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Propagation.tla
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-------------------------- MODULE Propagation --------------------------
EXTENDS Integers, TLAPS
CONSTANTS Replicas, Values, Coordinators
\* The following variables together constitute the system state
VARIABLES
msgs, \* The set of messages that have been delivered.
knowVal, \* val[x] is true is a replica x knows a value
allKnowVal, \* allKnowVal[a] is TRUE if replica knows that all
\*other replicas know the value
cordVal, \* cordVal[a] value ervery coordinator is trying to propagate
safeStateAttained, \* safeStateAttained[x] is true if an agent x knows that
\* the required state of knowledge has been attained
state_time \* time of the system state
(***************************************************************************)
(*Time *)
(***************************************************************************)
\* for now, representing time with natural numbers such as t_0, t_1, t_2,...
\* since using no other property but existence and greater than
\* time is discrete logical time
Time == Nat
(***************************************************************************)
(* Some temporal properties *)
(***************************************************************************)
\* If a message has been delivered from all replicas at different times,
\* there will be a time when the new system state will have that message
\* from all replicas
temporalProperty1 ==
(
(\A r \in Replicas : \E t \in Time : (state_time' = t)
/\ (\E m \in msgs' : (m.rep = r)))
=> (\E T \in Time : state_time' = T
/\ (\A r \in Replicas : (\E m \in msgs' : (m.rep = r))))
)
\* For all coordinators, there is a time when no messages have been sent to or from it
temporalProperty2 ==
\A c \in Coordinators:
(\E t \in Time : state_time = t
/\ (~(\E m \in msgs : m.cord = c)))
\*if some new state X exists at a time t, then X is the current state at some time
\* the following property includes multiple variations for pattern matching
temporalProperty3 ==
(
\A t \in Time:
\A c \in Coordinators:
(( state_time' = t
/\ \E v \in Values :
\E m \in msgs' : m.type = "1a"
/\ m.cord = c
/\ m.val = v)
=>
(\E t2 \in Time:
/\ state_time = t2
/\ \E v \in Values :
\E m \in msgs : m.type = "1a"
/\ m.cord = c
/\ m.val = v))
)
/\(
\A t \in Time:
\A c \in Coordinators:
((state_time' = t
/\ \A r \in Replicas :
\E m \in msgs' : m.type = "1b"
/\ m.cord = c
/\ m.rep = r)
=>
(\E t2 \in Time:
/\ state_time = t2
/\ \A r \in Replicas :
\E m \in msgs : m.type = "1b"
/\ m.cord = c
/\ m.rep = r))
)
/\(
\A t \in Time:
\A c \in Coordinators:
((state_time' = t
/\ \E m \in msgs' : m.cord = c
/\ m.type = "2a")
=>
(\E t2 \in Time:
/\ state_time = t2
/\ \E m \in msgs : m.cord = c
/\ m.type = "2a"))
)
/\(
\A t \in Time:
\A c \in Coordinators:
((state_time' = t
/\ \A r \in Replicas :
\E m \in msgs' : m.type = "2b"
/\ m.cord = c
/\ m.rep = r)
=>
(\E t2 \in Time:
/\ state_time = t2
/\ \A r \in Replicas :
\E m \in msgs : m.type = "2b"
/\ m.cord = c
/\ m.rep = r))
)
temporalProperties == temporalProperty1
/\ temporalProperty2
/\ temporalProperty3
(***************************************************************************)
(*Predicates for message sending and delivery *)
(* NOTE *)
(* *)
(* A message is added to msgs once delivered *)
(* Only the msgs is accessible by agents *)
(***************************************************************************)
\*sending a message does not change system state, so we leave out the definition
CONSTANT Send(_,_)
\*A message that is Delivered is added to msgs
\* A single msgs variable implies that if a message is delivered,
\* it is delivered to all acceptors at the same time, i.e., broadcast
Deliver(m,t) == (state_time' = t) /\ (msgs' = msgs \cup {m})
(***************************************************************************)
(* Phase 1a: A coordinator sends a "1a" message to all replicas *)
