From 5c6ce3398489d159cc66a6b964c79e3da23204b8 Mon Sep 17 00:00:00 2001 From: Stephan Merz Date: Fri, 28 Feb 2025 15:27:30 +0100 Subject: [PATCH 01/46] fixed proof in Paxos/Consensus.tla Signed-off-by: Stephan Merz --- specifications/Paxos/Consensus.tla | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/specifications/Paxos/Consensus.tla b/specifications/Paxos/Consensus.tla index 25aa9bda..a23c65ae 100644 --- a/specifications/Paxos/Consensus.tla +++ b/specifications/Paxos/Consensus.tla @@ -54,5 +54,16 @@ THEOREM Invariance == Spec => []Inv Success == <>(chosen # {}) LiveSpec == Spec /\ WF_chosen(Next) +ASSUME ValuesNonempty == Value # {} + THEOREM LivenessTheorem == LiveSpec => Success +<1>1. [][Next]_chosen /\ WF_chosen(Next) => [](Init => Success) + <2>1. Init' \/ (chosen # {})' + BY DEF Init + <2>2. Init /\ <>_chosen => (chosen # {})' + BY DEF Init, Next + <2>3. Init => ENABLED <>_chosen + BY ValuesNonempty, ExpandENABLED DEF Init, Next + <2>. QED BY <2>1, <2>2, <2>3, PTL DEF Success +<1>. QED BY <1>1, PTL DEF LiveSpec, Spec ============================================================================= From bb85239192776db9b82d4907b517a1ce7ef335b8 Mon Sep 17 00:00:00 2001 From: Stephan Merz Date: Fri, 7 Mar 2025 16:04:02 +0100 Subject: [PATCH 02/46] fixed proofs of Bakery-Boulangerie Signed-off-by: Stephan Merz --- .github/workflows/CI.yml | 8 +- specifications/Bakery-Boulangerie/Bakery.pdf | Bin 189563 -> 0 bytes specifications/Bakery-Boulangerie/Bakery.tla | 254 ++++++++++------- .../Bakery-Boulangerie/Boulanger.pdf | Bin 179422 -> 0 bytes .../Bakery-Boulangerie/Boulanger.tla | 267 +++++++++++++----- 5 files changed, 343 insertions(+), 186 deletions(-) delete mode 100644 specifications/Bakery-Boulangerie/Bakery.pdf delete mode 100644 specifications/Bakery-Boulangerie/Boulanger.pdf diff --git a/.github/workflows/CI.yml b/.github/workflows/CI.yml index f1c5c047..88711889 100644 --- a/.github/workflows/CI.yml +++ b/.github/workflows/CI.yml @@ -151,10 +151,10 @@ jobs: specifications/LearnProofs/FindHighest.tla specifications/LoopInvariance/BinarySearch.tla # Failing; see https://github.com/tlaplus/Examples/issues/67 - specifications/Bakery-Boulangerie/Bakery.tla - specifications/Bakery-Boulangerie/Boulanger.tla - specifications/Paxos/Consensus.tla - specifications/PaxosHowToWinATuringAward/Consensus.tla + # specifications/Bakery-Boulangerie/Bakery.tla + # specifications/Bakery-Boulangerie/Boulanger.tla + # specifications/Paxos/Consensus.tla + # specifications/PaxosHowToWinATuringAward/Consensus.tla specifications/lamport_mutex/LamportMutex_proofs.tla specifications/byzpaxos/VoteProof.tla # Long-running; see https://github.com/tlaplus/tlapm/issues/85 diff --git a/specifications/Bakery-Boulangerie/Bakery.pdf b/specifications/Bakery-Boulangerie/Bakery.pdf deleted file mode 100644 index e6666c762c4f033d20f3334d90e3c5d4df2adfb7..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 189563 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b/specifications/Bakery-Boulangerie/Bakery.tla index bfbae75b..847c2e5d 100644 --- a/specifications/Bakery-Boulangerie/Bakery.tla +++ b/specifications/Bakery-Boulangerie/Bakery.tla @@ -223,6 +223,47 @@ TypeOK == /\ num \in [Procs -> Nat] /\ pc \in [Procs -> {"ncs", "e1", "e2", "e3", "e4", "w1", "w2", "cs", "exit"}] +THEOREM TypeCorrect == Spec => []TypeOK +<1>. USE N \in Nat DEF ProcSet, Procs +<1>1. Init => TypeOK + BY DEF Init, TypeOK +<1>2. TypeOK /\ [Next]_vars => TypeOK' + <2>. SUFFICES ASSUME TypeOK, [Next]_vars + PROVE TypeOK' + OBVIOUS + <2>1. ASSUME NEW self \in Procs, ncs(self) + PROVE TypeOK' + BY <2>1 DEF TypeOK, ncs + <2>2. ASSUME NEW self \in Procs, e1(self) + PROVE TypeOK' + BY <2>2 DEF TypeOK, e1 + <2>3. ASSUME NEW self \in Procs, e2(self) + PROVE TypeOK' + BY <2>3 DEF TypeOK, e2 + <2>4. ASSUME NEW self \in Procs, e3(self) + PROVE TypeOK' + BY <2>4 DEF TypeOK, e3 + <2>5. ASSUME NEW self \in Procs, e4(self) + PROVE TypeOK' + BY <2>5 DEF TypeOK, e4 + <2>6. ASSUME NEW self \in Procs, w1(self) + PROVE TypeOK' + BY <2>6 DEF TypeOK, w1 + <2>7. ASSUME NEW self \in Procs, w2(self) + PROVE TypeOK' + BY <2>7 DEF TypeOK, w2 + <2>8. ASSUME NEW self \in Procs, cs(self) + PROVE TypeOK' + BY <2>8 DEF TypeOK, cs + <2>9. ASSUME NEW self \in Procs, exit(self) + PROVE TypeOK' + BY <2>9 DEF TypeOK, exit + <2>10. CASE UNCHANGED vars + BY <2>10 DEF TypeOK, vars + <2>. QED + BY <2>1, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7, <2>8, <2>9, <2>10 DEF Next, p +<1>. QED BY <1>1, <1>2, PTL DEF Spec + (***************************************************************************) (* Before(i, j) is a condition that implies that num[i] > 0 and, if j is *) (* trying to enter its critical section and i does not change num[i], then *) @@ -247,21 +288,21 @@ Before(i,j) == /\ num[i] > 0 (***************************************************************************) (* Inv is the complete inductive invariant. *) (***************************************************************************) -Inv == /\ TypeOK - /\ \A i \in Procs : +IInv == \A i \in Procs : \* /\ (pc[i] \in {"ncs", "e1", "e2"}) => (num[i] = 0) - /\ (pc[i] \in {"e4", "w1", "w2", "cs"}) => (num[i] # 0) - /\ (pc[i] \in {"e2", "e3"}) => flag[i] - /\ (pc[i] = "w2") => (nxt[i] # i) - /\ pc[i] \in {(*"e2",*) "w1", "w2"} => i \notin unchecked[i] - /\ (pc[i] \in {"w1", "w2"}) => - \A j \in (Procs \ unchecked[i]) \ {i} : Before(i, j) - /\ /\ (pc[i] = "w2") - /\ \/ (pc[nxt[i]] = "e2") /\ (i \notin unchecked[nxt[i]]) - \/ pc[nxt[i]] = "e3" - => max[nxt[i]] >= num[i] - /\ (pc[i] = "cs") => \A j \in Procs \ {i} : Before(i, j) + /\ (pc[i] \in {"e4", "w1", "w2", "cs"}) => (num[i] # 0) + /\ (pc[i] \in {"e2", "e3"}) => flag[i] + /\ (pc[i] = "w2") => (nxt[i] # i) + /\ pc[i] \in {"w1", "w2"} => i \notin unchecked[i] + /\ (pc[i] \in {"w1", "w2"}) => + \A j \in (Procs \ unchecked[i]) \ {i} : Before(i, j) + /\ /\ (pc[i] = "w2") + /\ \/ (pc[nxt[i]] = "e2") /\ (i \notin unchecked[nxt[i]]) + \/ pc[nxt[i]] = "e3" + => max[nxt[i]] >= num[i] + /\ (pc[i] = "cs") => \A j \in Procs \ {i} : Before(i, j) +Inv == TypeOK /\ IInv ----------------------------------------------------------------------------- (***************************************************************************) (* Proof of Mutual Exclusion *) @@ -274,35 +315,35 @@ Inv == /\ TypeOK THEOREM Spec => []MutualExclusion <1> USE N \in Nat DEFS Procs, TypeOK, Before, \prec, ProcSet <1>1. Init => Inv - BY DEF Init, Inv -<1>2. Inv /\ [Next]_vars => Inv' - <2> SUFFICES ASSUME Inv, + BY DEF Init, Inv, IInv +<1>2. Inv /\ [Next]_vars => IInv' + <2> SUFFICES ASSUME TypeOK, IInv, [Next]_vars - PROVE Inv' - OBVIOUS + PROVE IInv' + BY DEF Inv <2>1. ASSUME NEW self \in Procs, ncs(self) - PROVE Inv' - BY <2>1 DEF ncs, Inv + PROVE IInv' + BY <2>1 DEF ncs, IInv <2>2. ASSUME NEW self \in Procs, e1(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e1" /\ UNCHANGED <> BY <2>2 DEF e1 <3>1. CASE /\ flag' = [flag EXCEPT ![self] = ~ flag[self]] /\ pc' = [pc EXCEPT ![self] = "e1"] /\ UNCHANGED <> - BY <3>1 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ flag' = [flag EXCEPT ![self] = TRUE] /\ unchecked' = [unchecked EXCEPT ![self] = Procs \ {self}] /\ max' = [max EXCEPT ![self] = 0] /\ pc' = [pc EXCEPT ![self] = "e2"] - BY <3>2 DEF Inv + BY <3>2 DEF IInv <3>. QED BY <3>1, <3>2, <2>2 DEF e1 <2>3. ASSUME NEW self \in Procs, e2(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e2" /\ UNCHANGED << num, flag, nxt >> BY <2>3 DEF e2 @@ -311,117 +352,142 @@ THEOREM Spec => []MutualExclusion num[i] > max[self], max' = [max EXCEPT ![self] = num[i]], pc' = [pc EXCEPT ![self] = "e2"] - PROVE Inv' - BY <3>1, Z3T(10) DEF Inv + PROVE IInv' + \* BY <3>1, SMTT(30) DEF IInv -- works + <4>. SUFFICES ASSUME NEW p \in Procs + PROVE IInv!