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LeaderProof.scala
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LeaderProof.scala
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package distribution
import FifoNetwork._
import Networking._
import Quantifiers._
import Protocol._
import ListUtils._
import stainless.lang._
import stainless.collection._
import stainless.proof._
import stainless.annotation._
import stainless.lang.synthesis._
import scala.language.postfixOps
// This object contains lemma and auxiliary functions used in the proofs
object ProtocolProof {
import Protocol._
def areEqual[T](t1: T, t2: T) = t1 == t2
case class Params(n: BigInt, starterProcess: BigInt, ssns: BigInt => BigInt) extends Parameter
def validParam(p: Parameter) = {
val Params(n, starterProcess, ssns) = p
0 <= starterProcess && starterProcess < n &&
intForAll2(n, n, distinctSSN_fun(ssns))
}
def validId(net: VerifiedNetwork, id: ActorId) = {
require(networkInvariant(net.param, net.states, net.messages, net.getActor))
val UID(uid) = id
val Params(n, starterProcess, ssns) = net.param
0 <= uid && uid < n
}
/**
* Invariant stating that the getActor map is defined for all actors,
* and that getActor(i) has id "i"
*/
def getActorDefined(getActor: Map[ActorId, Actor])(i: BigInt) = {
getActor.contains(UID(i)) && getActor(UID(i)).myId == UID(i)
}
def init_getActorDefined(n: BigInt, ssns: BigInt => BigInt): Boolean = {
if (n <= 0) intForAll(n, getActorDefined(init_getActor(ssns)))
else init_getActorDefined(n - 1, ssns) && intForAll(n, getActorDefined(init_getActor(ssns)))
} holds
/**
* Invariant stating that the states map is defined for all actors
*/
def statesDefined(states: Map[ActorId, State]) = {
(i: BigInt) => states.contains(UID(i))
}
def init_statesDefined(n: BigInt): Boolean = {
if (n <= 0) intForAll(n, statesDefined(init_states))
else init_statesDefined(n - 1) && init_states.contains(UID(n - 1)) && intForAll(n, statesDefined(init_states))
} holds
def statesStillDefined(n: BigInt, states: Map[ActorId, State], id: ActorId, s: State): Boolean = {
require(intForAll(n, statesDefined(states)))
if (n <= 0)
intForAll(n, statesDefined(states.updated(id, s)))
else
states.updated(id, s).contains(UID(n - 1)) &&
statesStillDefined(n - 1, states, id, s) &&
intForAll(n, statesDefined(states.updated(id, s)))
} holds
/**
* Invariant stating that communication only happens in rings
*/
def ringChannels(n: BigInt, messages: Map[(ActorId, ActorId), List[Message]])(i: BigInt, j: BigInt) = {
0 <= i && i < n && (messages.contains(UID(i), UID(j)) ==> (j == increment(i, n)))
}
def init_ringChannels_aux(u: BigInt, v: BigInt, n: BigInt): Boolean = {
require(u <= n && v <= n)
if (u <= 0 || v <= 0)
intForAll2(u, v, ringChannels(n, init_messages))
else {
init_ringChannels_aux(u - 1, v, n) &&
init_ringChannels_aux(u, v - 1, n) &&
intForAll2(u, v, ringChannels(n, init_messages))
}
} holds
def init_ringChannels(n: BigInt): Boolean = {
init_ringChannels_aux(n, n, n) &&
intForAll2(n, n, ringChannels(n, init_messages))
} holds
def stillRingChannels(
n: BigInt, m1: BigInt, m2: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
m1 <= n && m2 <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
intForAll2(m1, m2, ringChannels(n, messages)) && ureceiver == increment(usender, n))
if (m1 <= 0 || m2 <= 0)
intForAll2(m1, m2, ringChannels(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
stillRingChannels(n, m1 - 1, m2, messages, usender, ureceiver, tt) &&