(***************************************************************************)
Phase1a(c,t ) ==
\E t2 \in Time :
(state_time' = t2
/\ Send([type |-> "1a", cord |-> c ,
val |-> cordVal[c]], t2))
(***************************************************************************)
(* Phase 1b: If a replica receives a "1a" message, it replies with a "1b" *)
(* message and learns the value *)
(***************************************************************************)
Phase1b_learn(r,t) ==
\E t2 \in Time :
(state_time' = t2
/\ knowVal' = [knowVal EXCEPT ![r] = TRUE])
Phase1b(r,m,t) ==
\E t2 \in Time :
(state_time' = t2
/\ Send([type |-> "1b", cord |-> m.cord, rep |-> r], t2))
(***************************************************************************)
(* Phase 2a: If the coordinator receives "1b" messages from all replicas *)
(* it sends a "2a" message to all replicas *)
(***************************************************************************)
Phase2a(c,t) ==
\E t2 \in Time :
(state_time' = t2
/\ Send([type |-> "2a", cord |-> c], t2))
(***************************************************************************)
(* Phase 2b: If areplica receives a "2a" message from a coordinator *)
(* it knows that all other replicas have also committed thesame value *)
(***************************************************************************)
Phase2b_learn(r,t) ==
\E t2 \in Time :
(state_time' = t2
/\ allKnowVal' = [allKnowVal EXCEPT ![r] = TRUE])
Phase2b(r,m,t) ==
\E t2 \in Time :
(state_time' = t2
/\ Send([type |-> "2b", cord |-> m.cord, rep |-> r], t2))
(***************************************************************************)
(* Learning: If 2b messages for all replica are delivered to a coordinator *)
(* then the coodinator knows that the `^$E^2_{G}\phi$ ^' has been attained *)
(***************************************************************************)
Phase2c_learn(c,t) ==
\E t2 \in Time :
(state_time' = t2
/\ safeStateAttained' = [safeStateAttained EXCEPT ![c] = TRUE])
(***************************************************************************)
(* The type of possible messages *)
(***************************************************************************)
Messages == [type : {"1a"}, cord : Coordinators, val : Values]
\cup [type : {"1b"}, cord : Coordinators, rep : Replicas]
\cup [type : {"2a"}, cord : Coordinators]
\cup [type : {"2b"}, cord : Coordinators, rep : Replicas]
(***************************************************************************)
(* The following section specifies rules and predicates that define *)
(* availability and the behaviour of available agents *)
(***************************************************************************)
\*Defining Availability as an abstract prediate that can either be true or false
CONSTANT Available(_, _)
ASSUME \A x, t : Available(x,t) \in BOOLEAN
\*Always Available
Always_available(x) == \A t \in Time : Available(x,t)
(***************************************************************************)
(*Rules for Coordinators *)
(***************************************************************************)
Rule_1a_msg(c,t) ==
(~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c))
<=> Phase1a(c,t)
Rule_2a_msg(c,t) ==
(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)
<=> Phase2a(c,t)
Rule_2c_learn(c,t) ==
(\A r \in Replicas : (\E m \in msgs : m.type = "2b" /\ m.rep = r /\ m.cord = c))
<=> (Phase2c_learn(c,t))
(***************************************************************************)
(*Rules for Replicas *)
(***************************************************************************)
Rule_1b_msg(r,t) ==
\A c \in Coordinators : \A m \in msgs : \A v \in Values :
((m.type = "1a" /\ m.cord = c /\ m.val = v) => Phase1b(r,m,t))
Rule_1b_learn(r,t) ==
(Phase1b_learn(r,t) <=> (\E m \in msgs : Send([type |-> "1b", cord |-> m.cord, rep |-> r], t)))
/\ (Phase1b_learn(r,t) <=> (\E c \in Coordinators: \E m \in msgs : m.type = "1a" /\ m.cord = c))
Rule_2b_msg(r,t) ==