(p)' + BY DEF IInv + <4>1. CASE p = self + BY <3>1, <4>1 DEF IInv + <4>2. CASE p # self + BY <3>1, <4>2 DEF IInv + <4>. QED BY <4>1, <4>2 <3>2. ASSUME NEW i \in unchecked[self], unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {i}], ~(num[i] > max[self]), max' = max, pc' = [pc EXCEPT ![self] = "e2"] - PROVE Inv' - <4>. TypeOK' BY <3>2 DEF Inv - <4>1. \A ii \in Procs : (pc'[ii] \in {"e4", "w1", "w2", "cs"}) => (num'[ii] # 0) - BY <3>2 DEF Inv - <4>2. \A ii \in Procs : (pc'[ii] \in {"e2", "e3"}) => flag'[ii] - BY <3>2 DEF Inv - <4>3. \A ii \in Procs : (pc'[ii] = "w2") => (nxt'[ii] # ii) - BY <3>2 DEF Inv - <4>4. \A ii \in Procs : pc'[ii] \in {(*"e2",*) "w1", "w2"} => ii \notin unchecked'[ii] - BY <3>2 DEF Inv - <4>5. \A ii \in Procs : (pc'[ii] \in {"w1", "w2"}) => - \A j \in (Procs \ unchecked'[ii]) \ {ii} : Before(ii, j)' - BY <3>2 DEF Inv - <4>6. \A ii \in Procs : - /\ (pc'[ii] = "w2") - /\ \/ (pc'[nxt'[ii]] = "e2") /\ (ii \notin unchecked'[nxt'[ii]]) - \/ pc'[nxt'[ii]] = "e3" - => max'[nxt'[ii]] >= num'[ii] - BY <3>2 DEF Inv - <4>7. \A ii \in Procs : (pc'[ii] = "cs") => \A j \in Procs \ {ii} : Before(ii, j)' - BY <3>2 DEF Inv - <4>. QED BY (*<4>0,*) <4>1, <4>2, <4>3, <4>4, <4>5, <4>6, <4>7 DEF Inv + PROVE IInv' + BY <3>2 DEF IInv <3>3. CASE /\ unchecked[self] = {} /\ pc' = [pc EXCEPT ![self] = "e3"] /\ UNCHANGED << unchecked, max >> - BY <3>3 DEF Inv + <4>. SUFFICES ASSUME NEW i \in Procs + PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>3, <4>1 DEF IInv + <4>2. CASE i # self + BY <3>3, <4>2 DEF IInv + <4>. QED BY <4>1, <4>2 <3>. QED BY <3>1, <3>2, <3>3, <2>3 DEF e2 <2>4. ASSUME NEW self \in Procs, e3(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e3" /\ UNCHANGED << flag, unchecked, max, nxt >> BY <2>4 DEF e3 <3>1. CASE /\ \E k \in Nat: num' = [num EXCEPT ![self] = k] /\ pc' = [pc EXCEPT ![self] = "e3"] - BY <3>1 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ \E i \in {j \in Nat : j > max[self]}: num' = [num EXCEPT ![self] = i] /\ pc' = [pc EXCEPT ![self] = "e4"] - BY <3>2, SMTT(60) DEF Inv + BY <3>2 DEF IInv <3>3. QED BY <3>1, <3>2, <2>4 DEF e3 <2>5. ASSUME NEW self \in Procs, e4(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e4" /\ UNCHANGED << num, max, nxt >> BY <2>5 DEF e4 <3>1. CASE /\ flag' = [flag EXCEPT ![self] = ~ flag[self]] /\ pc' = [pc EXCEPT ![self] = "e4"] /\ UNCHANGED unchecked - BY <3>1 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ flag' = [flag EXCEPT ![self] = FALSE] /\ unchecked' = [unchecked EXCEPT ![self] = Procs \ {self}] /\ pc' = [pc EXCEPT ![self] = "w1"] - BY <3>2, Z3T(30) DEF Inv + BY <3>2 DEF IInv <3>. QED BY <3>1, <3>2, <2>5 DEF e4 <2>6. ASSUME NEW self \in Procs, w1(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "w1" /\ UNCHANGED << num, flag, unchecked, max >> BY <2>6 DEF w1 - <3>1. CASE /\ unchecked[self] # {} - /\ \E i \in unchecked[self]: - nxt' = [nxt EXCEPT ![self] = i] - /\ ~ flag[nxt'[self]] - /\ pc' = [pc EXCEPT ![self] = "w2"] - BY <3>1, Z3 DEF Inv + <3>1. ASSUME NEW i \in unchecked[self], + nxt' = [nxt EXCEPT ![self] = i], + ~ flag[nxt'[self]], + pc' = [pc EXCEPT ![self] = "w2"] + PROVE IInv' + \* BY <3>1, SMTT(30) DEF IInv -- works + <4>. SUFFICES ASSUME NEW j \in Procs + PROVE IInv!(j)' + BY DEF IInv + <4>1. CASE j = self + BY <3>1, <4>1 DEF IInv + <4>2. CASE j # self + BY <3>1, <4>2, SMTT(10) DEF IInv + <4>. QED BY <4>1, <4>2 <3>2. CASE /\ unchecked[self] = {} /\ pc' = [pc EXCEPT ![self] = "cs"] /\ nxt' = nxt - BY <3>2, Z3 DEF Inv + <4>. SUFFICES ASSUME NEW j \in Procs + PROVE IInv!(j)' + BY DEF IInv + <4>1. CASE j = self + BY <3>2, <4>1 DEF IInv + <4>2. CASE j # self + BY <3>2, <4>2 DEF IInv + <4>. QED BY <4>1, <4>2 <3>. QED BY <3>1, <3>2, <2>6 DEF w1 <2>7. ASSUME NEW self \in Procs, w2(self) - PROVE Inv' - BY <2>7, Z3T(30) DEF w2, Inv + PROVE IInv' + <3>. /\ pc[self] = "w2" + /\ unchecked' = [unchecked EXCEPT ![self] = @ \ {nxt[self]}] + /\ pc' = [pc EXCEPT ![self] = "w1"] + /\ UNCHANGED << num, flag, max, nxt >> + BY <2>7 DEF w2 + <3>1. CASE num[nxt[self]] = 0 + BY <3>1 DEF IInv + <3>2. CASE num[self] < num[nxt[self]] + BY <3>2 DEF IInv + <3>3. CASE num[self] = num[nxt[self]] /\ self < nxt[self] + BY <3>3 DEF IInv + <3>. QED BY <2>7, <3>1, <3>2, <3>3 DEF w2 <2>8. ASSUME NEW self \in Procs, cs(self) - PROVE Inv' - BY <2>8, Z3 DEF cs, Inv + PROVE IInv' + \* BY <2>8, SMTT(30) DEF cs, IInv -works + <3>1. \A i,j \in Procs: Before(i,j) => Before(i,j)' + BY <2>8 DEF cs + <3>. HIDE DEF Before + <3>. QED BY <2>8, <3>1 DEF cs, IInv <2>9. ASSUME NEW self \in Procs, exit(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "exit" /\ UNCHANGED << flag, unchecked, max, nxt >> BY <2>9 DEF exit <3>1. CASE /\ \E k \in Nat: num' = [num EXCEPT ![self] = k] /\ pc' = [pc EXCEPT ![self] = "exit"] - BY <3>1 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ num' = [num EXCEPT ![self] = 0] /\ pc' = [pc EXCEPT ![self] = "ncs"] - BY <3>2 DEF Inv + BY <3>2 DEF IInv <3>. QED BY <3>1, <3>2, <2>9 DEF exit <2>10. CASE UNCHANGED vars - BY <2>10 DEF vars, Inv + BY <2>10 DEF vars, IInv <2>11. QED BY <2>1, <2>10, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7, <2>8, <2>9 DEF Next, p <1>3. Inv => MutualExclusion - BY SMT DEF MutualExclusion, Inv + BY SMT DEF MutualExclusion, Inv, IInv <1>4. QED - BY <1>1, <1>2, <1>3, PTL DEF Spec + BY <1>1, <1>2, <1>3, TypeCorrect, PTL DEF Spec, Inv ------------------------------------------------------------------------------ Trying(i) == pc[i] = "e1" InCS(i) == pc[i] = "cs" @@ -429,45 +495,15 @@ DeadlockFree == (\E i \in Procs : Trying(i)) ~> (\E i \in Procs : InCS(i)) StarvationFree == \A i \in Procs : Trying(i) ~> InCS(i) ----------------------------------------------------------------------------- -II == \A i \in Procs : -\* /\ (pc[i] \in {"ncs", "e1", "e2"}) => (num[i] = 0) \* not found Test 1 (21993 states) - /\ (pc[i] \in {"e4", "w1", "w2", "cs"}) => (num[i] # 0) \* found Test 1 - /\ (pc[i] \in {"e2", "e3"}) => flag[i] \* found Test 1 - /\ (pc[i] = "w2") => (nxt[i] # i) \* not found Test 1 (12115 states) or with N=2 - /\ pc[i] \in {"e2", "w1", "w2"} => i \notin unchecked[i] \* found Test 1 - /\ (pc[i] \in {"w1", "w2"}) => \* found Test 1 - \A j \in (Procs \ unchecked[i]) \ {i} : Before(i, j) - /\ /\ (pc[i] = "w2") \* found Test 1 - /\ \/ (pc[nxt[i]] = "e2") /\ (i \notin unchecked[nxt[i]]) - \/ pc[nxt[i]] = "e3" - => max[nxt[i]] >= num[i] - /\ (pc[i] = "cs") => \A j \in Procs \ {i} : Before(i, j) \* found Test 1 - -IInit == /\ num \in [Procs -> Nat] - /\ flag \in [Procs -> BOOLEAN] - /\ unchecked \in [Procs -> SUBSET Procs] - /\ max \in [Procs -> Nat] - /\ nxt \in [Procs -> Procs] - /\ pc \in [Procs -> {"ncs", "e1", "e2", "e3", - "e4", "w1", "w2", "cs", "exit"}] - /\ II - -ISpec == IInit /\ [][Next]_vars +(***************************************************************************) +(* The following spec can be used to check inductiveness of the invariant *) +(* with the help of TLC. *) +(***************************************************************************) +ISpec == Inv /\ [][Next]_vars ============================================================================= \* Modification History -\* Last modified Mon Mar 06 13:47:10 CET 2023 by merz -\* Last modified Tue Aug 27 12:23:10 PDT 2019 by loki -\* Last modified Sat May 19 16:40:23 CEST 2018 by merz -\* Last modified Thu May 17 07:02:45 PDT 2018 by lamport \* Created Thu Nov 21 15:54:32 PST 2013 by lamport Test 1: 5248 distinct initial states 151056 full initial states -IInit == /\ num \in [Procs -> Nat] - /\ flag \in [Procs -> BOOLEAN] - /\ unchecked \in [Procs -> SUBSET Procs] - /\ max \in [Procs -> {0}] \* Nat] - /\ nxt \in [Procs -> {1}] - /\ pc \in [Procs -> {"ncs", "e1", "e2", "e3", - "e4", "w1", "w2", "cs"}] - /\ II +IInit == TypeOK /\ IInv diff --git a/specifications/Bakery-Boulangerie/Boulanger.pdf b/specifications/Bakery-Boulangerie/Boulanger.pdf deleted file mode 100644 index 87005b03b43ed3f5a26843c74fa2daa5ac35c0a1..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 179422 zcmb5VQ0IRpWMs`XlZRA6M2wb^ zjunb@VP$j!ikX0cz~0CTiiZb^UdGhU+{J=`g`I_+;Qw7v^kSAaE~ZWd^kO!KE~X-; z#`Y$rP<(t)&Mr=-hPF^1TfN%a_B#`)esA>_*V0-M*@lgDBi&P?U0Z^?`0AEL(?%}>v~Ty#l9}k$ZZ8k3`gpMh4k1j6p=hAKYYv8Oq94ahnHrlCzLf}!v>y7mfc-jwZDCP?%l@P`rpTw@e5qJ zEo_!Xu5iPO&F#C5wQ%jO@wECs4y>9*VXdiC-_+JP;EbixQnh}Zl}jZJ67JvMhl+MW zV!5QAd9^xeq;Thu!^WX*lqS)sVe#58FPEn-BjVv-OZ3V6HBK7VRS5-BS0wJEQ<%ve zMFw4ZyR5%P#nV2xeW>}Nb7Z=#*BsI24~4v+chzchHe8^$OtV7WpA9#3Y?_^WtmxRS zd^<-VcDuXxBE!1945$hu?&afqE6~fCM2cf!ZT4H+b@pN1yzubLIdOKXNlPEf@%_3e zn_XA*HM;5x)BjFB<#X$QhE&#o>__8P4ci$yfKH`!EuQjM7gTu8!>!&|aMIWm$mFsY zj?IaFtcq=+l258(qw*pq&!DyIs{1P8(7ANs8SJ!>NFSn8LPQ$kT;<;M0S#r9H*PEI2sntk+h5cx%h*KE1|8Wtx=X^wZ**oj8=y1yO7>kv;Jv zjZTqL&-&%i(oO0moVuN;WR;3D0q(5#*`$d8xQZG~52D^?^4{JrJ`E)=1?r|1VFdow zX#I8i4TyC8ZnTaA!i7hpmktR!<=pE&EtB&%sL;%U&18;m`W<`j+;tDXZoIh~$kTi+ zxVQVZ?z2;Tbf8dlfZ0LuzIxmE@UUz{p`T}drLs!G6|tT8DPayEPE!#xOQxkB|2F&l z{N$sl13j7Ab_SdWN0H8B;h_K zOcTBsSGnwZ&+R?$<|&teFbY!nJOQSnoa-)F+W2-j6}d8OOE|hG%=zAk!rOu?n2(*m ztcn6wrBw&^8*Ts^O!&;v#z5NaFhK|FE{U=&yWx!iMP?z-qaF$a^T;nEPl1fZ(h>;IrlC-4ji zAL2kADqNSq0!|*vVOr(`*v+o5sdP%jl3B2Dnd&LMz=Zd8y|*0&(=zAG?z)zAxkfVH ziGG)jy#azD4})ftH3u=aWHQ0|{^TyBEUO2OzpT+9#sk2#w!4$~0~9?6snWwDE`8{} zkr0fM!y2o-XHmi|xK^8r%xqQ~fLth-uz~zUUPw}odJ7=OXsJICRBmG-pqfk}&?&*g5a(Cn#m6$i6X15Rnp|{g z_}Als=wA$x>&TLe{OTrupPT)FdorB<0fJ;N4j%Tnm=E+CkQ*gClt+LIeiU2J^H#`h z6Osv-kjaU0F5&89*G0?({a#?;n&1mHDjk4iNI~Hl64+3YXy-miWl)y*>ieUiV$qRh z?dz)Stf;@w=0`vfikkX1?rVXkvRz}na#LTvr^9gUK8t4FBRn*)S>$7A!~|`oICzwv zHH_n4sM?y0=3=n|a#~f{0H8ML>zHMO(u$LC=}qF*9109*6F =6UTvj5~y60?H>t zX(Hrbx<|`VvEG4C5HyQIh4w9i)+L5v!dyV4&co1#vE3qWDez{K`&0K)_-~8-Q=Z#eX%nUS&(D#;~ z>GXj4CZR9IU-42|t$p?DJ_kPg1*bfe1~O;MTEQui(IDzw2a^+O7T&MN;Hi~wG|FgS zi(eh7;X)uNVPm!La`@p9mV!EWw7I-AfSwkIEE)~&24{8yC|a|`#inHg%lQPvM)eU9 zJh>=zm%!)O`|a4b~@sPy3d5ea;F=ku}C@KoFCkHfV!zuoaO zi*Ike#(j2e{x%)YW0Z4 z3laPbC$EebYK2Huwh%1LX^^nbr0RkgSVSFM z9t&QO$~fbNf^iF0Dxy5{3I$trhsAcIA+mlBG$kJHRzUVDOyjpoZLlAY577r(z&OWL zEy^1xgn*nhz3JC2N1ga24<#O?XjA!5q|~{p`8grA^TsIWP@!QKWyR3zGJ;iBvH;NQ zf*na<79+>0IZRx0F5dMS#?E?ZbdR=7=xA~0q;k@twB2}HPuT;%sJBkUIykHZ4FSJM z^seLRaR6dk-M5h$_1FG?jtTS);Hf+u%KBsiwO5zuZHE#B=Yyy4wHfX6*oA;AcH z%W4#j((8Xh1DPbN`!i~m;8mR05A`%Ao_