stillRingChannels(n, m1, m2 - 1, messages, usender, ureceiver, tt) &&
intForAll2(m1, m2, ringChannels(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
/**
* Invariant stating that each channel contains at most one message
*/
def smallChannel(n: BigInt, messages: Map[(ActorId, ActorId), List[Message]])(i: BigInt) = {
0 <= i && i < n && messages.getOrElse((UID(i), UID(increment(i, n))), Nil()).size < 2
}
def init_smallChannel_aux(u: BigInt, n: BigInt): Boolean = {
require(u <= n)
if (u <= 0)
intForAll(u, smallChannel(n, init_messages))
else
init_smallChannel_aux(u - 1, n) &&
intForAll(u, smallChannel(n, init_messages))
} holds
def init_smallChannel(n: BigInt) = init_smallChannel_aux(n, n)
def stillSmallChannel(
n: BigInt, m: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
m <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
tt.size < 2 &&
intForAll(m, smallChannel(n, messages)))
if (m <= 0)
intForAll(m, smallChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
stillSmallChannel(n, m - 1, messages, usender, ureceiver, tt) &&
intForAll(m, smallChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
def emptyToSmallChannel(
n: BigInt, m: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
m <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
tt.size < 2 &&
intForAll(m, emptyChannel(n, messages))
)
if (m <= 0)
intForAll(m, smallChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
emptyToSmallChannel(n, m - 1, messages, usender, ureceiver, tt) &&
intForAll(m, smallChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
/**
* Invariant stating that there is at most channel which is not empty
*/
def onlyOneChannel(n: BigInt, messages: Map[(ActorId, ActorId), List[Message]])(i: BigInt, j: BigInt) = {
0 <= i && i < n &&
0 <= j && j < n && (
messages.getOrElse((UID(i), UID(increment(i, n))), Nil()).isEmpty ||
messages.getOrElse((UID(j), UID(increment(j, n))), Nil()).isEmpty ||
i == j
)
}
def init_onlyOneChannel_aux(u: BigInt, v: BigInt, n: BigInt): Boolean = {
require(u <= n && v <= n)
if (u <= 0 || v <= 0)
intForAll2(u, v, onlyOneChannel(n, init_messages))
else {
init_onlyOneChannel_aux(u - 1, v, n) &&
init_onlyOneChannel_aux(u, v - 1, n) &&
onlyOneChannel(n, init_messages)(u - 1, v - 1) &&
intForAll2(u, v, onlyOneChannel(n, init_messages))
}
} holds
//
def init_onlyOneChannel(n: BigInt) = init_onlyOneChannel_aux(n, n, n)
def stillOneChannel(
n: BigInt, m1: BigInt, m2: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
m1 <= n && m2 <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
!messages.getOrElse((UID(usender), UID(ureceiver)), Nil()).isEmpty &&
intForAll2(m1, m2, onlyOneChannel(n, messages)) &&
ureceiver == increment(usender, n))
if (m1 <= 0 || m2 <= 0)
intForAll2(m1, m2, onlyOneChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
stillOneChannel(n, m1 - 1, m2, messages, usender, ureceiver, tt) &&
stillOneChannel(n, m1, m2 - 1, messages, usender, ureceiver, tt) &&
intForAll2(m1, m2, onlyOneChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
def emptyToOneChannel(
n: BigInt, m1: BigInt, m2: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
m1 <= n && m2 <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
intForAll(n, emptyChannel(n, messages)) &&
ureceiver == increment(usender, n))
if (m1 <= 0 || m2 <= 0)
intForAll2(m1, m2, onlyOneChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
elimForAll(n, emptyChannel(n, messages), m1 - 1) &&
elimForAll(n, emptyChannel(n, messages), m2 - 1) &&
onlyOneChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt))(m1 - 1, m2 - 1) &&
emptyToOneChannel(n, m1 - 1, m2, messages, usender, ureceiver, tt) &&
emptyToOneChannel(n, m1, m2 - 1, messages, usender, ureceiver, tt) &&
intForAll2(m1, m2, onlyOneChannel(n, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
/**
* Invariant stating that if there is an Election message in transit, then
* no one can know the leader
*/
def noLeaderDuringElection(n: BigInt, states: Map[ActorId, State], messages: Map[(ActorId, ActorId), List[Message]]) = {
require(intForAll(n, statesDefined(states)))
(i: BigInt) =>
0 <= i && i < n && elimForAll(n, statesDefined(states), i) &&
(existsMessage(n, messages, isElectionMessage) ==> !isKnowLeaderState(states(UID(i))))
}
def init_noLeaderDuringElection_aux(u: BigInt, n: BigInt): Boolean = {
require(u <= n)
if (u <= 0 || n <= 0)
init_statesDefined(n) &&
intForAll(u, noLeaderDuringElection(n, init_states, init_messages))
else {
n > 0 &&
init_emptyChannel(n) &&
nothingExists(n, init_messages, isElectionMessage) &&
!existsMessage(n, init_messages, isElectionMessage) &&
init_statesDefined(n) &&
noLeaderDuringElection(n, init_states, init_messages)(u - 1) &&
init_noLeaderDuringElection_aux(u - 1, n) &&
intForAll(u, noLeaderDuringElection(n, init_states, init_messages))
}
} holds
def init_noLeaderDuringElection(n: BigInt) = init_noLeaderDuringElection_aux(n, n)
def stillnoLeaderDuringElection(
n: BigInt, u: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt, tt: List[Message]): Boolean = {
require(
u <= n &&
0 <= usender && usender < n &&
0 <= ureceiver && ureceiver < n &&
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeaderDuringElection(n, states, messages)) &&
ureceiver == increment(usender, n) &&
!contains(tt, isElectionMessage)
)
if (u <= 0)
intForAll(u, noLeaderDuringElection(n, states, messages.updated((UID(usender), UID(ureceiver)), tt)))
else if (existsMessage(n, messages, isElectionMessage))
elimForAll(n, noLeaderDuringElection(n, states, messages), u - 1) &&
stillnoLeaderDuringElection(n, u - 1, states, messages, usender, ureceiver, tt) &&
intForAll(u, noLeaderDuringElection(n, states, messages.updated((UID(usender), UID(ureceiver)), tt)))
else
updateChannel(n, messages, tt, isElectionMessage, usender, ureceiver) &&
stillnoLeaderDuringElection(n, u - 1, states, messages, usender, ureceiver, tt) &&
intForAll(u, noLeaderDuringElection(n, states, messages.updated((UID(usender), UID(ureceiver)), tt)))
} holds
/**
* Invariant stating if two people know the leader, they must agree on
* the identity
*/
def onlyOneLeader(n: BigInt, states: Map[ActorId, State]) = {
require(intForAll(n, statesDefined(states)))
(i: BigInt, j: BigInt) => {
0 <= i && i < n && elimForAll(n, statesDefined(states), i) &&
0 <= j && j < n && elimForAll(n, statesDefined(states), j) &&
((states(UID(i)), states(UID(j))) match {
case (KnowLeader(a), KnowLeader(b)) => a == b
case _ => true
})
}
}
/**
* Invariant stating that if someone knows the leader, then there are no
* NonParticipant's
*/
// def everyOneParticipated(n: BigInt, states: Map[ActorId, State], messages: Map[(ActorId, ActorId), List[Message]]) = {
// require(intForAll(n, statesDefined(states)))
//
//
// (i: BigInt, j: BigInt) =>
// 0 <= i && i < n && elimForAll(n, statesDefined(states), i) &&
// 0 <= j && j < n && elimForAll(n, statesDefined(states), j) &&
// (existsMessage(n, messages, Elected(i)) ==> (
// states(UID(i)) == KnowLeader(i) &&
// (states(UID(j)) match {
// case KnowLeader(i2) => i == i2