\A c \in Coordinators : \A m \in msgs :
((m.type = "2a" /\ m.cord = c) => Phase2b(r,m,t))
Rule_2b_learn(r,t) ==
(Phase2b_learn(r,t) <=> (\E m \in msgs : Send([type |-> "2b", cord |-> m.cord, rep |-> r], t)))
/\ (Phase2b_learn(r,t) <=> (\E c \in Coordinators: \E m \in msgs : m.type = "2a" /\ m.cord = c))
(***************************************************************************)
(*Non-Byzantine behavior of agents *)
(***************************************************************************)
\*A coordinator's state-dependent expected behavior
\*Each state has an associated time and depending on context (state or state'),
\*it can either be state_time or state_time'
Coordinator_behavior ==
(\A c \in Coordinators :
\A t \in Time : ( ( (\/ state_time = t \/ state_time' = t)
/\ Available(c,t)) =>
/\ Rule_1a_msg(c,t)
/\ Rule_2a_msg(c,t)
/\ Rule_2c_learn(c,t) ) )
\*A replica's state-dependent expected behavior
\*Each state has an associated time and depending on context(state or state'),
\*it can either be state_time or state_time'
Replica_behavior ==
(\A r \in Replicas :
\A t \in Time : ( ( (\/ state_time = t \/ state_time' = t)
/\ Available(r,t)) =>
/\ Rule_1b_msg(r,t)
/\ Rule_1b_learn(r,t)
/\ Rule_2b_msg(r,t)
/\ Rule_2b_learn(r,t) ) )
(***************************************************************************)
(*The following section specifies assumptions for proving progress *)
(***************************************************************************)
\*All replicas are always available
allReplicasAlwaysAvailable == \A r \in Replicas : Always_available(r)
\* A coordinator is always available
existsCoordinatorAlwaysAvailable == \E c \in Coordinators : Always_available(c)
aAvailability == allReplicasAlwaysAvailable
/\ existsCoordinatorAlwaysAvailable
aBehavior == Coordinator_behavior
/\ Replica_behavior
\* The set of replicas is non-empty
aNonEmptyReplicas == \E r \in Replicas : Always_available(r)
\* Guaranteed Message Delivery: all messages that are sent will be eventually delivered
aDelivery ==
\* 1a msg
/\ \A c \in Coordinators : \A v \in Values : \A t \in Time :
Send([type |-> "1a", cord |-> c, val |-> v], t) =>
\E t2 \in Time : (t2 > t) /\ Deliver([type |-> "1a", cord |-> c,
val |-> v],t2)
\* 1b msg Deliver([type |-> "1b", cord |-> m.cord, val |-> m.val, rep |-> r]
/\ \A r \in Replicas : \A m \in msgs : \A t \in Time :
Send([type |-> "1b", cord |-> m.cord, rep |-> r],t) =>
\E t2 \in Time : (t2 > t) /\ Deliver ([type |-> "1b",
cord |-> m.cord,
rep |-> r],t2)
/\ \* 2a msg Deliver([type |-> "2a", cord |-> c], t2)
\A c \in Coordinators : \A t \in Time :
Send([type |-> "2a", cord |-> c], t)
<=> \E t2 \in Time : (t2 > t) /\ Deliver([type |-> "2a", cord |-> c], t2)
/\ \* 2b message Send([type |-> "2b", cord |-> m.cord, rep |-> r]
\A r \in Replicas : \A m \in msgs : \A t \in Time :
Send([type |-> "2b", cord |-> m.cord, rep |-> r],t) =>
\E t2 \in Time : (t2 > t) /\ Deliver ([type |-> "2b",
cord |-> m.cord,
rep |-> r],t2)
\* Messages Cannot Be Corrupted: if a message exists in the system state,
\* then it must have been sent at some time
aNoCorruption ==
(
\A r \in Replicas: \A m \in msgs :
(
(m.type = "2b" /\ m.rep = r)
=> (\E t \in Time : state_time' = t /\ Available(r,t) /\ Send([type |-> "2b", cord |-> m.cord, rep |-> r],t))
)
)
/\ (
\A c \in Coordinators : \A m \in msgs :
(
(m.type = "2a" /\ m.cord = c)
=> (\E t2 \in Time : state_time' = t2 /\ Available(c,t2) /\ Send([type |-> "2a", cord |-> c],t2))
)
)
/\ (
\A r \in Replicas : \A m \in msgs :
(
((m.type = "1b" /\ m.rep = r))
=> (\E t \in Time : state_time' = t /\ Available(r,t) /\ Send([type |-> "1b", cord |-> m.cord, rep |-> r],t))
)
)
\* Additional system properties
aSysProps == msgs \in SUBSET Messages
/\ cordVal \in [Coordinators -> Values]
/\ temporalProperties
\* progress assertions
pAssertions == aAvailability
/\ aDelivery
/\ aNonEmptyReplicas
/\ aBehavior
/\ aSysProps
\* safety assertions
sAssertions == allReplicasAlwaysAvailable
/\ aNoCorruption