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USE N \in Nat DEF Procs, ProcSet +<1>1. Init => TypeOK + BY DEF Init, TypeOK +<1>2. TypeOK /\ [Next]_vars => TypeOK' + <2>. SUFFICES ASSUME TypeOK, [Next]_vars + PROVE TypeOK' + OBVIOUS + <2>1. ASSUME NEW self \in Procs, ncs(self) + PROVE TypeOK' + BY <2>1 DEF TypeOK, ncs + <2>2. ASSUME NEW self \in Procs, e1(self) + PROVE TypeOK' + BY <2>2 DEF TypeOK, e1 + <2>3. ASSUME NEW self \in Procs, e2(self) + PROVE TypeOK' + BY <2>3 DEF TypeOK, e2 + <2>4. ASSUME NEW self \in Procs, e3(self) + PROVE TypeOK' + BY <2>4 DEF TypeOK, e3 + <2>5. ASSUME NEW self \in Procs, e4(self) + PROVE TypeOK' + BY <2>5 DEF TypeOK, e4 + <2>6. ASSUME NEW self \in Procs, w1(self) + PROVE TypeOK' + BY <2>6 DEF TypeOK, w1 + <2>7. ASSUME NEW self \in Procs, w2(self) + PROVE TypeOK' + BY <2>7 DEF TypeOK, w2 + <2>8. ASSUME NEW self \in Procs, cs(self) + PROVE TypeOK' + BY <2>8 DEF TypeOK, cs + <2>9. ASSUME NEW self \in Procs, exit(self) + PROVE TypeOK' + BY <2>9 DEF TypeOK, exit + <2>10. CASE UNCHANGED vars + BY <2>10 DEF TypeOK, vars + <2>. QED + BY <2>1, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7, <2>8, <2>9, <2>10 DEF Next, p +<1>. QED BY <1>1, <1>2, PTL DEF Spec + (***************************************************************************) (* Before(i, j) is a condition that implies that num[i] > 0 and, if j is *) (* trying to enter its critical section and i does not change num[i], then *) @@ -327,8 +368,7 @@ Before(i,j) == /\ num[i] > 0 (***************************************************************************) (* Inv is the complete inductive invariant. *) (***************************************************************************) -Inv == /\ TypeOK - /\ \A i \in Procs : +IInv == \A i \in Procs : /\ (pc[i] \in {"ncs", "e1", "e2"}) => (num[i] = 0) /\ (pc[i] \in {"e4", "w1", "w2", "cs"}) => (num[i] # 0) /\ (pc[i] \in {"e2", "e3"}) => flag[i] @@ -346,6 +386,8 @@ Inv == /\ TypeOK /\ pc[nxt[i]] \in {"e4", "w1", "w2", "cs"} => Before(i, nxt[i]) /\ (pc[i] = "cs") => \A j \in Procs \ {i} : Before(i, j) + +Inv == TypeOK /\ IInv ----------------------------------------------------------------------------- (***************************************************************************) (* Proof of Mutual Exclusion *) @@ -360,36 +402,46 @@ THEOREM Spec => []MutualExclusion <1> USE N \in Nat DEFS Procs, TypeOK, Before, \prec, ProcSet <1>1. Init => Inv - BY SMT DEF Init, Inv - -<1>2. Inv /\ [Next]_vars => Inv' - <2> SUFFICES ASSUME Inv, + BY DEF Init, Inv, IInv + +<1>2. Inv /\ [Next]_vars => IInv' + <2> SUFFICES ASSUME TypeOK, IInv, [Next]_vars - PROVE Inv' - OBVIOUS + PROVE IInv' + BY DEF Inv <2>1. ASSUME NEW self \in Procs, ncs(self) - PROVE Inv' - BY <2>1, Z3 DEF ncs, Inv + PROVE IInv' + <3>. /\ pc[self] = "ncs" + /\ pc' = [pc EXCEPT ![self] = "e1"] + /\ UNCHANGED << num, flag, unchecked, max, nxt, previous >> + BY <2>1 DEF ncs + <3>. SUFFICES ASSUME NEW i \in Procs PROVE IInv!(i)' + BY DEF IInv + <3>1. CASE i = self + BY <3>1 DEF IInv + <3>2. CASE i # self + BY <3>2, Zenon DEF IInv + <3>. QED BY <3>1, <3>2 <2>2. ASSUME NEW self \in Procs, e1(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e1" /\ UNCHANGED << num, nxt, previous >> BY <2>2 DEF e1 <3>1. CASE /\ flag' = [flag EXCEPT ![self] = ~ flag[self]] /\ pc' = [pc EXCEPT ![self] = "e1"] /\ UNCHANGED <> - BY <3>1 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ flag' = [flag EXCEPT ![self] = TRUE] /\ unchecked' = [unchecked EXCEPT ![self] = Procs \ {self}] /\ max' = [max EXCEPT ![self] = 0] /\ pc' = [pc EXCEPT ![self] = "e2"] - BY <3>2 DEF Inv + BY <3>2 DEF IInv <3>. QED BY <3>1, <3>2, <2>2 DEF e1 <2>3. ASSUME NEW self \in Procs, e2(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e2" /\ UNCHANGED << num, flag, nxt, previous >> BY <2>3 DEF e2 @@ -398,75 +450,126 @@ THEOREM Spec => []MutualExclusion num[i] > max[self], max' = [max EXCEPT ![self] = num[i]], pc' = [pc EXCEPT ![self] = "e2"] - PROVE Inv' - BY <3>1, Z3 DEF Inv + PROVE IInv' + BY <3>1 DEF IInv <3>2. ASSUME NEW i \in unchecked[self], unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {i}], ~(num[i] > max[self]), max' = max, pc' = [pc EXCEPT ![self] = "e2"] - PROVE Inv' - BY <3>2, Z3 DEF Inv + PROVE IInv' + \* BY <3>2, SMTT(30) DEF IInv -works + <4>. SUFFICES ASSUME NEW j \in Procs, IInv!(j) PROVE IInv!(j)' + BY DEF IInv + <4>1. CASE j = self + BY <3>2, <4>1 + <4>2. CASE j # self + BY <3>2, <4>2 + <4>. QED BY <4>1, <4>2 <3>3. CASE /\ unchecked[self] = {} /\ pc' = [pc EXCEPT ![self] = "e3"] /\ UNCHANGED << unchecked, max >> - BY <3>3, Z3 DEF Inv + <4>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>3, <4>1 + <4>2. CASE i # self + BY <3>3, <4>2 + <4>. QED BY <4>1, <4>2 <3>. QED BY <3>1, <3>2, <3>3, <2>3 DEF e2 <2>4. ASSUME NEW self \in Procs, e3(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e3" /\ UNCHANGED << flag, unchecked, max, nxt, previous >> BY <2>4 DEF e3 <3>1. CASE /\ \E k \in Nat: num' = [num EXCEPT ![self] = k] /\ pc' = [pc EXCEPT ![self] = "e3"] - BY <3>1, Z3 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ num' = [num EXCEPT ![self] = max[self] + 1] /\ pc' = [pc EXCEPT ![self] = "e4"] - BY <3>2, Z3 DEF Inv + BY <3>2 DEF IInv <3>. QED BY <3>1, <3>2, <2>4 DEF e3 <2>5. ASSUME NEW self \in Procs, e4(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "e4" /\ UNCHANGED << num, max, nxt, previous >> BY <2>5 DEF e4 <3>1. CASE /\ flag' = [flag EXCEPT ![self] = ~ flag[self]] /\ pc' = [pc EXCEPT ![self] = "e4"] /\ UNCHANGED unchecked - BY <3>1, Z3 DEF Inv + BY <3>1 DEF IInv <3>2. CASE /\ flag' = [flag EXCEPT ![self] = FALSE] /\ num[self] = 1 /\ unchecked' = [unchecked EXCEPT ![self] = 1..(self-1)] /\ pc' = [pc EXCEPT ![self] = "w1"] - BY <3>2, Z3 DEF Inv + \* BY <3>2, SMTT(30) DEF IInv -- works + <4>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>2, <4>1 + <4>2. CASE i # self + \* BY <3>2, <4>2, SMTT(30) -- works + <5>1. \A k,l \in Procs : Before(k,l) => Before(k,l)' + BY <3>2 + <5>2. Before(i, nxt[i]) => Before(i, nxt[i])' + BY <5>1 + <5>. HIDE DEF Before + <5>. QED BY <3>2, <4>2, <5>1, <5>2 + <4>. QED BY <4>1, <4>2 <3>3. CASE /\ flag' = [flag EXCEPT ![self] = FALSE] /\ num[self] # 1 /\ unchecked' = [unchecked EXCEPT ![self] = Procs \ {self}] /\ pc' = [pc EXCEPT ![self] = "w1"] - BY <3>3, Z3 DEF Inv + \* BY <3>3, SMTT(30) DEF IInv -- works + <4>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>3, <4>1 + <4>2. CASE i # self + BY <3>3, <4>2 + <4>. QED BY <4>1, <4>2 <3>. QED BY <3>1, <3>2, <3>3, <2>5 DEF e4 <2>6. ASSUME NEW self \in Procs, w1(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "w1" /\ UNCHANGED << num, flag, unchecked, max >> BY <2>6 DEF w1 - <3>1. CASE /\ unchecked[self] # {} - /\ \E i \in unchecked[self]: - nxt' = [nxt EXCEPT ![self] = i] - /\ ~ flag[nxt'[self]] - /\ previous' = [previous EXCEPT ![self] = -1] - /\ pc' = [pc EXCEPT ![self] = "w2"] - BY <3>1, Z3 DEF Inv + <3>1. ASSUME NEW j \in unchecked[self], + nxt' = [nxt EXCEPT ![self] = j], + ~ flag[j], + previous' = [previous EXCEPT ![self] = -1], + pc' = [pc EXCEPT ![self] = "w2"] + PROVE IInv' + <4>. pc[j] \notin {"e2", "e3"} + BY <3>1 DEF IInv + <4>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>1, <4>1 + <4>2. CASE i # self + BY <3>1, <4>2 + <4>. QED BY <3>1 <3>2. CASE /\ unchecked[self] = {} /\ pc' = [pc EXCEPT ![self] = "cs"] /\ UNCHANGED << nxt, previous >> - BY <3>2, Z3 DEF Inv + \* BY <3>2, SMTT(30) DEF IInv -- works + <4>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <4>1. CASE i = self + BY <3>2, <4>1 + <4>2. CASE i # self + <5>1. \A j \in Procs : Before(i,j) => Before(i,j)' + BY <3>2, <4>2 + <5>. HIDE DEF Before + <5>. QED BY <3>2, <4>2, <5>1 + <4>. QED BY <4>1, <4>2 <3>. QED BY <3>1, <3>2, <2>6 DEF w1 <2>7. ASSUME NEW self \in Procs, w2(self) - PROVE Inv' + PROVE IInv' <3>. /\ pc[self] = "w2" /\ UNCHANGED << num, flag, max, nxt >> BY <2>7 DEF w2 @@ -475,64 +578,82 @@ THEOREM Spec => []MutualExclusion \/ /\ previous[self] # -1 /\ num[nxt[self]] # previous[self] /\ unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {nxt[self]}] - /\ unchecked'[self] = {} - /\ pc' = [pc EXCEPT ![self] = "cs"] /\ UNCHANGED previous - BY <3>1, Z3 DEF Inv - <3>2. CASE /\ \/ num[nxt[self]] = 0 - \/ <> \prec <> - \/ /\ previous[self] # -1 - /\ num[nxt[self]] # previous[self] - /\ unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {nxt[self]}] - /\ unchecked'[self] # {} - /\ pc' = [pc EXCEPT ![self] = "w1"] - /\ UNCHANGED previous - BY <3>2, Z3 DEF Inv - <3>3. CASE /\ ~ \/ num[nxt[self]] = 0 + <4>1. Before(self, nxt[self]) + <5>0. /\ IInv!