// case NonParticipant() => false
// case Participant() => true
// })))
// }
//
/**
* Invariant stating that ssn's are unique
*/
def distinctSSN(n: BigInt, getActor: Map[ActorId, Actor]) = {
require(intForAll(n, getActorDefined(getActor)))
(i: BigInt, j: BigInt) =>
0 <= i && i < n && elimForAll(n, getActorDefined(getActor), i) &&
0 <= j && j < n && elimForAll(n, getActorDefined(getActor), j) && {
val Process(id1, ssn1) = getActor(UID(i))
val Process(id2, ssn2) = getActor(UID(j))
ssn1 != ssn2 || i == j
}
}
def distinctSSN_fun(ssns: BigInt => BigInt) = {
(i: BigInt, j: BigInt) => ssns(i) != ssns(j) || i == j
}
def init_distinctSSN_aux(n: BigInt, u: BigInt, v: BigInt, ssns: BigInt => BigInt): Boolean = {
require(u <= n && v <= n && intForAll2(n, n, distinctSSN_fun(ssns)))
if (u <= 0 || v <= 0 || n <= 0)
init_getActorDefined(n, ssns) &&
intForAll2(u, v, distinctSSN(n, init_getActor(ssns)))
else {
init_getActorDefined(n, ssns) &&
init_distinctSSN_aux(n, u - 1, v, ssns) &&
init_distinctSSN_aux(n, u, v - 1, ssns) &&
elimForAll2(n, n, distinctSSN_fun(ssns), u - 1, v - 1) &&
intForAll2(u, v, distinctSSN(n, init_getActor(ssns)))
}
} holds
def init_distinctSSN(n: BigInt, ssns: BigInt => BigInt): Boolean = {
require(intForAll2(n, n, distinctSSN_fun(ssns)))
init_distinctSSN_aux(n, n, n, ssns)
} holds
/**
* Property (not invariant) stating that all channels are empty
*/
def emptyChannel(n: BigInt, messages: Map[(ActorId, ActorId), List[Message]])(i: BigInt) = {
0 <= i && i < n && messages.getOrElse((UID(i), UID(increment(i, n))), Nil()).isEmpty
}
def init_emptyChannel_aux(u: BigInt, n: BigInt): Boolean = {
require(u <= n)
if (u <= 0)
intForAll(u, emptyChannel(n, init_messages))
else
init_emptyChannel_aux(u - 1, n) &&
intForAll(u, emptyChannel(n, init_messages))
} holds
def init_emptyChannel(n: BigInt) = init_emptyChannel_aux(n, n)
def channelsBecomeEmpty(
n: BigInt, m: BigInt, messages: Map[(ActorId, ActorId), List[Message]],
usender: BigInt, ureceiver: BigInt): Boolean = {
require(
intForAll2(n, n, onlyOneChannel(n, messages)) &&
0 <= usender && usender < n &&
ureceiver == increment(usender, n) &&
messages.contains((UID(usender), UID(ureceiver))) &&
!messages((UID(usender), UID(ureceiver))).isEmpty &&
m <= n
)
if (m <= 0)
intForAll(m, emptyChannel(n, messages.updated((UID(usender), UID(ureceiver)), Nil())))
else if (usender == m - 1)
emptyChannel(n, messages.updated((UID(usender), UID(ureceiver)), Nil()))(m - 1) &&
channelsBecomeEmpty(n, m - 1, messages, usender, ureceiver) &&
intForAll(m, emptyChannel(n, messages.updated((UID(usender), UID(ureceiver)), Nil())))
else
elimForAll2(n, n, onlyOneChannel(n, messages), usender, m - 1) &&
emptyChannel(n, messages.updated((UID(usender), UID(ureceiver)), Nil()))(m - 1) &&
channelsBecomeEmpty(n, m - 1, messages, usender, ureceiver) &&
intForAll(m, emptyChannel(n, messages.updated((UID(usender), UID(ureceiver)), Nil())))
} holds
/**
* Property stating that there is no leader (yet)
*/
def noLeader(n: BigInt, states: Map[ActorId, State]) = {
require(intForAll(n, statesDefined(states)))
(i: BigInt) =>
0 <= i && i < n && elimForAll(n, statesDefined(states), i) &&
!isKnowLeaderState(states(UID(i)))
}
def init_noLeader_aux(u: BigInt, n: BigInt): Boolean = {
require(u <= n && init_statesDefined(n))
if (u <= 0)
intForAll(u, noLeader(n, init_states))
else
init_noLeader_aux(u - 1, n) &&
intForAll(u, noLeader(n, init_states))
} holds
def init_noLeader(n: BigInt) = init_noLeader_aux(n, n)
