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HPaxos 2.0 live #84

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HPaxos 2.0 live #84

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b127bec
Introduce 2b message type
karbyshev Dec 17, 2024
e68b174
Update message proofs
karbyshev Dec 17, 2024
1fe592c
Add lemma about non-proposal messages
karbyshev Dec 17, 2024
340fc8f
New algorithm formulation
karbyshev Dec 18, 2024
94dafa6
Update relational algorithm spec
karbyshev Dec 19, 2024
4616dc6
Fix proofs of auxiliary lemmas
karbyshev Dec 20, 2024
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21 changes: 9 additions & 12 deletions hpaxos-tla/HMessage.tla
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Original file line number Diff line number Diff line change
Expand Up @@ -29,11 +29,11 @@ FINSUBSET(S, R) == { Range(seq) : seq \in [R -> S] }
NoMessage == [ type |-> "null" ]

MessageRec0 ==
[ type : {"proposer"}, bal : Ballot, prev : {NoMessage}, refs : {{}} ]
[ type : {"1a"}, bal : Ballot, prev : {NoMessage}, refs : {{}} ]

MessageRec1(M, n) ==
M \cup
[ type : {"acceptor"},
[ type : {"1b", "2a", "2b"},
acc : Acceptor,
prev : M \cup {NoMessage},
refs : FINSUBSET(M, RefCardinality),
Expand All @@ -55,18 +55,15 @@ Message == UNION { MessageRec[n] : n \in MessageDepthRange }
-----------------------------------------------------------------------------
(* Message types *)

Proposal(m) == m.type = "proposer"
NonProposal(m) == m.type = "acceptor"
Proposal(m) == m.type = "1a"

OneA(m) == m.type = "proposer"
OneA(m) == m.type = "1a"

OneB(m) ==
/\ m.type = "acceptor"
/\ \E r \in m.refs : OneA(r)
OneB(m) == m.type = "1b"

TwoA(m) ==
/\ m.type = "acceptor"
/\ \A r \in m.refs : ~OneA(r)
TwoA(m) == m.type = "2a"

TwoB(m) == m.type = "2b"

-----------------------------------------------------------------------------
(* Transitive references *)
Expand Down Expand Up @@ -106,5 +103,5 @@ PrevTran(m) == UNION {PrevTranBound[n][m] : n \in PrevTranDepthRange}

=============================================================================
\* Modification History
\* Last modified Fri Nov 22 20:52:10 CET 2024 by karbyshev
\* Last modified Tue Dec 17 17:59:35 CET 2024 by karbyshev
\* Created Tue May 14 16:39:44 CEST 2024 by karbyshev
58 changes: 40 additions & 18 deletions hpaxos-tla/HMessage_proof.tla
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Expand Up @@ -63,7 +63,7 @@ PROOF
<1> DEFINE P(m) == MessageRec[m] # {}
<1> SUFFICES ASSUME NEW j \in Nat PROVE P(j) OBVIOUS
<1>0. P(0)
<2> [type |-> "proposer", bal |-> 0, prev |-> NoMessage, refs |-> {}] \in MessageRec[0]
<2> [type |-> "1a", bal |-> 0, prev |-> NoMessage, refs |-> {}] \in MessageRec[0]
BY MessageRec_eq0 DEF MessageRec0, Ballot
<2> QED OBVIOUS
<1>1. ASSUME NEW m \in Nat, P(m) PROVE P(m+1)
Expand Down Expand Up @@ -104,14 +104,14 @@ LEMMA Message_nontriv == Message # {}
PROOF BY MessageRec_nontriv DEF Message, MessageDepthRange

LEMMA Message_1a_ref ==
\A m \in Message : m.type = "proposer" <=> m.refs = {}
\A m \in Message : OneA(m) <=> m.refs = {}
PROOF
<1> DEFINE P(j) == \A mm \in MessageRec[j] : mm.type = "proposer" <=> mm.refs = {}
<1> SUFFICES ASSUME NEW j \in Nat PROVE P(j) BY DEF Message, MessageDepthRange
<1> DEFINE P(j) == \A mm \in MessageRec[j] : mm.type = "1a" <=> mm.refs = {}
<1> SUFFICES ASSUME NEW j \in Nat PROVE P(j) BY DEF Message, MessageDepthRange, OneA
<1>0. P(0) BY MessageRec_eq0 DEF MessageRec0
<1>1. ASSUME NEW m \in Nat, P(m) PROVE P(m+1)
<2> SUFFICES ASSUME NEW mm \in MessageRec[m+1]
PROVE mm.type = "proposer" <=> mm.refs = {}
PROVE mm.type = "1a" <=> mm.refs = {}
BY DEF Message
<2>3. QED BY <1>1, MessageRec_eq1, MessageRec_nontriv, FinSubset_sub_nontriv,
RefCardinalitySpec DEF MessageRec1
Expand Down Expand Up @@ -194,11 +194,13 @@ PROOF