/\ aBehavior
/\ aSysProps
-----------------------------------------------------------------------------
(***************************************************************************)
(*The proof *)
(***************************************************************************)
\* Lemma - if a coordinator is available, 1a messages from it will be eventually
\* delivered if it has not received 1b messsages from all replicas
LEMMA Lemma1 ==
\A c \in Coordinators :
(
(\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
/\ pAssertions
=>
(\E t \in Time :
/\ state_time' = t
/\ (\E m \in msgs' : \E v \in Values : m.cord = c /\ m.type = "1a" /\ m.val = v))
)
<1> SUFFICES ASSUME NEW c \in Coordinators, pAssertions,
(\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
PROVE
(\E t \in Time :
/\ state_time' = t
/\ (\E m \in msgs' : \E v \in Values : m.cord = c /\ m.type = "1a" /\ m.val = v))
OBVIOUS
<1>1. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
OBVIOUS
<1>2. aBehavior /\ aDelivery /\ aSysProps
BY DEF pAssertions
<1>3. Coordinator_behavior
BY <1>2 DEF aBehavior
<1>4. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c))
/\ Rule_1a_msg(c,t) )
BY <1>1, <1>3 DEF Coordinator_behavior
<1>5. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c))
/\ Rule_1a_msg(c,t)
/\ ( \E t2 \in Time : Phase1a(c,t2)))
BY <1>4 DEF Rule_1a_msg
<1>6. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c))
/\ Rule_1a_msg(c,t)
/\ ( \E t2 \in Time : Send([type |-> "1a", cord |-> c ,
val |-> cordVal[c]], t2)))
BY <1>5 DEF Phase1a
<1>6a. cordVal[c] \in Values
BY <1>2 DEF aSysProps
<1>7. ( \E t2 \in Time : Deliver([type |-> "1a", cord |-> c , val |-> cordVal[c]], t2))
BY <1>6, <1>6a, <1>2 DEF aDelivery
<1>8. ( \E t2 \in Time : (state_time' = t2)
/\ (msgs' = msgs \cup {[type |-> "1a", cord |-> c , val |-> cordVal[c]]}) )
BY <1>7 DEF Deliver
<1>9. ( \E t2 \in Time : (state_time' = t2)
/\ \E v \in Values : ([type |-> "1a", cord |-> c , val |-> v] \in msgs') )
BY <1>8, <1>6a
<1>10. ( \E t2 \in Time : (state_time' = t2)
/\ \E v \in Values : \E m \in msgs' : m.type = "1a" /\ m.cord = c /\ m.val = v )
BY <1>9
<1>200. QED
BY <1>10
\* Lemma - If 1a messages from a coordinator have been delivered, eventually
\* 1b messages for P from all replicas will be delivered
LEMMA Lemma2 ==
\A c \in Coordinators :
(
( \E t2 \in Time : state_time = t2
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
/\ pAssertions
=>
( \E t2 \in Time : state_time' = t2
/\ \A r \in Replicas : \E m \in msgs' : m.type = "1b" /\ m.cord = c /\ m.rep = r)
)
<1> SUFFICES ASSUME NEW c \in Coordinators, pAssertions,
(\E t2 \in Time : state_time = t2
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
PROVE (\E t2 \in Time : state_time' = t2
/\ \A r \in Replicas : \E m \in msgs' : m.type = "1b" /\ m.cord = c /\ m.rep = r)
OBVIOUS
<1>1. (\E t2 \in Time : state_time = t2
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
OBVIOUS
<1>2. aBehavior /\ aDelivery /\ aSysProps /\ aAvailability
BY DEF pAssertions
<1>3. Coordinator_behavior
BY <1>2 DEF aBehavior
<1>4. \A r \in Replicas : Always_available(r)
BY <1>2 DEF aAvailability, allReplicasAlwaysAvailable
<1>5. (\E t2 \in Time : state_time = t2
/\ (\E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
/\ (\A r \in Replicas : Available(r,t2)))
BY <1>4, <1>1 DEF Always_available
<1>5a. (\A r \in Replicas :
\A t \in Time : (state_time = t /\ Available(r,t)) =>
/\ Rule_1b_msg(r,t)
/\ Rule_2b_msg(r,t) )
BY <1>2 DEF aBehavior, Replica_behavior
<1>6. (\E t2 \in Time : state_time = t2
/\ (\E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
/\ \A r \in Replicas : Available(r,t2))
BY <1>5
<1>7.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ Available(r,t2)
/\ (\E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
BY <1>6
<1>8.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ Rule_1b_msg(r,t2)
/\ (\E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v)
BY <1>7, <1>5a