def stillNoLeader_aux(n: BigInt, u: BigInt, states: Map[ActorId, State], id: BigInt, s: State): Boolean = {
require(
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeader(n, states)) &&
!isKnowLeaderState(s) &&
u <= n
)
if (u <= 0 || n <= 0)
statesStillDefined(n, states, UID(id), s) &&
intForAll(u, noLeader(n, states.updated(UID(id), s)))
else
elimForAll(n, noLeader(n, states), u - 1) &&
statesStillDefined(n, states, UID(id), s) &&
noLeader(n, states.updated(UID(id), s))(u - 1) &&
stillNoLeader_aux(n, u - 1, states, id, s) &&
intForAll(u, noLeader(n, states.updated(UID(id), s)))
} holds
def stillNoLeader(n: BigInt, states: Map[ActorId, State], id: BigInt, s: State) = {
require(
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeader(n, states)) &&
!isKnowLeaderState(s)
)
stillNoLeader_aux(n, n, states, id, s)
} holds
def noLeaderImpliesNoLeaderDuringElection_aux(
n: BigInt, u: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]]
): Boolean = {
require(
0 <= n && u <= n &&
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeader(n, states))
)
if (u <= 0 || n <= 0)
intForAll(u, noLeaderDuringElection(n, states, messages))
else
elimForAll(n, noLeader(n, states), u - 1) &&
noLeaderImpliesNoLeaderDuringElection_aux(n, u - 1, states, messages) &&
intForAll(u, noLeaderDuringElection(n, states, messages))
} holds
def noLeaderImpliesNoLeaderDuringElection(
n: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]]
) = {
require(
0 <= n &&
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeader(n, states)))
noLeaderImpliesNoLeaderDuringElection_aux(n, n, states, messages)
}
def noElectionImpliesNoLeaderDuringElection_aux(
n: BigInt, u: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]]
): Boolean = {
require(
0 <= n && u <= n &&
intForAll(n, statesDefined(states)) &&
!existsMessage(n, messages, isElectionMessage)
)
if (u <= 0 || n <= 0)
intForAll(u, noLeaderDuringElection(n, states, messages))
else
noElectionImpliesNoLeaderDuringElection_aux(n, u - 1, states, messages) &&
intForAll(u, noLeaderDuringElection(n, states, messages))
} holds
def noElectionImpliesNoLeaderDuringElection(
n: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]]
): Boolean = {
require(
0 <= n &&
intForAll(n, statesDefined(states)) &&
!existsMessage(n, messages, isElectionMessage)
)
noElectionImpliesNoLeaderDuringElection_aux(n, n, states, messages)
} holds
def electionImpliesNoLeader_aux(
n: BigInt, u: BigInt,
states: Map[ActorId, State],
messages: Map[(ActorId, ActorId), List[Message]]): Boolean = {
require(
n >= 0 && u <= n &&
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeaderDuringElection(n, states, messages)) &&
existsMessage(n, messages, isElectionMessage)
)
if (u <= 0 || n <= 0)
intForAll(u, noLeader(n, states))
else
elimForAll(n, noLeaderDuringElection(n, states, messages), u - 1) &&
electionImpliesNoLeader_aux(n, u - 1, states, messages) &&
intForAll(u, noLeader(n, states))
} holds
def electionImpliesNoLeader(n: BigInt, states: Map[ActorId, State], messages: Map[(ActorId, ActorId), List[Message]]) = {
require(
n >= 0 &&
intForAll(n, statesDefined(states)) &&
intForAll(n, noLeaderDuringElection(n, states, messages)) &&
existsMessage(n, messages, isElectionMessage)
)
electionImpliesNoLeader_aux(n, n, states, messages)
}
/**
* Making initial network
*/
def init_states_fun(id: ActorId): Option[State] = Some(NonParticipant())
def init_getActor_fun(ssns: BigInt => BigInt) = {
require(true)
(id: ActorId) => {
val UID(uid) = id
Some(Process(id, ssns(uid))): Option[Actor]
}
}