LEMMA MessageSpec ==
ASSUME NEW m \in Message
PROVE \/ /\ m.type = "proposer"
PROVE \/ /\ m.type = "1a"
/\ m.bal \in Ballot
/\ m.prev = NoMessage
/\ m.refs = {}
\/ /\ m.type = "acceptor"
\/ /\ \/ m.type = "1b"
\/ m.type = "2a"
\/ m.type = "2b"
/\ m.acc \in Acceptor
/\ m.prev \in Message \cup {NoMessage}
/\ m.refs # {}
Expand All @@ -207,11 +209,13 @@ LEMMA MessageSpec ==
PROOF
<1> DEFINE P(n) ==
\A x \in MessageRec[n] :
\/ /\ x.type = "proposer"
\/ /\ x.type = "1a"
/\ x.bal \in Ballot
/\ x.prev = NoMessage
/\ x.refs = {}
\/ /\ x.type = "acceptor"
\/ /\ \/ x.type = "1b"
\/ x.type = "2a"
\/ x.type = "2b"
/\ x.acc \in Acceptor
/\ x.prev \in Message \cup {NoMessage}
/\ x.refs # {}
Expand All @@ -221,11 +225,13 @@ PROOF
<1>0. P(0) BY MessageRec_eq0 DEF MessageRec0
<1>1. ASSUME NEW k \in Nat, P(k) PROVE P(k + 1)
<2> SUFFICES ASSUME NEW x \in MessageRec[k + 1]
PROVE \/ /\ x.type = "proposer"
PROVE \/ /\ x.type = "1a"
/\ x.bal \in Ballot
/\ x.prev = NoMessage
/\ x.refs = {}
\/ /\ x.type = "acceptor"
\/ /\ \/ x.type = "1b"
\/ x.type = "2a"
\/ x.type = "2b"
/\ x.acc \in Acceptor
/\ x.prev \in Message \cup {NoMessage}
/\ x.refs # {}
Expand All @@ -234,13 +240,14 @@ PROOF
OBVIOUS
<2>1. CASE x \in MessageRec[k]
BY <1>1, <2>1
<2>3. CASE x \in [ type : {"acceptor"},
lrns : SUBSET Learner,
<2>3. CASE x \in [ type : {"1b", "2a", "2b"},
acc : Acceptor,
prev : MessageRec[k] \cup {NoMessage},
refs : FINSUBSET(MessageRec[k], RefCardinality) ]
BY <2>3, Message_spec, MessageRec_nontriv, FinSubset_sub,
FinSubset_sub_nontriv, RefCardinalitySpec
refs : FINSUBSET(MessageRec[k], RefCardinality),
lrns : SUBSET Learner ]
BY <2>3, Message_spec, MessageRec_nontriv,
FinSubset_sub, FinSubset_sub_nontriv,
RefCardinalitySpec
<2> QED BY <1>1, <2>1, <2>3, MessageRec_eq1 DEF MessageRec1
<1>2. HIDE DEF P
<1>3. QED BY <1>0, <1>1, NatInduction, Isa
Expand All @@ -250,13 +257,28 @@ LEMMA MessageTypeSpec ==
PROVE \/ /\ OneA(m)
/\ ~OneB(m)
/\ ~TwoA(m)
/\ ~TwoB(m)
\/ /\ ~OneA(m)
/\ OneB(m)
/\ ~TwoA(m)
/\ ~TwoB(m)
\/ /\ ~OneA(m)
/\ ~OneB(m)
/\ TwoA(m)
PROOF BY MessageSpec DEF OneA, OneB, TwoA
/\ ~TwoB(m)
\/ /\ ~OneA(m)
/\ ~OneB(m)
/\ ~TwoA(m)
/\ TwoB(m)
PROOF BY MessageSpec DEF OneA, OneB, TwoA, TwoB

LEMMA MessageNonProposalSpec ==
ASSUME NEW m \in Message,
~Proposal(m)
PROVE \/ OneB(m)
\/ TwoA(m)
\/ TwoB(m)
PROOF BY MessageTypeSpec DEF Proposal, OneA

-----------------------------------------------------------------------------
(* Transitive references *)
Expand Down Expand Up @@ -756,5 +778,5 @@ PROOF BY Message_prev_PrevTranBound1, Zenon

=============================================================================
\* Modification History
\* Last modified Mon Dec 09 16:10:10 CET 2024 by karbyshev
\* Last modified Tue Dec 17 19:23:14 CET 2024 by karbyshev
\* Created Tue May 14 16:44:53 CEST 2024 by karbyshev
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