<1>9.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ \E m \in msgs : (\E v \in Values : m.type = "1a" /\ m.cord = c /\ m.val = v)
/\ Rule_1b_msg(r,t2)
BY <1>8
<1>10.\A r \in Replicas :
/\ \E m \in msgs : (\E v \in Values : m.type = "1a" /\ m.cord = c /\ m.val = v)
/\ \E t \in Time : Phase1b(r,m,t)
BY <1>9 DEF Rule_1b_msg
<1>11.\A r \in Replicas :
/\ \E m \in msgs : (m.type = "1a" /\ m.cord = c)
/\ \E t \in Time : Send([type |-> "1b", cord |-> m.cord, rep |-> r], t)
BY <1>10 DEF Phase1b
<1>12.\A r \in Replicas :
/\ \E m \in msgs : m.type = "1a" /\ m.cord = c
/\ \E t \in Time : Deliver([type |-> "1b", cord |-> m.cord, rep |-> r], t)
BY <1>11, <1>2 DEF aDelivery
<1>13.\A r \in Replicas :
/\ \E m \in msgs : m.type = "1a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ msgs' = msgs \cup {[type |-> "1b", cord |-> m.cord, rep |-> r]}
BY <1>12 DEF Deliver
<1>13a.\A r \in Replicas :
/\ \E m \in msgs : m.type = "1a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ msgs' = msgs \cup {[type |-> "1b", cord |-> c, rep |-> r]}
BY <1>13
<1>14.\A r \in Replicas :
/\ \E m \in msgs : m.type = "1a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ [type |-> "1b", cord |-> c, rep |-> r] \in msgs'
BY <1>13
<1>15.\A r \in Replicas :
/\ \E t \in Time : state_time' = t
/\ \E m \in msgs' : m.type = "1b" /\ m.cord = c /\ m.rep = r
BY <1>14
<1>16. \E t \in Time : state_time' = t
/\ \A r \in Replicas : \E m \in msgs' : m.type = "1b" /\ m.cord = c /\ m.rep = r
BY <1>2, <1>15 DEF aSysProps, temporalProperties, temporalProperty1
<1>200. QED
BY <1>16
\* Lemma - If 1b messages from all replicas have been delivered for a coordinator
\* which is available, eventually 2a messages from it will be delivered
LEMMA Lemma3 ==
\A c \in Coordinators :
(
(\E t \in Time : Available(c,t)
/\ state_time = t
/\ ((\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord=c)))
/\ pAssertions
=>
(\E t \in Time :
state_time' = t
/\ (\E m \in msgs' : m.cord = c /\ m.type = "2a"))
)
<1> SUFFICES ASSUME NEW c \in Coordinators, pAssertions,
(\E t \in Time : Available(c,t)
/\ state_time = t
/\ ((\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord=c)))
PROVE (\E t \in Time :
state_time' = t
/\ (\E m \in msgs' : m.cord = c /\ m.type = "2a"))
OBVIOUS
<1>1. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ ((\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord=c)))
OBVIOUS
<1>2. aBehavior /\ aDelivery /\ aSysProps /\ aAvailability
BY DEF pAssertions
<1>3. Coordinator_behavior
BY <1>2 DEF aBehavior
<1>4. (\E t \in Time : Available(c,t)
/\ state_time = t
/\ Rule_2a_msg(c,t)
/\ ((\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord=c)))
BY <1>1, <1>2 DEF aBehavior, Coordinator_behavior
<1>5. \E t \in Time :
state_time = t
/\ Rule_2a_msg(c,t)
/\ (\E t2 \in Time : Phase2a(c,t2))
BY <1>4 DEF Rule_2a_msg
<1>6. (\E t2 \in Time : Send([type |-> "2a", cord |-> c], t2))
BY <1>5 DEF Phase2a
<1>7. (\E t2 \in Time : Deliver([type |-> "2a", cord |-> c], t2))
BY <1>6, <1>2 DEF aDelivery
<1>8. (\E t2 \in Time : state_time' = t2 /\ msgs' = msgs \cup{[type |-> "2a", cord |-> c]})
BY <1>7 DEF Deliver
<1>9. (\E t2 \in Time : state_time' = t2 /\ [type |-> "2a", cord |-> c] \in msgs')
BY <1>8
<1>10. (\E t \in Time :
state_time' = t
/\ (\E m \in msgs' : m.cord = c /\ m.type = "2a"))
BY <1>9
<1>200. QED
BY <1>10
\* Lemma - If 2a messages from a coordinator have been delivered, eventually
\* 2b messages for P from all replicas will be delivered
LEMMA Lemma4 ==
\A c \in Coordinators :
(
( \E t2 \in Time : state_time = t2
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c)
/\ pAssertions
=>
( \E t2 \in Time : state_time' = t2
/\ \A r \in Replicas : \E m \in msgs' : m.type = "2b" /\ m.cord = c /\ m.rep = r)
)
<1> SUFFICES ASSUME NEW c \in Coordinators, pAssertions,
( \E t2 \in Time : state_time = t2
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c)
PROVE
( \E t2 \in Time : state_time' = t2
/\ \A r \in Replicas : \E m \in msgs' : m.type = "2b" /\ m.cord = c /\ m.rep = r)