(self) /\ IInv!(nxt[self]) + /\ num[self] > 0 + BY DEF IInv + <5>. HIDE DEF Before + <5>1. pc[nxt[self]] \in {"e4", "w1", "w2", "cs"} + => \/ <> \prec <> + \/ Before(self, nxt[self]) + BY <3>1 DEF IInv + <5>2. ASSUME pc[nxt[self]] \in {"w1", "w2", "cs"}, + Before(nxt[self], self) + PROVE FALSE + <6>1. /\ num[nxt[self]] > 0 + /\ <> \prec <> + BY <5>0, <5>2 DEF Before + <6>2. Before(self, nxt[self]) + BY <3>1, <5>1, <5>2, <6>1 DEF IInv + <6>. QED BY <5>2, <6>2 DEF Before + <5>3. /\ pc[nxt[self]] \in {"w1", "w2"} => self \in unchecked[nxt[self]] + /\ pc[nxt[self]] # "cs" + BY <5>0, <5>2 + <5>. QED BY <5>0, <5>1, <5>3 DEF Before + <4>2. CASE /\ unchecked'[self] = {} + /\ pc' = [pc EXCEPT ![self] = "cs"] + <5>1. unchecked[self] \subseteq {nxt[self]} + BY <3>1, <4>2, Zenon + <5>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <5>. QED BY <3>1, <4>1, <4>2, <5>1 + <4>3. CASE /\ unchecked'[self] # {} + /\ pc' = [pc EXCEPT ![self] = "w1"] + <5>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <5>. QED BY <3>1, <4>1, <4>3 + <4>. QED BY <2>7, <3>1, <4>2, <4>3 DEF w2 + <3>2. CASE /\ ~ \/ num[nxt[self]] = 0 \/ <> \prec <> \/ /\ previous[self] # -1 /\ num[nxt[self]] # previous[self] /\ previous' = [previous EXCEPT ![self] = num[nxt[self]]] /\ pc' = [pc EXCEPT ![self] = "w2"] /\ UNCHANGED unchecked - BY <3>3, Z3 DEF Inv - <3>. QED BY <3>1, <3>2, <3>3, <2>7 DEF w2 + BY <3>2 DEF IInv + <3>. QED BY <3>1, <3>2, <2>7 DEF w2 <2>8. ASSUME NEW self \in Procs, cs(self) - PROVE Inv' - BY <2>8, Z3 DEF cs, Inv + PROVE IInv' + \* BY <2>8, SMTT(30) DEF cs, IInv -- works + <3>1. \A i,j \in Procs : Before(i,j) => Before(i,j)' + BY <2>8 DEF cs + <3>. HIDE DEF Before + <3>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <3>. QED BY <2>8, <3>1 DEF cs <2>9. ASSUME NEW self \in Procs, exit(self) - PROVE Inv' - <3>. /\ pc[self] = "exit" - /\ UNCHANGED << flag, unchecked, max, nxt, previous >> - BY <2>9 DEF exit - <3>1. CASE /\ \E k \in Nat: num' = [num EXCEPT ![self] = k] - /\ pc' = [pc EXCEPT ![self] = "exit"] - BY <3>1, Z3 DEF Inv - <3>2. CASE /\ num' = [num EXCEPT ![self] = 0] - /\ pc' = [pc EXCEPT ![self] = "ncs"] - BY <3>2, Z3 DEF Inv - <3>. QED BY <3>1, <3>2, <2>9 DEF exit + PROVE IInv' + \* BY <2>9, SMTT(30) DEF exit, IInv --works + <3>. SUFFICES ASSUME NEW i \in Procs, IInv!(i) PROVE IInv!(i)' + BY DEF IInv + <3>. QED BY <2>9 DEF exit <2>10. CASE UNCHANGED vars - BY <2>10, Z3 DEF vars, Inv + BY <2>10 DEF vars, IInv <2>11. QED - BY <2>1, <2>10, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7, <2>8, <2>9 DEF Next, p + BY <2>1, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7, <2>8, <2>9, <2>10 DEF Next, p <1>3. Inv => MutualExclusion - BY SMT DEF Inv, MutualExclusion + BY DEF Inv, IInv, MutualExclusion <1>4. QED - BY <1>1, <1>2, <1>3, PTL DEF Spec + BY <1>1, <1>2, <1>3, TypeCorrect, PTL DEF Spec, Inv ------------------------------------------------------------------------------ Trying(i) == pc[i] = "e1" InCS(i) == pc[i] = "cs" DeadlockFree == (\E i \in Procs : Trying(i)) ~> (\E i \in Procs : InCS(i)) StarvationFree == \A i \in Procs : Trying(i) ~> InCS(i) ============================================================================= -\* Modification History -\* Last modified Tue Aug 27 12:22:38 PDT 2019 by loki -\* Last modified Thu May 24 20:03:58 CEST 2018 by merz -\* Last modified Thu May 24 13:49:22 CEST 2018 by merz -\* Last modified Tue Jul 21 17:55:23 PDT 2015 by lamport \* Created Thu Nov 21 15:54:32 PST 2013 by lamport From f3feb888f0341dcf99d721e957ee3b80e6f17c57 Mon Sep 17 00:00:00 2001 From: Stephan Merz Date: Sat, 22 Mar 2025 14:30:59 +0100 Subject: [PATCH 03/46] fixed proofs in lamport_mutex Signed-off-by: Stephan Merz --- .../lamport_mutex/LamportMutex_proofs.tla | 140 ++++++++++++------ 1 file changed, 93 insertions(+), 47 deletions(-) diff --git a/specifications/lamport_mutex/LamportMutex_proofs.tla b/specifications/lamport_mutex/LamportMutex_proofs.tla index ef9157ef..11994e79 100644 --- a/specifications/lamport_mutex/LamportMutex_proofs.tla +++ b/specifications/lamport_mutex/LamportMutex_proofs.tla @@ -59,15 +59,7 @@ LEMMA TypeCorrect == Spec => []TypeOK /\ req' \in [Proc -> [Proc -> Nat]] BY <3>1, <3>3 DEF ReqMessage <3>5. network' \in [Proc -> [Proc -> Seq(Message)]] - <4>. DEFINE nw == [network EXCEPT ![q][p] = Tail(@)] - <4>1. nw \in [Proc -> [Proc -> Seq(Message)]] - BY <3>1 - <4>. HIDE DEF nw - <4>2. AckMessage \in Message - BY DEF Message - <4>3. [nw EXCEPT ![p][q] = Append(@, AckMessage)] \in [Proc -> [Proc -> Seq(Message)]] - BY <4>1, <4>2 - <4>. QED BY <3>1, <4>3, Zenon DEF nw + BY <3>1 DEF Message <3>6. /\ ack' \in [Proc -> SUBSET Proc] /\ crit' \in SUBSET Proc BY <3>1 @@ -81,7 +73,13 @@ LEMMA TypeCorrect == Spec => []TypeOK NEW q \in Proc \ {p}, ReceiveRelease(p,q) PROVE TypeOK' - BY <2>6 DEF ReceiveRelease + \* works -- BY <2>6, SMTT(30) DEF ReceiveRelease + <3>1. /\ network[q][p] # << >> + /\ req' = [req EXCEPT ![p][q] = 0] + /\ network' = [network EXCEPT ![q][p] = Tail(@)] + /\ UNCHANGED <> + BY <2>6 DEF ReceiveRelease + <3>. QED BY <3>1 <2>7. CASE UNCHANGED vars BY <2>7 DEF vars <2>8. QED BY <2>1, <2>2, <2>3, <2>4, <2>5, <2>6, <2>7 DEF Next @@ -140,10 +138,7 @@ LEMMA AtMostOneHead == ASSUME NEW s \in Seq(Message), NEW mtype, AtMostOne(s,mtype), s # << >>, Head(s).type = mtype PROVE ~ Contains(Tail(s), mtype) -<1>. SUFFICES ASSUME NEW i \in 1 .. Len(Tail(s)), Tail(s)[i].type = mtype - PROVE FALSE - BY Tail(s) \in Seq(Message), Isa DEF Contains -<1>. QED BY HeadTailProperties DEF AtMostOne +BY DEF AtMostOne, Contains LEMMA ContainsSend == ASSUME NEW s \in Seq(Message), NEW mtype, NEW m \in Message @@ -169,9 +164,20 @@ LEMMA PrecedesSend == BY DEF Precedes LEMMA PrecedesTail == - ASSUME NEW s \in Seq(Message), NEW mt1, NEW mt2, Precedes(s, mt1, mt2) + ASSUME NEW s \in Seq(Message), s # << >>, + NEW mt1, NEW mt2, Precedes(s, mt1, mt2) PROVE Precedes(Tail(s), mt1, mt2) -BY DEF Precedes +<1>. SUFFICES + ASSUME NEW i \in 1 .. Len(Tail(s)), NEW j \in 1 .. Len(Tail(s)), + Tail(s)[i].type = mt1, Tail(s)[j].type = mt2 + PROVE i < j + BY DEF Precedes +<1>1. /\ i+1 \in 1 .. Len(s) /\ s[i+1].type = mt1 + /\ j+1 \in 1 .. Len(s) /\ s[j+1].type = mt2 + OBVIOUS +<1>2. i+1 < j+1 + BY <1>1 DEF Precedes +<1>. QED BY <1>2 LEMMA PrecedesInTail == ASSUME NEW s \in Seq(Message), s # << >>, @@ -179,7 +185,21 @@ LEMMA PrecedesInTail == Head(s).type = mt1 \/ Head(s).type \notin {mt1, mt2}, Precedes(Tail(s), mt1, mt2) PROVE Precedes(s, mt1, mt2) -BY SMTT(30) DEF Precedes +<1>. SUFFICES + ASSUME NEW i \in 1 .. Len(s), NEW j \in 1 .. Len(s), + s[i].type = mt1, s[j].type = mt2 + PROVE i < j + BY DEF Precedes +<1>1. CASE i = 1 + BY <1>1 +<1>2. CASE i > 1 + <2>1. /\ i-1 \in 1 .. Len(Tail(s)) + /\ Tail(s)[i-1].type = mt1 + /\ j-1 \in 1 .. Len(Tail(s)) + /\ Tail(s)[j-1].type = mt2 + BY <1>2 + <2>. QED BY <2>1 DEF Precedes +<1>. QED BY <1>1, <1>2 ----------------------------------------------------------------------------- (***************************************************************************) @@ -273,8 +293,9 @@ THEOREM BasicInvariant == Spec => []BasicInv BY <3>1, <4>2 <5>. \A q \in Proc : UNCHANGED network[p][q] BY <3>1, <4>2 - <5>. /\ \A q \in Proc \ {n} : UNCHANGED network[q][p] - /\ p = n => UNCHANGED network[n][p] + <5>. \A q \in Proc \ {n} : UNCHANGED network[q][p] + BY <3>1 + <5>. p = n => UNCHANGED network[n][p] BY <3>1, <4>2 DEF Broadcast <5>. n # p => Contains(network'[n][p], "ack") <=> Contains(network[n][p], "ack") BY <3>1, <4>2, ContainsSend DEF Broadcast @@ -315,10 +336,7 @@ THEOREM BasicInvariant == Spec => []BasicInv BY <3>1, <4>1 DEF BasicInv, NetworkInv <5>. QED BY DEF NetworkInv <4>2. CASE p # n - <5>. /\ network'[p][p] = network[p][p] - /\ network'[p][q] = network[p][q] - BY <3>1, <4>2 - <5>. QED BY DEF BasicInv, NetworkInv + BY <3>1, <4>2 DEF BasicInv, NetworkInv <4>. QED BY <4>1, <4>2 <3>3. ASSUME NEW p \in Proc PROVE CommInv(p)' @@ -333,9 +351,7 @@ THEOREM BasicInvariant == Spec => []BasicInv <5>. ASSUME NEW q \in Proc PROVE Contains(network'[q][p], "ack") <=> Contains(network[q][p], "ack") <6>1. CASE n = q - <7>. network'[q][p] = Append(network[q][p], RelMessage) - BY <3>1, <4>2, <6>1 - <7>. QED BY ContainsSend + BY <3>1, <4>2, <6>1, ContainsSend <6>2. CASE n # q BY <3>1, <6>2 <6>. QED BY <6>1, <6>2 @@ -368,14 +384,16 @@ THEOREM BasicInvariant == Spec => []BasicInv BY <3>1, AtMostOneTail, AtMostOneSend, Zenon DEF BasicInv, NetworkInv <4>2. AtMostOne(network'[p][q], "ack") <5>. DEFINE nw == [network EXCEPT ![k][n] = Tail(network[k][n])] - <5>1. /\ nw \in [Proc -> [Proc -> Seq(Message)]] - /\ AtMostOne(nw[p][q], "ack") - /\ ~ Contains(nw[n][k], "ack") - BY <3>1, <3>3, AtMostOneTail DEF BasicInv, NetworkInv + <5>1. nw \in [Proc -> [Proc -> Seq(Message)]] + BY <3>1 + <5>2. AtMostOne(nw[p][q], "ack") + BY <3>1, AtMostOneTail DEF BasicInv, NetworkInv + <5>3. ~ Contains(nw[n][k], "ack") + BY <3>1, <3>3 DEF BasicInv, NetworkInv <5>. HIDE DEF nw <5>. DEFINE nw2 == [nw EXCEPT ![n][k] = Append(network[n][k], AckMessage)] <5>5. AtMostOne(nw2[p][q], "ack") - BY <3>3, <5>1, NotContainsSend + BY <3>3, <5>1, <5>2, <5>3, NotContainsSend <5>. QED BY <3>1, <5>5 DEF nw <4>3. AtMostOne(network'[p][q], "rel") BY <3>1, AtMostOneTail, AtMostOneSend, Zenon DEF BasicInv, NetworkInv @@ -556,7 +574,9 @@ THEOREM ClockInvariant == Spec => []ClockInv <4>1. CASE p = n <5>1. ASSUME NEW i \in 1 .. Len(network'[p][q]), network'[p][q][i].type = "req" PROVE network'[p][q][i].clock = req'[p][p] - BY <3>1, <4>1, <5>1 DEF Broadcast, Contains + <6>. network'[p][q] = Append(network[p][q], ReqMessage(clock[n])) + BY <3>1, <4>1 DEF Broadcast + <6>. QED BY <3>1, <4>1, <5>1 DEF Contains <5>2. ~ Contains(network'[q][p], "ack") /\ q \notin ack'[p] BY <3>1, <4>1 DEF Broadcast <5>3. p \notin crit' @@ -569,14 +589,23 @@ THEOREM ClockInvariant == Spec => []ClockInv BY <3>1, <4>2 <5>. DEFINE qp == network[q][p] <5>3. ASSUME Contains(qp', "ack") \/ q \in ack'[p] - PROVE /\ req'[q][p] = req[p][p] + PROVE /\ req'[q][p] = req'[p][p] /\ clock'[q] > req'[p][p] - /\ Precedes(qp', "ack", "req") => - \A i \in 1 .. Len(qp') : qp'[i].type = "req" => qp'[i].clock > req'[p][p] - <6>. Contains(qp, "ack") \/ q \in ack[p] + /\ \A i \in 1 .. Len(qp') : qp'[i].type = "req" => qp'[i].clock > req'[p][p] + <6>1. Contains(qp, "ack") \/ q \in ack[p] BY <3>1, <4>2, <5>3, ContainsSend DEF Broadcast - <6>. QED - BY <3>1, <4>2, <5>1 DEF ClockInvInner, Broadcast, Contains + <6>3. /\ req'[q][p] = req'[p][p] + /\ clock'[q] > req'[p][p] + BY <3>1, <4>2, <5>1, <6>1 DEF ClockInvInner \* <3>1, <4>2, <6>2 + <6>4. SUFFICES + ASSUME NEW i \in 1 .. Len(qp'), qp'[i].type = "req" + PROVE qp'[i].clock > req'[p][p] + BY <6>3 + <6>5. qp' = Append(qp, ReqMessage(clock[n])) + BY <3>1, <4>2 DEF Broadcast + <6>7. qp'[i] = ReqMessage(clock[n]) + BY <3>1, <4>2, <6>4, <6>5 DEF Contains + <6>. QED BY <3>1, <4>2, <6>3, <6>7 DEF ReqMessage <5>. QED BY <5>1, <5>2, <5>3 DEF ClockInvInner, beats <4>3. CASE n \notin {p,q} \* all relevant variables unchanged BY <3>1, <4>3 DEF ClockInv, ClockInvInner, beats @@ -607,8 +636,17 @@ THEOREM ClockInvariant == Spec => []ClockInv BY DEF RelMessage, Message <3>2. ASSUME NEW p \in Proc, NEW q \in Proc \ {p} PROVE ClockInvInner(p,q)' - <4>1. CASE n = p - BY <3>1, <4>1 DEF Broadcast, ClockInvInner, Contains + <4>. DEFINE pq == network[p][q] qp == network[q][p] + <4>1. CASE p = n + \* BY <3>1, <4>1 DEF ClockInvInner, Broadcast, Contains, RelMessage -- works + <5>1. ASSUME NEW i \in 1 .. Len(pq'), pq'[i].type = "req" + PROVE pq'[i].clock = req'[p][p] + BY <3>1, <4>1, <5>1 DEF Broadcast, Contains + <5>2. ~(Contains(qp, "ack")' \/ q \in ack'[p]) + BY <3>1, <4>1 DEF Broadcast, Contains, RelMessage + <5>3. p \in crit' => beats(p,q)' + BY <3>1, <4>1 + <5>. QED BY <5>1, <5>2, <5>3 DEF ClockInvInner <4>2. CASE n # p <5>1. /\ UNCHANGED << network[p][q], req[p][p], req[q][p], req[p][q], ack[p], clock >> /\ p \in crit' <=> p \in crit @@ -619,7 +657,7 @@ THEOREM ClockInvariant == Spec => []ClockInv /\ Precedes(network'[n][p], "ack", "req") <=> Precedes(network[n][p], "ack", "req") BY <3>1, <4>2, ContainsSend, PrecedesSend DEF Broadcast <5>4. \A i \in 1 .. Len(network'[n][p]) : network'[n][p][i].type # "req" - BY <3>1 DEF Broadcast, Contains + BY <3>1 DEF Broadcast, Contains, RelMessage <5>. QED BY <5>1, <5>2, <5>3, <5>4 DEF ClockInv, ClockInvInner, beats <4>. QED BY <4>1, <4>2 <3>. QED BY <3>2 DEF ClockInv @@ -636,8 +674,10 @@ THEOREM ClockInvariant == Spec => []ClockInv /\ UNCHANGED <> /\ Contains(network[k][n], "req") BY <2>4 DEF ReceiveRequest, ClockInv, ClockInvInner, Contains + <3>a. ClockInvInner(k,n) + BY DEF ClockInv <3>2. m.clock = req[k][k] - BY <3>1 DEF ClockInv, ClockInvInner, Contains + BY <3>1, <3>a DEF ClockInv, ClockInvInner <3>3. /\ req[k][k] > 0 /\ n \notin ack[k] /\ k \notin crit BY <3>1 DEF BasicInv, CommInv @@ -661,19 +701,25 @@ THEOREM ClockInvariant == Spec => []ClockInv /\ q \notin ack'[p] /\ p \notin crit' BY <3>1, <3>2, <3>3, <4>1 - <5>5. ASSUME Precedes(qp', "ack", "req"), + <5>5. ASSUME Precedes(qp', "ack", "req"), NEW i \in 1 .. Len(qp'), qp'[i].type = "req" PROVE FALSE - BY <3>1, <4>1, <5>5 DEF Precedes + <6>1. ~(Len(qp') < i) + BY <3>1, <4>1, <5>5 + <6>2. /\ Len(qp') \in 1 .. Len(qp') + /\ qp'[Len(qp')].type = "ack" + BY <3>1, <4>1 + <6>. QED BY <5>5, <6>1, <6>2 DEF Precedes <5>. QED BY <5>2, <5>3, <5>4, <5>5 DEF ClockInvInner, Contains <4>2. CASE p = k /\ q # n BY <3>1, <4>2 DEF ClockInvInner, beats <4>3. CASE p = n /\ q = k - <5>1. UNCHANGED << req[q][p], clock[q], ack >> + <5>1. /\ pq' = Append(pq, AckMessage) + /\ UNCHANGED << req[q][p], clock[q], ack >> BY <3>1, <4>3 <5>2. ASSUME NEW i \in 1 .. Len(pq'), pq'[i].type = "req" PROVE i \in 1 .. Len(pq) /\ pq'[i] = pq[i] - BY <3>1, <4>3, <5>2 + BY <5>1, <5>2 <5>3. qp' = Tail(qp) /\ Head(qp).type = "req" /\ qp # << >> BY <3>1, <4>3, Zenon <5>4. Contains(qp', "ack") <=> Contains(qp, "ack") @@ -750,7 +796,7 @@ THEOREM ClockInvariant == Spec => []ClockInv /\ network' = [network EXCEPT ![k][n] = Tail(@)] /\ UNCHANGED << clock, ack, crit >> /\ Contains(network[k][n], "rel") - BY <2>6 DEF ReceiveRelease, Contains\*, BasicInv, CommInv, Contains + BY <2>6 DEF ReceiveRelease, Contains <3>2. /\ ~ Contains(network[n][k], "ack") /\ n \notin ack[k] BY <3>1, Zenon DEF BasicInv, CommInv From 4b3e413863b07423ff60a91d0d64b62d9f06c965 Mon Sep 17 00:00:00 2001 From: Stephan Merz Date: Mon, 31 Mar 2025 15:59:06 +0200 Subject: [PATCH 04/46] fixed proofs in byzpaxos Signed-off-by: Stephan Merz --- .github/workflows/CI.yml | 8 +- specifications/byzpaxos/BPConProof.tla | 149 +++++++++++++------- specifications/byzpaxos/Consensus.tla | 108 +++------------ specifications/byzpaxos/VoteProof.tla | 185 ++++++++++++++----------- 4 files changed, 224 insertions(+), 226 deletions(-) diff --git a/.github/workflows/CI.yml b/.github/workflows/CI.yml index 88711889..cf038981 100644 --- a/.github/workflows/CI.yml +++ b/.github/workflows/CI.yml @@ -147,21 +147,19 @@ jobs: specifications/ewd998/AsyncTerminationDetection_proof.tla specifications/ewd998/EWD998_proof.tla ## Regressions? - specifications/byzpaxos/Consensus.tla specifications/LearnProofs/FindHighest.tla specifications/LoopInvariance/BinarySearch.tla # Failing; see https://github.com/tlaplus/Examples/issues/67 - # specifications/Bakery-Boulangerie/Bakery.tla - # specifications/Bakery-Boulangerie/Boulanger.tla - # specifications/Paxos/Consensus.tla # specifications/PaxosHowToWinATuringAward/Consensus.tla specifications/lamport_mutex/LamportMutex_proofs.tla - specifications/byzpaxos/VoteProof.tla # Long-running; see https://github.com/tlaplus/tlapm/issues/85 + specifications/Bakery-Boulangerie/Bakery.tla + specifications/Bakery-Boulangerie/Boulanger.tla specifications/LoopInvariance/Quicksort.tla specifications/LoopInvariance/SumSequence.tla specifications/bcastByz/bcastByz.tla specifications/MisraReachability/ReachabilityProofs.tla + specifications/byzpaxos/VoteProof.tla specifications/byzpaxos/BPConProof.tla # Takes about 30 minutes ) python $SCRIPT_DIR/check_proofs.py \ diff --git a/specifications/byzpaxos/BPConProof.tla b/specifications/byzpaxos/BPConProof.tla index a55259cc..1e1c49c5 100644 --- a/specifications/byzpaxos/BPConProof.tla +++ b/specifications/byzpaxos/BPConProof.tla @@ -582,7 +582,7 @@ LEMMA NextDef == \/ Phase1c(self) \/ \E self \in FakeAcceptor : FakingAcceptor(self) <1>1. \A self : acceptor(self) <=> NextDef!2!1!(self) - BY DEF acceptor, Phase1b, Phase2av, Phase2b, LearnsSent + BY Zenon DEF acceptor, Phase1b, Phase2av, Phase2b, LearnsSent <1>2. \A self : leader(self) <=> NextDef!2!2!(self) BY DEF leader, Phase1a, Phase1c <1>3. \A self : facceptor(self) <=> NextDef!2!3!