothesis Hnin : cn \notin pts.
Hypothesis PtsNonEmpty : pts != [::].
Definition tpc := TwoPhaseCommitProtocol others Hnin l.
(* Take the transitions *)
Notation sts := (snd_trans tpc).
Notation rts := (rcv_trans tpc).
Notation loc z d := (getLocal z d).
Definition cn_state (d : dstatelet) (c : CStateT) (l : Log) : Prop :=
loc cn d = st :-> c \+ log :-> l.
Definition pt_state (d : dstatelet) (p : PStateT) (l : Log) (pt : nid) : Prop :=
loc pt d = st :-> p \+ log :-> l.
(*****************************************************)
(* Phase Zero *)
Definition EverythingInit (d : dstatelet) (round : nat) (l : Log) : Prop :=
[/\ cn_state d (round, CInit) l &
forall pt, pt \in pts ->
[/\ pt_state d (round, PInit) l pt,
no_msg_from_to pt cn (dsoup d) &
no_msg_from_to cn pt (dsoup d)]].
(* Phase One *)
Definition pt_PhaseOne (d : dstatelet) (round : nat)
(next_data : data) (l : Log) (pt : nid) : Prop :=
[\/ [/\ pt_state d (round, PInit) l pt,
no_msg_from_to pt cn (dsoup d) &
msg_spec cn pt prep_req (round :: next_data) (dsoup d)]
,  [/\ pt_state d (round, PGotRequest next_data) l pt,
no_msg_from_to pt cn (dsoup d) &
no_msg_from_to cn pt (dsoup d)]
,  [/\ pt_state d (round, PRespondedYes next_data) l pt,
no_msg_from_to cn pt (dsoup d) &
msg_spec pt cn prep_yes [::round] (dsoup d)]
|  [/\ pt_state d (round, PRespondedNo next_data) l pt,
no_msg_from_to cn pt (dsoup d) &
msg_spec pt cn prep_no [::round] (dsoup d)]].
Definition pt_PhaseOneResponded (d : dstatelet) (round : nat) (next_data : data)
(l : Log) (committed : bool) (pt : nid) : Prop :=
[/\ no_msg_from_to cn pt (dsoup d), no_msg_from_to pt cn (dsoup d) &
if committed
then pt_state d (round, PRespondedYes next_data) l pt
else pt_state d (round, PRespondedNo next_data) l pt].
Definition pt_Init d round l pt :=
[/\ pt_state d (round, PInit) l pt,
no_msg_from_to pt cn (dsoup d) &
no_msg_from_to cn pt (dsoup d)].
Definition cn_PhaseOneSend d round next_data l sent :=
[/\ cn_state d (round, CSentPrep next_data sent) l,
uniq sent, {subset sent <= pts} &
forall pt, pt \in pts ->
if pt \in sent
then pt_PhaseOne d round next_data l pt
else pt_Init d round l pt].
Definition cn_PhaseOneReceive d round next_data l recvd :=
let rps := map fst recvd in
[/\ cn_state d (round, CWaitPrepResponse next_data recvd) l,
uniq rps, {subset rps <= pts} ,
forall pt b, pt \in pts -> (pt, b) \in recvd ->
pt_PhaseOneResponded d round next_data l b pt &
forall pt,   pt \in pts -> pt \notin rps ->
pt_PhaseOne d round next_data l pt].
Definition PhaseOne (d : dstatelet) (round : nat) (next_data : data) (l : Log) :=
(exists sent, cn_PhaseOneSend d round next_data l sent) \/
(exists recvd, cn_PhaseOneReceive d round next_data l recvd).
(* Phase Two *)
Definition pt_PhaseTwoCommit d round next_data l pt :=
[\/ [/\ pt_state d (round, PRespondedYes next_data) l pt,
msg_spec cn pt commit_req [::round] (dsoup d) &
no_msg_from_to pt cn (dsoup d)]
, [/\ pt_state d (round, PCommitted next_data) (rcons l (true, next_data)) pt,
no_msg_from_to cn pt (dsoup d) & no_msg_from_to pt cn (dsoup d)]
| [/\ pt_state d (round.+1, PInit) (rcons l (true, next_data)) pt,
no_msg_from_to cn pt (dsoup d) &
msg_spec pt cn commit_ack [::round] (dsoup d)
]].
Definition pt_PhaseTwoAbort d round next_data l pt :=
[\/ [/\ (pt_state d (round, PRespondedYes next_data) l pt \/
pt_state d (round, PRespondedNo next_data) l pt),
msg_spec cn pt abort_req [::round] (dsoup d) &
no_msg_from_to pt cn (dsoup d)]
, [/\ pt_state d (round, PAborted next_data) (rcons l (false, next_data)) pt,
no_msg_from_to cn pt (dsoup d) & no_msg_from_to pt cn (dsoup d)]
| [/\ pt_state d (round.+1, PInit) (rcons l (false, next_data)) pt,
no_msg_from_to cn pt (dsoup d) &
msg_spec pt cn abort_ack [::round] (dsoup d)
]].