OBVIOUS
<1>1. (\E t2 \in Time : state_time = t2
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c)
OBVIOUS
<1>2. aBehavior /\ aDelivery /\ aSysProps /\ aAvailability
BY DEF pAssertions
<1>3. Coordinator_behavior
BY <1>2 DEF aBehavior
<1>4. \A r \in Replicas : Always_available(r)
BY <1>2 DEF aAvailability, allReplicasAlwaysAvailable
<1>5. (\E t2 \in Time : state_time = t2
/\ (\E m \in msgs : m.type = "2a" /\ m.cord = c)
/\ (\A r \in Replicas : Available(r,t2)))
BY <1>4, <1>1 DEF Always_available
<1>5a. (\A r \in Replicas :
\A t \in Time : (state_time = t /\ Available(r,t)) =>
/\ Rule_1b_msg(r,t)
/\ Rule_2b_msg(r,t) )
BY <1>2 DEF aBehavior, Replica_behavior
<1>6. (\E t2 \in Time : state_time = t2
/\ (\E m \in msgs : m.type = "2a" /\ m.cord = c )
/\ \A r \in Replicas : Available(r,t2))
BY <1>5
<1>7.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ Available(r,t2)
/\ (\E m \in msgs : m.type = "2a" /\ m.cord = c )
BY <1>6
<1>8.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ Rule_2b_msg(r,t2)
/\ (\E m \in msgs : m.type = "2a" /\ m.cord = c)
BY <1>7, <1>5a
<1>9.\A r \in Replicas : \E t2 \in Time :
state_time = t2
/\ \E m \in msgs : ( m.type = "2a" /\ m.cord = c )
/\ Rule_2b_msg(r,t2)
BY <1>8
<1>10.\A r \in Replicas :
/\ \E m \in msgs : ( m.type = "2a" /\ m.cord = c )
/\ \E t \in Time : Phase2b(r,m,t)
BY <1>9 DEF Rule_2b_msg
<1>11.\A r \in Replicas :
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
/\ \E t \in Time : Send([type |-> "2b", cord |-> m.cord, rep |-> r], t)
BY <1>10 DEF Phase2b
<1>12.\A r \in Replicas :
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
/\ \E t \in Time : Deliver([type |-> "2b", cord |-> m.cord, rep |-> r], t)
BY <1>11, <1>2 DEF aDelivery
<1>13.\A r \in Replicas :
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ msgs' = msgs \cup {[type |-> "2b", cord |-> m.cord, rep |-> r]}
BY <1>12 DEF Deliver
<1>13a.\A r \in Replicas :
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ msgs' = msgs \cup {[type |-> "2b", cord |-> c, rep |-> r]}
BY <1>13
<1>14.\A r \in Replicas :
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
/\ \E t \in Time : state_time' = t
/\ [type |-> "2b", cord |-> c, rep |-> r] \in msgs'
BY <1>13
<1>15.\A r \in Replicas :
/\ \E t \in Time : state_time' = t
/\ \E m \in msgs' : m.type = "2b" /\ m.cord = c /\ m.rep = r
BY <1>14
<1>16. \E t \in Time : state_time' = t
/\ \A r \in Replicas : \E m \in msgs' : m.type = "2b" /\ m.cord = c /\ m.rep = r
BY <1>2, <1>15 DEF aSysProps, temporalProperties, temporalProperty1
<1>200. QED
BY <1>16
(***************************************************************************)
(*The intermediate states *)
(***************************************************************************)
\* after a coordinator has sent 1a msgs
msgsPost1a(c) ==
Always_available(c)
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v
\* after a coordinator has received 1b messages from all replicas
msgsPost1b(c) ==
Always_available(c)
/\ \A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.cord = c /\ m.rep = r
\* after a coordinator has sent 2a msgs
msgsPost2a(c) ==
Always_available(c)
/\ \E m \in msgs : m.cord = c /\ m.type = "2a"
\* after a coordinator has received 2b messages from all replicas
msgsPost2b(c) ==
Always_available(c)
/\ \A r \in Replicas : \E m \in msgs : m.type = "2b" /\ m.cord = c /\ m.rep = r
----------------------------------------------------------------------------
\* Eventually, there will be 1a messages from a coordinator
LEMMA Lemma5 ==
pAssertions =>
\E c \in Coordinators:
\E t \in Time: state_time = t
/\ msgsPost1a(c)
<1> SUFFICES ASSUME pAssertions
PROVE \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost1a(c)
OBVIOUS
<1>1. aAvailability /\ aSysProps /\ aNonEmptyReplicas
BY DEF pAssertions
<1>2. existsCoordinatorAlwaysAvailable
BY <1>1 DEF aAvailability
<1>3. \E c \in Coordinators : Always_available(c)
BY <1>2 DEF existsCoordinatorAlwaysAvailable