def init_messages_fun(ids: (ActorId, ActorId)): Option[List[Message]] = None()
val init_states = Map(init_states_fun)
val init_messages = Map(init_messages_fun)
def init_getActor(ssns: BigInt => BigInt) = {
require(true)
Map(init_getActor_fun(ssns))
}
def makeNetwork(p: Parameter) = {
require {
val Params(n, starterProcess, ssns) = p
validParam(p) &&
init_statesDefined(n) &&
init_getActorDefined(n, ssns) &&
init_ringChannels(n) &&
init_smallChannel(n) &&
init_emptyChannel(n) &&
init_onlyOneChannel(n) &&
init_noLeaderDuringElection(n) &&
init_noLeader(n) &&
init_distinctSSN(n, ssns) &&
init_electingMax(n, starterProcess, ssns) &&
init_electedMax(n, starterProcess, ssns) &&
init_electionParticipants_aux(n, starterProcess) &&
noLeaderImpliesknowTrueLeader(n, starterProcess, init_states, init_getActor(ssns)) &&
intForAll(n, statesDefined(init_states)) &&
intForAll(n, getActorDefined(init_getActor(ssns))) &&
intForAll2(n, n, ringChannels(n, init_messages)) &&
intForAll(n, smallChannel(n, init_messages)) &&
intForAll(n, emptyChannel(n, init_messages)) &&
intForAll2(n, n, onlyOneChannel(n, init_messages)) &&
intForAll(n, noLeaderDuringElection(n, init_states, init_messages)) &&
intForAll(n, noLeader(n, init_states)) &&
intForAll2(n, n, distinctSSN(n, init_getActor(ssns))) &&
intForAll(n, electingMax(n, starterProcess, init_messages, init_getActor(ssns))) &&
intForAll(n, electedMax(n, starterProcess, init_messages, init_getActor(ssns))) &&
intForAll(n, electionParticipants(n, starterProcess, init_states, init_messages)) &&
intForAll(n, knowTrueLeader(n, starterProcess, init_states, init_getActor(ssns)))
}
val Params(n, starterProcess, ssns) = p
VerifiedNetwork(p, init_states, init_messages, init_getActor(ssns))
} ensuring (res => networkInvariant(res.param, res.states, res.messages, res.getActor))
def inBetween(n: BigInt, i: BigInt, j: BigInt, k: BigInt) = {
0 <= i && i < n &&
0 <= j && j < n &&
0 <= k && k < n && (
(i <= k && k <= j) ||
(i > j && k >= i) ||
(i > j && k <= j)
)
}
def incrementBetween(n: BigInt, i: BigInt, j: BigInt) = {
require(0 <= j && j < n && inBetween(n, i, j, increment(j, n)))
increment(j, n) == i
} holds
def maxGreater(n: BigInt, i: BigInt, j: BigInt, k: BigInt, getActor: Map[ActorId, Actor]): Boolean = {
require(
intForAll(n, getActorDefined(getActor)) &&
inBetween(n, i, j, k) && elimForAll(n, getActorDefined(getActor), k))
val Process(_, ssn) = getActor(UID(k))
if (i == j) ssn <= collectMaxSSN(n, i, j, getActor)
else if (k == j) ssn <= collectMaxSSN(n, i, j, getActor)
else
maxGreater(n, i, decrement(j, n), k, getActor) &&
ssn <= collectMaxSSN(n, i, j, getActor)
} holds
def maxExists(n: BigInt, i: BigInt, j: BigInt, getActor: Map[ActorId, Actor]): BigInt = {
require(
0 <= i && i < n &&
0 <= j && j < n &&
intForAll(n, getActorDefined(getActor)) &&
elimForAll(n, getActorDefined(getActor), j)
)
val Process(_, ssn) = getActor(UID(j))
if (i == j) j
else {
val oldval = collectMaxSSN(n, i, decrement(j, n), getActor)
if (ssn > oldval) j
else maxExists(n, i, decrement(j, n), getActor)
}
} ensuring (res => elimForAll(n, getActorDefined(getActor), res) && {
val Process(_, ssn) = getActor(UID(res))
ssn == collectMaxSSN(n, i, j, getActor) &&
inBetween(n, i, j, res)
})
/**
* Invariant stating that when there is an Election(ssn) message in
* the channel from i to increment(i,n), then ssn must the the maximum
* of all ssns of Actors from starterProcess to i included.
* We only look at the head of the channel because there is at most one
* message in each channel.