Definition pt_PhaseTwoResponded d round next_data l b pt :=
[/\ pt_state d (round.+1, PInit) (rcons l (b, next_data)) pt,
no_msg_from_to cn pt (dsoup d) & no_msg_from_to pt cn (dsoup d)].
Definition cn_PhaseTwoSendCommits d round next_data l sent :=
[/\ cn_state d (round, CSentCommit next_data sent) l,
uniq sent, {subset sent <= pts} &
forall pt, pt \in pts ->
if pt \in sent
then pt_PhaseTwoCommit d round next_data l pt
else pt_PhaseOneResponded d round next_data l true pt].
Definition cn_PhaseTwoSendAborts d round next_data l sent :=
[/\ cn_state d (round, CSentAbort next_data sent) l,
uniq sent, {subset sent <= pts} &
forall pt, pt \in pts ->
if pt \in sent
then pt_PhaseTwoAbort d round next_data l pt
else exists b, pt_PhaseOneResponded d round next_data l b pt].
Definition cn_PhaseTwoReceiveCommits d round next_data l recvd :=
[/\ cn_state d (round, CWaitAckCommit next_data recvd) l,
uniq recvd, {subset recvd <= pts} &
forall pt, pt \in pts ->
if pt \in recvd
then pt_PhaseTwoResponded d round next_data l true pt
else pt_PhaseTwoCommit d round next_data l pt].
Definition cn_PhaseTwoReceiveAborts d round next_data l recvd :=
[/\ cn_state d (round, CWaitAckAbort next_data recvd) l,
uniq recvd, {subset recvd <= pts} &
forall pt, pt \in pts ->
if pt \in recvd
then pt_PhaseTwoResponded d round next_data l false pt
else pt_PhaseTwoAbort d round next_data l pt].
Definition PhaseTwoCommit d round next_data lg :=
[\/ exists sent : seq nid, cn_PhaseTwoSendCommits d round next_data lg sent |
exists recvd : seq nid, cn_PhaseTwoReceiveCommits d round next_data lg recvd ].
Definition PhaseTwoAbort d round next_data lg :=
[\/ exists sent : seq nid, cn_PhaseTwoSendAborts d round next_data lg sent |
exists recvd : seq nid, cn_PhaseTwoReceiveAborts d round next_data lg recvd ].
Definition PhaseTwo (d : dstatelet) (round : nat) (next_data : data) (l : Log) :=
[\/ exists sent, cn_PhaseTwoSendCommits d round next_data l sent,
exists sent, cn_PhaseTwoSendAborts d round next_data l sent,
exists recvd, cn_PhaseTwoReceiveCommits d round next_data l recvd |
exists recvd, cn_PhaseTwoReceiveAborts d round next_data l recvd].
Definition Inv (d : dstatelet) :=
exists round l,
[\/ EverythingInit d round l
, exists next_data, PhaseOne d round next_data l
| exists next_data, PhaseTwo d round next_data l].
(***********************************************************)
(* Elimination lemmas for the invariant to deal with cases *)
(***********************************************************)
Notation coh d := (coh tpc d).
Notation PI := pf_irr.
Export TPCProtocol.
(***********************************************************)
(* Lemmas about specific sub-predicates *)
(************************************************************)
Definition coord_msg tag (from to : nid) : bool :=
[&& from == cn, to \in pts & tagFromCoordinator tag].
Definition participant_msg tag (from to : nid) : bool :=
[&& from \in pts, to == cn & tagFromParticipant tag].
Definition internal_msg (tag : nat) (from to : nid) : bool :=
coord_msg tag from to || participant_msg tag from to.
Definition E_consume_external d i this tag m from P :=
let: d' := {| dstate := upd this (loc this d) (dstate d);
dsoup := consume_msg (dsoup d) i |} in
coh d ->
find i (dsoup d) =
Some {| content := {| tag := tag; tms_cont := m |};
from := from;
to := this;
active := true |} ->
~~ internal_msg tag from this ->
P d ->
P d'.

Lemma rp_step_external d (C : coh d) this (Hthis : this \in nodes cn pts others) tag from m i :
~~ internal_msg tag from this ->
find i (dsoup d) =
Some {| content := {| tag := tag; tms_cont := m |};
from := from;
to := this;
active := true |} ->
rp_step Hnin tag from m C Hthis = loc this d.