<1>4. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ (~(\E m \in msgs : m.cord = c)))
BY <1>3, <1>1 DEF aSysProps, temporalProperties, temporalProperty2
<1>5. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ Available(c,t)
/\ (~(\E m \in msgs : m.cord = c)))
BY <1>4 DEF Always_available
<1>5a. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ Available(c,t)
/\ \E r \in Replicas : Always_available(r)
/\ (~(\E m \in msgs : m.cord = c)))
BY <1>5, <1>1 DEF aNonEmptyReplicas
<1>5b. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ Available(c,t)
/\ (\E r \in Replicas : Always_available(r))
/\ (\E r \in Replicas : ~(\E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
BY <1>5a
<1>5c. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ Available(c,t)
/\ (\E r \in Replicas : Always_available(r))
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
BY <1>5b
<1>5d. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : Available(c,t)
/\ state_time = t
/\ (~(\A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.rep = r /\ m.cord = c)))
BY <1>5c
<1>6. \E t \in Time : \E c \in Coordinators : Always_available(c)
/\ state_time' = t
/\ \E v \in Values : \E m \in msgs' : m.type = "1a" /\ m.cord = c /\ m.val = v
BY <1>5d, Lemma1
<1>7. \E t \in Time : \E c \in Coordinators :
\E t2 \in Time: state_time = t2
/\ Always_available(c)
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v
BY <1>6, <1>1 DEF aSysProps, temporalProperties, temporalProperty3
<1>8. \E t \in Time : \E c \in Coordinators : state_time = t
/\ msgsPost1a(c)
BY <1>7 DEF msgsPost1a
<1>200. QED
BY <1>8
LEMMA Lemma6 ==
pAssertions =>
\E c \in Coordinators:
\E t \in Time: state_time = t
/\ msgsPost1b(c)
<1> SUFFICES ASSUME pAssertions
PROVE \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost1b(c)
OBVIOUS
<1>1. aAvailability /\ aSysProps /\ aNonEmptyReplicas
BY DEF pAssertions
<1>4. \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost1a(c)
BY Lemma5
<1>5. \E c \in Coordinators :
/\ \E t \in Time : state_time = t
/\ Always_available(c)
/\ \E v \in Values : \E m \in msgs : m.type = "1a" /\ m.cord = c /\ m.val = v
BY <1>4 DEF msgsPost1a
<1>5a. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time' = t
/\ \A r \in Replicas : \E m \in msgs' : m.type = "1b" /\ m.cord = c /\ m.rep = r)
BY <1>5, Lemma2
<1>6. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ \A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.cord = c /\ m.rep = r)
BY <1>5a, <1>1 DEF aSysProps, temporalProperties, temporalProperty3
<1>8. \E t \in Time : \E c \in Coordinators : state_time = t
/\ msgsPost1b(c)
BY <1>6 DEF msgsPost1b
<1>200. QED
BY <1>8
LEMMA Lemma7 ==
pAssertions =>
\E c \in Coordinators:
\E t \in Time: state_time = t
/\ msgsPost2a(c)
<1> SUFFICES ASSUME pAssertions
PROVE \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost2a(c)
OBVIOUS
<1>1. aAvailability /\ aSysProps /\ aNonEmptyReplicas
BY DEF pAssertions
<1>4. \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost1b(c)
BY Lemma6
<1>5. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ \A r \in Replicas : \E m \in msgs : m.type = "1b" /\ m.cord = c /\ m.rep = r)
BY <1>4 DEF msgsPost1b
<1>6. \E c \in Coordinators : Always_available(c)
/\ \E t \in Time : state_time' = t
/\ \E m \in msgs' : m.cord = c /\ m.type = "2a"
BY <1>5, Lemma3 DEF Always_available
<1>7. \E t \in Time : \E c \in Coordinators : Always_available(c)
/\ state_time = t
/\ \E m \in msgs : m.cord = c /\ m.type = "2a"
BY <1>6, <1>1 DEF aSysProps, temporalProperties, temporalProperty3
<1>8. \E t \in Time : \E c \in Coordinators : state_time = t
/\ msgsPost2a(c)
BY <1>7 DEF msgsPost2a
<1>200. QED
BY <1>8
LEMMA Lemma8 ==
pAssertions =>
\E c \in Coordinators:
\E t \in Time: state_time = t
/\ msgsPost2b(c)
<1> SUFFICES ASSUME pAssertions
PROVE \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost2b(c)
OBVIOUS
<1>1. aAvailability /\ aSysProps /\ aNonEmptyReplicas
BY DEF pAssertions
<1>4. \E c \in Coordinators:
\E t \in Time: state_time = t
/\msgsPost2a(c)