*/
def electingMax(
n: BigInt,
starterProcess: BigInt,
messages: Map[(ActorId, ActorId), List[Message]],
getActor: Map[ActorId, Actor]) = {
require(
0 <= starterProcess && starterProcess < n &&
intForAll(n, getActorDefined(getActor))
)
(i: BigInt) =>
0 <= i && i < n && (
messages.getOrElse((UID(i), UID(increment(i, n))), Nil()) match {
case Cons(Election(ssn), xs) =>
ssn == collectMaxSSN(n, starterProcess, i, getActor) ||
ssn == collectMaxSSN(n, starterProcess, decrement(starterProcess, n), getActor)
case _ => true
})
}
def init_electingMax_aux(
n: BigInt, u: BigInt, starterProcess: BigInt,
ssns: BigInt => BigInt): Boolean = {
require(
0 <= starterProcess && starterProcess < n && u <= n
)
if (u <= 0 || n <= 0)
init_getActorDefined(n, ssns) &&
intForAll(u, electingMax(n, starterProcess, init_messages, init_getActor(ssns)))
else
init_getActorDefined(n, ssns) &&
init_emptyChannel(n) &&
elimForAll(n, emptyChannel(n, init_messages), u - 1) &&
init_electingMax_aux(n, u - 1, starterProcess, ssns) &&
intForAll(u, electingMax(n, starterProcess, init_messages, init_getActor(ssns)))
} holds
def init_electingMax(
n: BigInt, starterProcess: BigInt,
ssns: BigInt => BigInt): Boolean = {
require(
0 <= starterProcess && starterProcess < n
)
init_electingMax_aux(n, n, starterProcess, ssns)
} holds
def emptyChannelsImplyElectingMax(
n: BigInt, m: BigInt,
starterProcess: BigInt,
messages: Map[(ActorId, ActorId), List[Message]],
getActor: Map[ActorId, Actor]
): Boolean = {
require(
0 <= starterProcess && starterProcess < n &&
m <= n &&
intForAll(n, emptyChannel(n, messages)) &&
intForAll(n, getActorDefined(getActor))
)
if (m <= 0 || n <= 0)
intForAll(m, electingMax(n, starterProcess, messages, getActor))
else
emptyChannelsImplyElectingMax(n, m - 1, starterProcess, messages, getActor) &&
elimForAll(n, emptyChannel(n, messages), m - 1) &&
intForAll(m, electingMax(n, starterProcess, messages, getActor))
} holds
def max(i: BigInt, j: BigInt) = if (i > j) i else j
def collectMaxSSN(n: BigInt, from: BigInt, to: BigInt, getActor: Map[ActorId, Actor]): BigInt = {
require(
0 <= from && from < n &&
0 <= to && to < n &&
intForAll(n, getActorDefined(getActor)) &&
elimForAll(n, getActorDefined(getActor), to)
)
val Process(uid, ssn) = getActor(UID(to))
if (from == to) {
ssn
} else {
max(collectMaxSSN(n, from, decrement(to, n), getActor), ssn)
}
}
def incrDecr(u: BigInt, n: BigInt) = {
require(0 <= u && u < n)
increment(decrement(u, n), n) == u
} holds
def stillMax(
n: BigInt,
starterProcess: BigInt,
usender: BigInt,
getActor: Map[ActorId, Actor]): Boolean = {
require {
0 <= starterProcess && starterProcess < n &&
0 <= usender && usender < n &&
intForAll(n, getActorDefined(getActor)) &&
elimForAll(n, getActorDefined(getActor), increment(usender, n))
}
val Process(_, ssn2) = getActor(UID(increment(usender, n)))
printing(intToString(collectMaxSSN(n, starterProcess, usender, getActor)))
printing(intToString(collectMaxSSN(n, starterProcess, increment(usender, n), getActor)))
if (starterProcess == increment(usender, n)) true
else {
max(collectMaxSSN(n, starterProcess, usender, getActor), ssn2) == collectMaxSSN(n, starterProcess, increment(usender, n), getActor)
}
} holds
def noElectionImpliesElectingMax(
n: BigInt, m: BigInt,
starterProcess: BigInt,
messages: Map[(ActorId, ActorId), List[Message]],
getActor: Map[ActorId, Actor]
): Boolean = {
require(
m <= n &&
0 <= starterProcess && starterProcess < n &&
intForAll(n, getActorDefined(getActor)) &&
!existsMessage(n, messages, isElectionMessage))
if (m <= 0)
intForAll(m, electingMax(n, starterProcess, messages, getActor))
else
noElectionImpliesElectingMax(n, m - 1, starterProcess, messages, getActor) && (