(self) @@ -605,7 +605,7 @@ Quorum == {S \cap Acceptor : S \in ByzQuorum} THEOREM QuorumTheorem == /\ \A Q1, Q2 \in Quorum : Q1 \cap Q2 # {} /\ \A Q \in Quorum : Q \subseteq Acceptor -BY BQA DEF Quorum +BY BQA, IsaM("blast") DEF Quorum, ByzAcceptor (***************************************************************************) (* We now define refinement mapping under which our algorithm implements *) @@ -719,7 +719,7 @@ LEMMA MaxBallotLemma1 == /\ y >= MaxBallot(S) BY MaxBallotProp <1>3. MaxBallot(S) \in Int /\ y \in Int - BY <1>1, <1>2, Isa DEF Ballot + BY <1>1, <1>2 DEF Ballot <1>. QED BY <1>1, <1>2, <1>3 LEMMA MaxBallotLemma2 == @@ -758,7 +758,7 @@ LEMMA MaxBallotLemma2 == <1>3. CASE ~(MaxBallot(S) >= MaxBallot(T)) <2>. SUFFICES ASSUME S # {} PROVE MaxBallot(S \cup T) = MaxBallot(T) - BY <1>3 + BY <1>3, Zenon <2>1. /\ MaxBallot(S) \in S /\ \A x \in S : MaxBallot(S) >= x BY MaxBallotProp @@ -844,7 +844,11 @@ bmsgsFinite == IsFiniteSet(1bOr2bMsgs) LEMMA FiniteMsgsLemma == ASSUME NEW m, bmsgsFinite, bmsgs' = bmsgs \cup {m} PROVE bmsgsFinite' -BY FS_AddElement DEF bmsgsFinite, 1bOr2bMsgs +<1>1. CASE m.type \in {"1b", "2b"} + BY <1>1, FS_AddElement, Isa DEF bmsgsFinite, 1bOr2bMsgs +<1>2. CASE m.type \notin {"1b", "2b"} + BY <1>2, FS_AddElement, Isa DEF bmsgsFinite, 1bOr2bMsgs +<1>. QED BY <1>1, <1>2 (***************************************************************************) (* Invariant 1bInv1 asserts that if (good) acceptor `a' has mCBal[a] # -1, *) @@ -1126,7 +1130,7 @@ TypeOK => /\ 1cmsgs' = 1cmsgs /\ 2amsgs' = 2amsgs /\ acceptorMsgsOfType("2b")' = acceptorMsgsOfType("2b") - BY <2>1 DEF msgsOfType, 1bmsgs, acceptorMsgsOfType, KnowsSafeAt, 1cmsgs, 2amsgs + BY <2>1, Isa DEF msgsOfType, 1bmsgs, acceptorMsgsOfType, KnowsSafeAt, 1cmsgs, 2amsgs <2>. QED BY <2>a DEF msgs, 1bRestrict @@ -1157,9 +1161,12 @@ TypeOK => /\ 1bmsgs' = 1bmsgs /\ 1cmsgs' = 1cmsgs /\ acceptorMsgsOfType("2b")' = acceptorMsgsOfType("2b") - BY <2>1, <3>1 DEF msgsOfType, 1bmsgs, 1bRestrict, acceptorMsgsOfType, KnowsSafeAt, 1cmsgs - <3>. QED - BY <3>1, <3>2, <2>1, <2>3 DEF msgs, 2amsgs, msgsOfType, acceptorMsgsOfType + BY <2>1, <3>1, Isa DEF msgsOfType, 1bmsgs, 1bRestrict, acceptorMsgsOfType, KnowsSafeAt, 1cmsgs + <3>3. 2amsgs \subseteq 2amsgs' + BY <2>1 DEF 2amsgs, acceptorMsgsOfType, msgsOfType + <3>4. \A m2a \in 2amsgs' \ 2amsgs : m2a = ma + BY <2>1 DEF 2amsgs, acceptorMsgsOfType, msgsOfType + <3>. QED BY <2>3, <3>2, <3>3, <3>4, Zenon DEF msgs <2>6. QED BY <2>4, <2>5 @@ -1190,7 +1197,7 @@ TypeOK => /\ 1cmsgs' = 1cmsgs /\ 2amsgs' = 2amsgs /\ acceptorMsgsOfType("2b")' = acceptorMsgsOfType("2b") \cup {bm} - BY <2>1 DEF msgsOfType, 1bmsgs, 1bRestrict, 1cmsgs, KnowsSafeAt, 2amsgs, acceptorMsgsOfType + BY <2>1, Isa DEF msgsOfType, 1bmsgs, 1bRestrict, 1cmsgs, KnowsSafeAt, 2amsgs, acceptorMsgsOfType <2>4. msgs' = msgs \cup {bm} BY <2>2 DEF msgs <2>. QED @@ -1208,11 +1215,15 @@ TypeOK => <3>1. bmsgs' = bmsgs BY <1>4 DEF LearnsSent <3>2. PICK S \in SUBSET sentMsgs("1b", b): - knowsSent' = [knowsSent EXCEPT ![self] = knowsSent[self] \cup S] + knowsSent' = [knowsSent EXCEPT ![self] = @ \cup S] BY <1>4, Zenon DEF LearnsSent + <3>ks. ASSUME NEW ac \in Acceptor, NEW bb, NEW vv, + KnowsSafeAt(ac,bb,vv) + PROVE KnowsSafeAt(ac,bb,vv)' + BY <3>2, <3>ks DEF TypeOK, KnowsSafeAt <3>3. ASSUME NEW m \in 1cmsgs PROVE m \in 1cmsgs' - BY <3>1, <3>2 DEF TypeOK, KnowsSafeAt, 1cmsgs, msgsOfType + BY <3>1, <3>ks DEF 1cmsgs, msgsOfType <3>4. ASSUME NEW m \in 1cmsgs', m \notin 1cmsgs PROVE m \in msgsOfType("1c") /\ m.bal = b <4>1. m \in msgsOfType("1c") @@ -1241,8 +1252,9 @@ TypeOK => <1>5. ASSUME NEW self \in Ballot, Phase1a(self) PROVE msgs' = msgs \cup {[type |-> "1a", bal |-> self]} - BY <1>5 DEF Phase1a, msgs, msgsOfType, 1bmsgs, 1bRestrict, 1cmsgs, KnowsSafeAt, - 2amsgs, acceptorMsgsOfType + BY <1>5, Isa + DEF Phase1a, msgs, msgsOfType, 1bmsgs, 1bRestrict, 1cmsgs, KnowsSafeAt, + 2amsgs, acceptorMsgsOfType <1>6. ASSUME NEW self \in Ballot, Phase1c(self) PROVE \E S \in SUBSET [type : {"1c"}, bal : {self}, val : Value]: @@ -1258,17 +1270,41 @@ TypeOK => BY <2>1, Zenon <2>2. /\ msgsOfType("1a")' = msgsOfType("1a") /\ 1bmsgs' = 1bmsgs - /\ 1cmsgs' = 1cmsgs \cup SS - /\ 2amsgs' = 2amsgs /\ acceptorMsgsOfType("2b")' = acceptorMsgsOfType("2b") - BY <2>1 DEF msgsOfType, 1bmsgs, 1bRestrict, 1cmsgs, KnowsSafeAt, 2amsgs, acceptorMsgsOfType - <2>3. QED - BY <2>2 DEF msgs + BY <2>1 DEF msgsOfType, 1bmsgs, 1bRestrict, acceptorMsgsOfType + <2>3. 1cmsgs' = 1cmsgs \cup SS + <3>1. ASSUME NEW ac \in Acceptor, NEW bb, NEW vv + PROVE KnowsSafeAt(ac, bb, vv)' <=> KnowsSafeAt(ac, bb, vv) + BY <2>1 DEF KnowsSafeAt + <3>. QED BY <2>1, <3>1 DEF 1cmsgs, msgsOfType + <2>4. 2amsgs' = 2amsgs + BY <2>1 DEF 2amsgs, acceptorMsgsOfType, msgsOfType + <2>. QED + BY <2>2, <2>3, <2>4 DEF msgs <1>7. ASSUME NEW self \in FakeAcceptor, FakingAcceptor(self) PROVE msgs' = msgs - BY <1>7, BQA DEF FakingAcceptor, msgs, 1bMessage, 2avMessage, 2bMessage, - msgsOfType, 1cmsgs, KnowsSafeAt, 1bmsgs, 2amsgs, acceptorMsgsOfType, msgsOfType + <2>1. PICK m \in 1bMessage \cup 2avMessage \cup 2bMessage : + /\ m.acc = self + /\ bmsgs' = bmsgs \cup {m} + /\ UNCHANGED knowsSent + BY <1>7 DEF FakingAcceptor + <2>2. m.type \in {"1b", "2av", "2b"} + BY DEF 1bMessage, 2avMessage, 2bMessage + <2>3. msgsOfType("1a")' = msgsOfType("1a") + BY <2>1, <2>2 DEF msgsOfType + <2>4. /\ 1bmsgs' = 1bmsgs + /\ acceptorMsgsOfType("2b")' = acceptorMsgsOfType("2b") + /\ acceptorMsgsOfType("2av")' = acceptorMsgsOfType("2av") + BY <2>1, BQA DEF 1bmsgs, acceptorMsgsOfType, msgsOfType + <2>5. ASSUME NEW ac \in Acceptor, NEW bb, NEW vv + PROVE KnowsSafeAt(ac, bb, vv)' <=> KnowsSafeAt(ac, bb, vv) + BY <2>1 DEF KnowsSafeAt + <2>6. 1cmsgs' = 1cmsgs + BY <2>1, <2>2, <2>5 DEF 1cmsgs, msgsOfType + <2>7. 2amsgs' = 2amsgs + BY <2>4 DEF 2amsgs + <2>. QED BY <2>3, <2>4, <2>6, <2>7 DEF msgs <1>9. QED BY <1>1, <1>2, <1>3, <1>4, <1>5, <1>6, <1>7, Zenon @@ -1279,7 +1315,8 @@ TypeOK => (***************************************************************************) THEOREM Invariance == Spec => []Inv <1>1. Init => Inv - BY FS_EmptySet DEF Init, Inv, TypeOK, bmsgsFinite, 1bOr2bMsgs, 1bInv1, 1bInv2, + BY FS_EmptySet, Zenon + DEF Init, Inv, TypeOK, bmsgsFinite, 1bOr2bMsgs, 1bInv1, 1bInv2, maxBalInv, 2avInv1, 2avInv2, 2avInv3, accInv, knowsSentInv <1>2. Inv /\ [Next]_vars => Inv' @@ -1301,12 +1338,14 @@ THEOREM Invariance == Spec => []Inv mbal |-> maxVBal[self], mval |-> maxVVal[self]] <4>1. msgs' = msgs \cup {mc} BY <3>1, MsgsLemma DEF Inv + <4>. mb \in 1bMessage + BY Zenon DEF Inv, TypeOK, 1bMessage, ByzAcceptor <4>2. TypeOK' - BY <3>1 DEF Inv, TypeOK, BMessage, 1bMessage, ByzAcceptor, Phase1b + BY <3>1 DEF Inv, TypeOK, BMessage, Phase1b <4>3. bmsgsFinite' BY <3>1, FiniteMsgsLemma, Zenon DEF Inv, bmsgsFinite, Phase1b <4>4. 1bInv1' - BY <3>1, <4>1, IsaT(100) DEF Phase1b, 1bInv1, Inv, accInv + BY <3>1, <4>1, Isa DEF Phase1b, 1bInv1, Inv, accInv <4>5. 1bInv2' BY <3>1 DEF Phase1b, 1bInv2, Inv, maxBalInv, TypeOK, 1bMessage, Ballot <4>6. maxBalInv' @@ -1332,7 +1371,7 @@ THEOREM Invariance == Spec => []Inv <4>11. knowsSentInv' BY <3>1 DEF Phase1b, Inv, knowsSentInv, msgsOfType <4>12. QED - BY <4>2, <4>3, <4>4, <4>5, <4>6, <4>7, <4>8, <4>9, <4>10, <4>11 DEF Inv + BY <4>2, <4>3, <4>4, <4>5, <4>6, <4>7, <4>8, <4>9, <4>10, <4>11, Zenon DEF Inv <3>2. CASE Phase2av(self, b) <4>1. PICK mc \in sentMsgs("1c", b) : /\ KnowsSafeAt(self, b, mc.val) @@ -1342,6 +1381,7 @@ THEOREM Invariance == Spec => []Inv /\ 2avSent' = [2avSent EXCEPT ![self] = {r \in 2avSent[self] : r.val # mc.val} \cup {[val |-> mc.val, bal |-> b]}] + /\ maxBal' = [maxBal EXCEPT ![self] = b] BY <3>2, Zenon DEF Phase2av <4>2. mc = [type |-> "1c", bal |-> mc.bal, val |-> mc.val] BY <4>1, BMessageLemma DEF sentMsgs, Inv, TypeOK, 1cMessage @@ -1362,7 +1402,7 @@ THEOREM Invariance == Spec => []Inv <4>6. bmsgsFinite' BY <4>1, FiniteMsgsLemma, Zenon DEF Inv, bmsgsFinite <4>7. 1bInv1' - BY <3>2, <4>1, <4>3, IsaT(100) DEF Phase2av, 1bInv1, Inv + BY <3>2, <4>1, <4>3 DEF Phase2av, 1bInv1, Inv <4>8. 1bInv2' BY <4>1 DEF Inv, 1bInv2 <4>9. maxBalInv' @@ -1378,7 +1418,7 @@ THEOREM Invariance == Spec => []Inv BY DEF 2avInv2 <5>2. CASE m.acc = self <6>1. CASE m = mb - BY <4>1, <6>1, Isa DEF Inv, TypeOK, Ballot + BY <4>1, <6>1 DEF Inv, TypeOK, Ballot <6>2. CASE m # mb <7>1. m \in bmsgs BY <4>1, <6>2 @@ -1415,8 +1455,16 @@ THEOREM Invariance == Spec => []Inv <5>2. CASE r \in 2avSent[a] BY <5>2, <4>4, <4>5, <3>2 DEF Phase2av, Inv, TypeOK, accInv, Ballot <5>3. CASE r \notin 2avSent[a] - BY <5>3, <3>2, <4>1, <4>2, <4>4 - DEF Phase2av, Inv, TypeOK, sentMsgs, msgsOfType, msgs, 1cmsgs, Ballot + <6>1. /\ a = self + /\ r = [val |-> mc.val, bal |-> b] + /\ KnowsSafeAt(self, b, mc.val) + /\ maxBal' = [maxBal EXCEPT ![a] = b] + BY <4>1, <5>3 DEF Inv, TypeOK + <6>2. r.bal <= maxBal'[a] + BY <6>1 DEF Inv, TypeOK, Ballot + <6>3. [type |-> "1c", bal |-> r.bal, val |-> r.val] \in msgs + BY <4>2, <6>1 DEF msgs, 1cmsgs, sentMsgs, msgsOfType + <6>. QED BY <4>4, <6>2, <6>3 <5>4. QED BY <5>2, <5>3 <4>14. knowsSentInv' @@ -1441,7 +1489,7 @@ THEOREM Invariance == Spec => []Inv <4>3. bmsgsFinite' BY <4>1, FiniteMsgsLemma, Zenon DEF Inv, bmsgsFinite <4>4. 