BY Lemma7
<1>5. \E c \in Coordinators :
/\ \E t \in Time : state_time = t
/\ Always_available(c)
/\ \E m \in msgs : m.type = "2a" /\ m.cord = c
BY <1>4 DEF msgsPost2a
<1>5a. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time' = t
/\ \A r \in Replicas : \E m \in msgs' : m.type = "2b" /\ m.cord = c /\ m.rep = r)
BY <1>5, Lemma4
<1>6. \E c \in Coordinators : Always_available(c)
/\ (\E t \in Time : state_time = t
/\ \A r \in Replicas : \E m \in msgs : m.type = "2b" /\ m.cord = c /\ m.rep = r)
BY <1>5a, <1>1 DEF aSysProps, temporalProperties, temporalProperty3
<1>8. \E t \in Time : \E c \in Coordinators : state_time = t
/\ msgsPost2b(c)
BY <1>6 DEF msgsPost2b
<1>200. QED
BY <1>8
\* Safety Lemma1 - A Coordinator will not learn `^$E^2_{G}\phi$ ^' if there
\* is at least one replica which does not know `^$E_{G}\phi$ ^'
THEOREM Lemma9 ==
sAssertions =>
(
\A t \in Time : \A c \in Coordinators :
(( state_time' = t
/\ Available(c,t)
/\ safeStateAttained' = [safeStateAttained EXCEPT ![c] = TRUE])
=>
(\A r \in Replicas : allKnowVal' = [allKnowVal EXCEPT ![r] = TRUE]))
)
<1> SUFFICES ASSUME sAssertions,
NEW t \in Time, NEW c \in Coordinators,
( state_time' = t
/\ Available(c,t)
/\ safeStateAttained' = [safeStateAttained EXCEPT ![c] = TRUE])
PROVE (\A r \in Replicas : allKnowVal' = [allKnowVal EXCEPT ![r] = TRUE])
OBVIOUS
<1>1.( state_time' = t
/\ Available(c,t)
/\ safeStateAttained'= [safeStateAttained EXCEPT ![c] = TRUE])
OBVIOUS
<1>2. aBehavior /\ allReplicasAlwaysAvailable
BY DEF sAssertions
<1>3. state_time' = t
/\ Available(c,t)
/\ Phase2c_learn(c,t)
BY <1>1, <1>2 DEF aBehavior, Coordinator_behavior, Rule_2c_learn, Phase2c_learn
<1>3a. state_time = t /\ Available(c,t) => ((\A r \in Replicas : (\E m \in msgs : m.type = "2b" /\ m.rep = r /\ m.cord = c)) <=> Phase2c_learn(c,t))
BY <1>2 DEF aBehavior, Coordinator_behavior, Rule_2c_learn
<1>4.
state_time' = t
/\ Available(c,t)
/\ Phase2c_learn(c,t)
/\ (\A r \in Replicas : (\E m \in msgs : m.type = "2b" /\ m.rep = r /\ m.cord = c))
BY <1>1, <1>2 DEF aBehavior, Coordinator_behavior, Rule_2c_learn, Phase2c_learn
<1>5. (\A r \in Replicas : (\E m \in msgs : m.type = "2b" /\ m.rep = r /\ m.cord = c))
BY <1>4
<1>5a1. \A r \in Replicas : (\E m \in msgs : m.type = "2b" /\ m.rep = r )
BY <1>5
<1>5a.
\A r \in Replicas:
(
(\E m \in msgs : (m.type = "2b" /\ m.rep = r))
=> (\E m \in msgs : \E t2 \in Time : state_time ' = t2 /\ Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2))
)
BY DEF sAssertions, aNoCorruption
<1>5b.
\A r \in Replicas : (\E m \in msgs : (m.type = "2b" /\ m.rep = r ))
/\ (\E m \in msgs : \E t2 \in Time : state_time ' = t2 /\ Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2))
BY <1>5, <1>5a
<1>5c.
\A r \in Replicas :
(\E m \in msgs : \E t2 \in Time : state_time ' = t2 /\ Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2))
BY <1>5b
<1>5d. \A r \in Replicas :
\E t2 \in Time : state_time ' = t2
/\ \E m \in msgs : Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2)
BY <1>5c
<1>5e. \A r \in Replicas :
\E t2 \in Time :
/\ state_time ' = t2
/\ Available(r,t2)
/\ (\E m \in msgs : Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2))
BY <1>5d, <1>2 DEF allReplicasAlwaysAvailable, Always_available
<1>5h. \A r \in Replicas :
\E t2 \in Time :
/\ state_time ' = t2
/\ Available(r,t2)
/\ (\E m \in msgs : Send([type |-> "2b", cord |-> m.cord, rep |-> r],t2))
/\ Phase2b_learn(r,t2)
BY <1>5e, <1>2 DEF aBehavior, Replica_behavior, Rule_2b_learn
<1>6. ASSUME (~ \A r \in Replicas : allKnowVal' = [allKnowVal EXCEPT ![r] = TRUE])
PROVE FALSE
<2>1. (~ \A r \in Replicas : allKnowVal' = [allKnowVal EXCEPT ![r] = TRUE])
BY <1>6
<2>2. ~(\A r \in Replicas : \E t2 \in Time: Phase2b_learn(r,t2))
BY <2>1 DEF Phase2b_learn
<2>9. FALSE
BY <1>5h, <2>2
<2>200. QED
BY <2>9
<1>200. QED
BY <1>6
\* Safety Lemma2 - Forall replicas, `^$k_rE_{G}\phi$ ^' implies that
\* all replicas know `$\phi$'
THEOREM Lemma10 ==
sAssertions =>
(
\A r \in Replicas : \A t2 \in Time :