messages.getOrElse((UID(m - 1), UID(increment(m - 1, n))), Nil()) match {
case Cons(Election(_), _) =>
witnessImpliesExists(n, hasMessage(n, messages, isElectionMessage), m - 1) &&
false
case _ => true
}) &&
intForAll(m, electingMax(n, starterProcess, messages, getActor))
} holds
def stillElectingMax(
n: BigInt, m: BigInt,
starterProcess: BigInt,
messages: Map[(ActorId, ActorId), List[Message]],
getActor: Map[ActorId, Actor],
myuid: BigInt,
value: BigInt): Boolean = {
require(
intForAll(n, getActorDefined(getActor)) &&
intForAll(n, emptyChannel(n, messages)) &&
0 <= myuid && myuid < n &&
elimForAll(n, emptyChannel(n, messages), myuid) &&
m <= n &&
0 <= starterProcess && starterProcess < n && (
value == collectMaxSSN(n, starterProcess, decrement(starterProcess, n), getActor) ||
value == collectMaxSSN(n, starterProcess, myuid, getActor)
)
)
if (m <= 0)
intForAll(m, electingMax(n, starterProcess, messages.updated((UID(myuid), UID(increment(myuid, n))), List(Election(value))), getActor))
else
stillElectingMax(n, m - 1, starterProcess, messages, getActor, myuid, value) &&
elimForAll(n, emptyChannel(n, messages), m - 1) &&
intForAll(m, electingMax(n, starterProcess, messages.updated((UID(myuid), UID(increment(myuid, n))), List(Election(value))), getActor))
} holds
/**
* Invariant about participant state
*/
def isParticipant(n: BigInt, states: Map[ActorId, State]) = {
require(intForAll(n, statesDefined(states)))
(i: BigInt) =>
0 <= i && i < n && elimForAll(n, statesDefined(states), i) &&
states(UID(i)) == Participant()
}
def stillParticipating(
n: BigInt,
starterProcess: BigInt,
usender: BigInt,
myuid: BigInt,
states: Map[ActorId, State]
): Boolean = {
require(
0 <= starterProcess && starterProcess < n &&
0 <= usender && usender < n &&
0 <= myuid && myuid < n &&
intForAll(n, statesDefined(states)) &&
forAllModulo(n, starterProcess, usender, isParticipant(n, states))
)
if (starterProcess == usender)
elimForAllModulo(n, starterProcess, usender, isParticipant(n, states), usender) &&
statesStillDefined(n, states, UID(myuid), Participant()) &&
forAllModulo(n, starterProcess, usender, isParticipant(n, states.updated(UID(myuid), Participant())))
else
stillParticipating(n, starterProcess, decrement(usender, n), myuid, states) &&
elimForAllModulo(n, starterProcess, usender, isParticipant(n, states), usender) &&
isParticipant(n, states)(usender) &&
statesStillDefined(n, states, UID(myuid), Participant()) &&
forAllModulo(n, starterProcess, usender, isParticipant(n, states.updated(UID(myuid), Participant())))
} holds
def f(
n: BigInt,
starterProcess: BigInt,
usender: BigInt,
myuid: BigInt,
states: Map[ActorId, State]
): Boolean = {
require(
0 <= starterProcess && starterProcess < n &&
0 <= usender && usender < n &&
0 <= myuid && myuid < n &&
intForAll(n, statesDefined(states)) &&
forAllModulo(n, starterProcess, usender, isParticipant(n, states)) &&
elimForAllModulo(n, starterProcess, usender, isParticipant(n, states), usender) &&
statesStillDefined(n, states, UID(myuid), Participant())
)
stillParticipating(n, starterProcess, usender, myuid, states)
} holds
def oneMoreParticipant(
n: BigInt,
starterProcess: BigInt,
usender: BigInt,
states: Map[ActorId, State]
): Boolean = {
require(
0 <= starterProcess && starterProcess < n &&
0 <= usender && usender < n &&
intForAll(n, statesDefined(states)) &&
// elimForAll(n, statesDefined(states), increment(usender,n)) &&
isParticipant(n, states)(increment(usender, n)) &&
forAllModulo(n, starterProcess, usender, isParticipant(n, states))
)
forAllModulo(n, starterProcess, increment(usender, n), isParticipant(n, states))
} holds