1bInv1' - BY <4>1, Isa DEF Inv, 1bInv1 + BY <4>1 DEF Inv, 1bInv1 <4>5. 1bInv2' BY <4>1 DEF Inv, 1bInv2 <4>6. maxBalInv' @@ -1544,7 +1592,7 @@ THEOREM Invariance == Spec => []Inv <4>3. TypeOK' BY <3>2, <4>1 DEF Phase1c, Inv, TypeOK, BMessage, 1cMessage <4>4. bmsgsFinite' - BY <4>1 DEF Inv, bmsgsFinite, 1bOr2bMsgs + BY <4>1, Zenon DEF Inv, bmsgsFinite, 1bOr2bMsgs <4>5. 1bInv1' BY <3>2, <4>2, Zenon DEF Phase1c, Inv, 1bInv1 <4>6. 1bInv2' @@ -1600,7 +1648,7 @@ THEOREM Invariance == Spec => []Inv PROVE Inv' <3> USE UNCHANGED vars DEF Inv, vars <3> msgs = msgs' - BY DEF msgs, msgsOfType, 1bmsgs, 1bRestrict, acceptorMsgsOfType, 1cmsgs, + BY Isa DEF msgs, msgsOfType, 1bmsgs, 1bRestrict, acceptorMsgsOfType, 1cmsgs, KnowsSafeAt, 2amsgs <3> QED BY DEF TypeOK, bmsgsFinite, 1bOr2bMsgs, 1bInv1, 1bInv2, @@ -1624,19 +1672,25 @@ THEOREM Spec => P!Spec <2>1. MaxBallot({}) = -1 BY MaxBallotProp, FS_EmptySet <2>2. P!Init!1 /\ P!Init!2 /\ P!Init!3 - BY <2>1 DEF Init, PmaxBal, 1bOr2bMsgs, None, P!None + BY <2>1, Zenon DEF Init, PmaxBal, 1bOr2bMsgs, None, P!None <2>3. msgs = {} - BY BQA DEF Init, msgsOfType, acceptorMsgsOfType, 1bmsgs, 1cmsgs, 2amsgs, Quorum, msgs + \* BY BQA DEF Init, msgsOfType, acceptorMsgsOfType, 1bmsgs, 1cmsgs, 2amsgs, Quorum, msgs + <3>1. bmsgs = {} + BY DEF Init + <3>2. \A Q \in Quorum : Q # {} + BY BQA DEF Quorum + <3>. QED + BY <3>1, <3>2 DEF msgsOfType, acceptorMsgsOfType, 1bmsgs, 1cmsgs, 2amsgs, msgs <2>4. QED BY <2>2, <2>3 DEF P!Init <1>2. Inv /\ Inv' /\ [Next]_vars => [P!Next]_P!vars - <2> InvP == Inv' + <2> DEFINE InvP == Inv' <2> SUFFICES ASSUME Inv, InvP, Next PROVE P!TLANext \/ P!vars' = P!vars <3> UNCHANGED vars => UNCHANGED P!vars - BY DEF vars, P!vars, PmaxBal, 1bOr2bMsgs, msgs, msgsOfType, acceptorMsgsOfType, - 1bmsgs, 2amsgs, 1cmsgs, KnowsSafeAt + BY Isa DEF vars, P!vars, PmaxBal, 1bOr2bMsgs, msgs, msgsOfType, acceptorMsgsOfType, + 1bmsgs, 2amsgs, 1cmsgs, KnowsSafeAt <3> QED BY PNextDef DEF Inv, P!ProcSet, P!Init, Ballot, P!Ballot <2> HIDE DEF InvP @@ -1670,7 +1724,7 @@ THEOREM Spec => P!Spec BY DEF PmaxBal, 1bOr2bMsgs <4> HIDE DEF mA <4>4. S(self)' = S(self) \cup {b} - BY <4>2, Isa + BY <4>2 <4>5. MaxBallot(S(self) \cup {b}) = b <5> DEFINE SS == S(self) \cup {b} <5>1. IsFiniteSet(S(self)) @@ -1683,7 +1737,7 @@ THEOREM Spec => P!Spec <5>3. S(self) \subseteq Ballot \cup {-1} BY BMessageLemma DEF mA, Inv, TypeOK, 1bMessage, 2bMessage <5>4. \A x \in SS : b >= x - BY <3>4, <4>3, <5>1, <5>3, MaxBallotProp, Z3T(10) DEF Ballot + BY <3>4, <4>3, <5>1, <5>3, MaxBallotProp DEF Ballot <5>5. QED BY <5>2, <5>3, <5>4, MaxBallotLemma1 <4>6. \A a \in Acceptor : a # self => S(a)' = S(a) @@ -1717,7 +1771,7 @@ THEOREM Spec => P!Spec <4>2. P!sentMsgs("2a", b) = {} <5>1. SUFFICES ASSUME NEW m \in P!sentMsgs("2a", b) PROVE m = [type |-> "2a", bal |-> b, val |-> v] - BY <3>3, <4>1 DEF P!sentMsgs + BY <3>3, <4>1, Zenon DEF P!sentMsgs <5>2. /\ m \in 2amsgs /\ m.type = "2a" /\ m.bal = b @@ -1735,7 +1789,7 @@ THEOREM Spec => P!Spec /\ m2av.acc = a /\ m2av.bal = b /\ m2av.val = v - BY <4>1, MsgsTypeLemmaPrime, Isa DEF 2amsgs + BY <4>1, MsgsTypeLemmaPrime DEF 2amsgs <5>5. PICK a \in Q \cap Q2 : a \in Acceptor BY QuorumTheorem <5>6. PICK mav \in acceptorMsgsOfType("2av") : @@ -1880,7 +1934,7 @@ THEOREM Spec => P!Spec BY <4>4 <5>2. \A a \in Q(BQ) : \E m \in SQ(BQ) : /\ m.acc = a /\ m.mbal = -1 - BY <5>1, Isa DEF 1bRestrict + BY <5>1 DEF 1bRestrict <5>3. \A m \in SQ(BQ) : m.mbal = -1 BY <4>2, <5>2 DEF InvP, Inv, knowsSentInv, msgsOfType, 1bRestrict, 1bInv2 @@ -1964,7 +2018,7 @@ THEOREM Spec => P!Spec <5>7. P!ShowsSafeAt(Q(BQ), b, v)!1!2!2!(m1c) BY <5>5, <5>6 <5>. QED - BY <5>4, <5>7, Isa DEF P!ShowsSafeAt, Quorum + BY <5>4, <5>7 DEF P!ShowsSafeAt, Quorum <4>6. QED BY <3>1, <4>1, <4>4, <4>5 DEF KnowsSafeAt <3>6. QED @@ -1976,7 +2030,7 @@ THEOREM Spec => P!Spec <3>1. msgs' = msgs \cup {[type |-> "1a", bal |-> self]} BY <2>7, MsgsLemma DEF Inv <3>2. UNCHANGED << PmaxBal, maxVBal, maxVVal >> - BY <2>7, Isa DEF Phase1a, PmaxBal, 1bOr2bMsgs + BY <2>7 DEF Phase1a, PmaxBal, 1bOr2bMsgs <3>. QED BY <3>1, <3>2 DEF P!Phase1a, P!TLANext, Ballot, P!Ballot <2>8. ASSUME NEW self \in Ballot, @@ -2118,10 +2172,3 @@ THEOREM chosen \subseteq P!chosen BY Isa DEF chosen, P!chosen, Quorum, Ballot, P!Ballot ============================================================================== -\* Modification History -\* Last modified Sat Jul 25 17:34:50 PDT 2020 by lamport -\* Last modified Fri Jul 24 17:51:34 CEST 2020 by merz -\* Last modified Wed Apr 15 15:16:26 CEST 2020 by doligez -\* Last modified Mon Aug 18 14:57:27 CEST 2014 by tomer -\* Last modified Mon Mar 04 17:24:05 CET 2013 by doligez -\* Last modified Wed Dec 01 11:35:29 PST 2010 by lamport diff --git a/specifications/byzpaxos/Consensus.tla b/specifications/byzpaxos/Consensus.tla index bee82550..91348085 100644 --- a/specifications/byzpaxos/Consensus.tla +++ b/specifications/byzpaxos/Consensus.tla @@ -147,7 +147,7 @@ LEMMA InductiveInvariance == OBVIOUS <1>1. CASE Next \* In the following BY proof, <1>1 denotes the case assumption Next - BY <1>1, FS_EmptySet, FS_AddElement DEF Inv, TypeOK, Next + BY <1>1, FS_Singleton DEF Inv, TypeOK, Next <1>2. CASE vars' = vars BY <1>2 DEF Inv, TypeOK, vars <1>3. QED @@ -177,81 +177,24 @@ Success == <>(chosen # {}) ASSUME ValueNonempty == Value # {} (***************************************************************************) -(* TLAPS does not yet reason about ENABLED. Therefore, we must omit all *) -(* proofs that involve ENABLED formulas. To perform as much of the proof *) -(* as possible, as much as possible we restrict the use of an ENABLED *) -(* expression to a step asserting that it equals its definition. ENABLED *) -(* A is true of a state s iff there is a state t such that the step s -> t *) -(* satisfies A. It follows from this semantic definition that ENABLED A *) -(* equals the formula obtained by *) -(* *) -(* 1. Expanding all definitions of defined symbols in A until all primes *) -(* are priming variables. *) -(* *) -(* 2. For each primed variable, replacing every instance of that primed *) -(* variable by a new symbol (the same symbol for each primed *) -(* variable). *) -(* *) -(* 3. Existentially quantifying over those new symbols. *) +(* Proving liveness requires reasoning about fairness assumptions, which *) +(* are defined in terms of enabledness of actions. It is usually a good *) +(* idea to prove a lemma that reduces ENABLED to a simple state predicate. *) (***************************************************************************) -LEMMA EnabledDef == - TypeOK => - ((ENABLED <>_vars) <=> (chosen = {})) -<1> DEFINE E == - \E chosenp : - /\ /\ chosen = {} - /\ \E v \in Value: chosenp = {v} - /\ ~ (<> = <>) -<1>1. E = ENABLED <>_vars - \* BY DEF Next, vars (* and def of ENABLED *) - PROOF OMITTED -<1>2. SUFFICES ASSUME TypeOK - PROVE E = (chosen = {}) - BY <1>1, Zenon -<1>3. E = \E chosenp : E!(chosenp)!1 - BY <1>2, Isa DEF TypeOK -<1>4. @ = (chosen = {}) - BY <1>2, ValueNonempty, Zenon DEF TypeOK -<1>5. QED - BY <1>3, <1>4, Zenon +LEMMA EnabledDef == (ENABLED <>_vars) <=> (chosen = {}) +BY ValueNonempty, ExpandENABLED DEF Next, vars -(***************************************************************************) -(* Here is our proof that Livespec implies Success. It uses the standard *) -(* TLA proof rules. For example RuleWF1 is defined in the TLAPS module to *) -(* be the rule WF1 discussed in *) -(* *) -(* `. AUTHOR = "Leslie Lamport", *) -(* TITLE = "The Temporal Logic of Actions", *) -(* JOURNAL = toplas, *) -(* volume = 16, *) -(* number = 3, *) -(* YEAR = 1994, *) -(* month = may, *) -(* PAGES = "872--923" .' *) -(* *) -(* PTL stands for propositional temporal logic reasoning. We expect that, *) -(* when TLAPS handles temporal reasoning, it will use a decision procedure *) -(* for PTL. *) -(***************************************************************************) -THEOREM LiveSpec => Success -<1>1. []Inv /\ [][Next]_vars /\ WF_vars(Next) => (chosen = {} ~> chosen # {}) - <2>. DEFINE P == chosen = {} - Q == chosen # {} - <2>1. SUFFICES [][Next]_vars /\ WF_vars(Next) => ((Inv /\ P) ~> Q) - BY PTL - <2>2. (Inv /\ P) /\ [Next]_vars => ((Inv' /\ P') \/ Q') - BY InductiveInvariance - <2>3. (Inv /\ P) /\ <>_vars => Q' - BY DEF Inv, Next, vars - <2>4. (Inv /\ P) => ENABLED <>_vars - BY EnabledDef DEF Inv - <2>. HIDE DEF P,Q - <2>. QED - BY <2>2, <2>3, <2>4, PTL -<1>2. (chosen = {} ~> chosen # {}) => ((chosen = {}) => <>(chosen # {})) - BY PTL -<1>3. QED - BY Invariance, <1>1, <1>2, PTL DEF LiveSpec, Spec, Init, Success +THEOREM Liveness == LiveSpec => Success +<1>. DEFINE P == chosen = {} + Q == chosen # {} +<1>1. P /\ [Next]_vars => P' \/ Q' + OBVIOUS +<1>2. P /\ <>_vars => Q' + BY DEF Next +<1>3. P => ENABLED <>_vars + BY EnabledDef +<1>. QED + BY <1>1, <1>2, <1>3, PTL DEF LiveSpec, Spec, Init, Success ----------------------------------------------------------------------------- (***************************************************************************) @@ -260,19 +203,6 @@ THEOREM LiveSpec => Success (***************************************************************************) THEOREM LiveSpecEquals == LiveSpec <=> Spec /\ ([]<><>_vars \/ []<>(chosen # {})) -<1>1. /\ Spec <=> Spec /\ []TypeOK - /\ LiveSpec <=> LiveSpec /\ []TypeOK - BY Invariance, PTL DEF LiveSpec, Inv -<1>2. (chosen # {}) <=> ~(chosen = {}) - OBVIOUS -<1>3. []TypeOK => (([]<>~ENABLED <>_vars) <=> []<>(chosen # {})) - BY <1>2, EnabledDef, PTL -<1>4. QED - BY <1>1, <1>3, PTL DEF LiveSpec +BY (chosen # {}) <=> ~(chosen = {}), EnabledDef, PTL DEF LiveSpec, Spec + ============================================================================= -\* Modification History -\* Last modified Mon May 11 18:36:27 CEST 2020 by merz -\* Last modified Mon Aug 18 15:00:45 CEST 2014 by tomer -\* Last modified Mon Aug 18 14:58:57 CEST 2014 by tomer -\* Last modified Tue Feb 14 13:35:49 PST 2012 by lamport -\* Last modified Mon Feb 07 14:46:59 PST 2011 by lamport diff --git a/specifications/byzpaxos/VoteProof.tla b/specifications/byzpaxos/VoteProof.tla index c7108327..cd611e0f 100644 --- a/specifications/byzpaxos/VoteProof.tla +++ b/specifications/byzpaxos/VoteProof.tla @@ -24,7 +24,7 @@ CONSTANT Value, \* As in module Consensus, the set of choosable values. (***************************************************************************) ASSUME QA == /\ \A Q \in Quorum : Q \subseteq Acceptor /\ \A Q1, Q2 \in Quorum : Q1 \cap Q2 # {} - + THEOREM QuorumNonEmpty == \A Q \in Quorum : Q # {} PROOF BY QA ----------------------------------------------------------------------------- @@ -298,7 +298,7 @@ THEOREM SafeAtProp == /\ \A d \in (c+1)..(bb-1), a \in Q : DidNotVoteIn(a, d) SA[bb \in Ballot] == Def(SA, bb) <1>2. \A b : SafeAt(b, v) = SA[b] - BY DEF SafeAt + BY Zenon DEF SafeAt <1>3. ASSUME NEW n \in Nat, NEW g, NEW h, \A i \in 0..(n-1) : g[i] = h[i] PROVE Def(g, n) = Def(h, n) @@ -400,7 +400,9 @@ LEMMA SafeLemma == <3>5. QED BY <3>3, <3>4 <2>8. CASE c < cc - BY <2>8, <1>2, <2>5 + <3>. cc \in 0 .. b /\ c \in 0 .. cc-1 + BY <2>8 + <3>. QED BY <1>2, <2>5 <2>9. QED BY <2>6, <2>7, <2>8 <1>3. \A b \in Ballot : P(b) @@ -491,7 +493,9 @@ LEMMA VT0 == /\ TypeOK <3> PICK aa \in QQ \cap Q : TRUE BY QA <3>4. c \leq d - BY <3>1, <3>2, <3>3 DEF DidNotVoteIn + <4>. SUFFICES ASSUME c \in d+1 .. b-1 PROVE FALSE + BY <3>1 + <4>. QED BY <3>2, <3>3 DEF DidNotVoteIn <3>5. CASE c = d BY <3>2, <3>3, <3>4, <3>5 <3>6. CASE d > c @@ -775,6 +779,8 @@ THEOREM InductiveInvariance == VInv /\ [Next]_vars => VInv' PROVE /\ SafeAt(cc, w)' /\ \A ax \in Q : \A z \in Value : VotedFor(ax, cc, z)' => (z = w) + <6>. cc \in 0 .. e-1 + BY <5>3 <6>1. /\ SafeAt(cc, w) /\ \A ax \in Q : \A z \in Value : VotedFor(ax, cc, z) => (z = w) @@ -786,7 +792,7 @@ THEOREM InductiveInvariance == VInv /\ [Next]_vars => VInv' <7>1. CASE VotedFor(ax, cc, z) BY <6>1, <7>1 <7>2. CASE ~ VotedFor(ax, cc, z) - BY <7>2, <6>3, <5>1, <5>3 + BY <7>2, <6>3, <5>1 <7>3. QED BY <7>1, <7>2 <6>4. QED @@ -951,7 +957,7 @@ THEOREM VT3 == Spec => C!Spec BY <2>1, NextDef DEF VInv <2>3. ASSUME IncreaseMaxBal(self, b) PROVE C!vars' = C!vars - BY <2>3 DEF IncreaseMaxBal, C!vars, chosen, ChosenIn, VotedFor + BY <2>3, Zenon DEF IncreaseMaxBal, C!vars, chosen, ChosenIn, VotedFor <2>4. ASSUME NEW v \in Value, VoteFor(self, b, v) PROVE [C!Next]_C!vars @@ -985,7 +991,7 @@ THEOREM VT3 == Spec => C!Spec <4>6. QED BY <4>4, <4>5 <3>. QED - BY <3>3, <3>1, <3>2 DEF C!Next, C!vars + BY <3>3, <3>1, <3>2, Zenon DEF C!Next, C!vars <2>5. QED BY <2>2, <2>3, <2>4 DEF BallotAction <1>3. QED @@ -1057,7 +1063,11 @@ THEOREM VT4 == TypeOK /\ VInv2 /\ VInv3 => /\ \A d \in (c+1)..(b-1), a \in Q : DidNotVoteIn(a, d) BY <1>1, <1>2, SafeAtProp <1>5. CASE \A a \in Q, c \in 0..(b-1) : DidNotVoteIn(a, c) - BY <1>5, ValueNonempty + <2>. PICK v \in Value : TRUE + BY ValueNonempty + <2>. WITNESS v \in Value + <2>. WITNESS -1 \in -1 .. (b-1) + <2>. QED BY <1>5 <1>6. CASE \E a \in Q, c \in 0..(b-1) : ~DidNotVoteIn(a, c) <2>1. PICK c \in 0..(b-1) : /\ \E a \in Q : ~DidNotVoteIn(a, c) @@ -1241,10 +1251,40 @@ THEOREM Liveness == LiveSpec => C!LiveSpec <1>5. Spec /\ LiveAssumption!(Q, b) => <>[](\A self \in Q : maxBal[self] = b) + <2>. DEFINE MB(s) == maxBal[s] = b + <2>0. (\A self \in Q : <>[]MB(self)) => <>[](\A self \in Q : MB(self)) + \* BY <1>a, EventuallyAlwaysForall \* fails, even when hiding the definition of MB + <3>. HIDE DEF MB + <3>. DEFINE A(x) == <>[]MB(x) + L(T) == \A self \in T : A(self) \* NB: changing the names of the bound vars makes the QED step fail! + R(T) == \A self \in T : MB(self) + I(T) == L(T) => <>[]R(T) + <3>1. I({}) + <4>1. R({}) OBVIOUS + <4>. QED BY <4>1, PTL + <3>2. ASSUME NEW T, NEW x + PROVE I(T) => I(T \cup {x}) + <4>1. L(T \cup {x}) => A(x) + <5>. HIDE DEF A + <5>. QED OBVIOUS + <4>2. L(T \cup {x}) /\ I(T) => <>[]R(T) + OBVIOUS + <4>3. <>[]R(T) /\ A(x) => <>[](R(T) /\ MB(x)) + BY PTL + <4>4. R(T) /\ MB(x) => R(T \cup {x}) + OBVIOUS + <4>5. <>[](R(T) /\ MB(x)) => <>[]R(T \cup {x}) + BY <4>4, PTL + <4>. QED BY <4>1, <4>2, <4>3, <4>5 + <3>. HIDE DEF I + <3>3. \A T : IsFiniteSet(T) => I(T) + BY <3>1, <3>2, FS_Induction, IsaM("blast") + <3>4. I(Q) + BY <1>a, <3>3 + <3>. QED BY <3>4 DEF I <2>1. SUFFICES ASSUME NEW self \in Q - PROVE Spec /\ LiveAssumption!(Q, b) => <>[](maxBal[self] = b) -\* BY <1>a, EventuallyAlwaysForall \* doesn't check, even when introducing definitions - PROOF OMITTED + PROVE Spec /\ LiveAssumption!(Q, b) => <>[]MB(self) + BY <2>0, Isa <2> DEFINE P == LInv1 /\ ~(maxBal[self] = b) QQ == LInv1 /\ (maxBal[self] = b) A == BallotAction(self, b) @@ -1264,31 +1304,14 @@ THEOREM Liveness == LiveSpec => C!LiveSpec <4>5. QED BY <4>1, <4>3, <4>4 DEF BallotAction <3>3. P => ENABLED <>_vars - <4>1. (ENABLED <>_vars) <=> - \E votesp, maxBalp: - /\ \/ /\ b > maxBal[self] - /\ maxBalp = [maxBal EXCEPT ![self] = b] - /\ votesp = votes - \/ \E v \in Value : - /\ maxBal[self] \leq b - /\ DidNotVoteIn(self, b) - /\ \A p \in Acceptor \ {self} : - \A w \in Value : VotedFor(p, b, w) => (w = v) - /\ SafeAt(b, v) - /\ votesp = [votes EXCEPT ![self] = votes[self] - \cup {<>}] - /\ maxBalp = [maxBal EXCEPT ![self] = b] - /\ <> # <> -\* BY DEF BallotAction, IncreaseMaxBal, VoteFor, vars, SafeAt, -\* DidNotVoteIn, VotedFor - PROOF OMITTED <4>. SUFFICES ASSUME P PROVE \E votesp, maxBalp: /\ b > maxBal[self] /\ maxBalp = [maxBal EXCEPT ![self] = b] /\ votesp = votes /\ <> # <> - BY <4>1 + BY ExpandENABLED + DEF BallotAction, IncreaseMaxBal, VoteFor, vars, SafeAt, DidNotVoteIn, VotedFor <4> WITNESS votes, [maxBal EXCEPT ![self] = b] <4>. QED BY QA DEF VInv, TypeOK, Ballot <3>. QED BY <3>1, <3>2, <3>3, PTL @@ -1309,7 +1332,7 @@ THEOREM Liveness == LiveSpec => C!LiveSpec <2>. QED BY <2>2, <2>3, <2>4, <2>5, <2>6, <1>2, PTL -<1> DEFINE LNInv2 == \A a \in Q : maxBal[a] = b +<1> DEFINE LNInv2 == \A self \in Q : maxBal[self] = b \* again, the bound var must be called self! LInv2 == VInv /\ LNInv2 <1>6. LInv2 /\ [LNext]_vars => LInv2' @@ -1319,10 +1342,9 @@ THEOREM Liveness == LiveSpec => C!LiveSpec <2> DEFINE Voted(a) == \E v \in Value : VotedFor(a, b, v) <2>1. Spec /\ LiveAssumption!(Q, b) => <>[]LInv2 <3>1. Spec /\ LiveAssumption!(Q,b) => <>[]LNInv2 -\* BY <1>5 \* doesn't check - PROOF OMITTED + BY <1>5, Isa <3>. QED BY <3>1, VT2, PTL - <2>2. LInv2 /\ (\A a \in Q : Voted(a)) => (chosen # {}) + <2>2. LInv2 /\ (\A self \in Q : Voted(self)) => (chosen # {}) <3>1. SUFFICES ASSUME LInv2, \A a \in Q : Voted(a) PROVE chosen # {} @@ -1337,11 +1359,10 @@ THEOREM Liveness == LiveSpec => C!LiveSpec BY <4>3 <3>3. QED BY <3>2 DEF chosen, ChosenIn - <2>3. Spec /\ LiveAssumption!(Q, b) => (\A a \in Q : <>[]Voted(a)) + <2>3. Spec /\ LiveAssumption!(Q, b) => (\A self \in Q : <>[]Voted(self)) <3>1. SUFFICES ASSUME NEW self \in Q PROVE Spec /\ LiveAssumption!(Q, b) => <>[]Voted(self) -\* OBVIOUS \* doesn't check?! - PROOF OMITTED + BY Isa <3>2. Spec /\ LiveAssumption!(Q, b) => <>Voted(self) <4>2. [][LNext]_vars /\ WF_vars(BallotAction(self, b)) => ((LInv2 /\ ~Voted(self)) ~> LInv2 /\ Voted(self)) @@ -1363,40 +1384,21 @@ THEOREM Liveness == LiveSpec => C!LiveSpec <6>4. QED BY <6>1, <6>2, <6>3 DEF BallotAction <5>3. P => ENABLED <>_vars - <6>1. SUFFICES ASSUME P - PROVE ENABLED <>_vars - OBVIOUS - <6>2. (ENABLED <>_vars) <=> - \E votesp, maxBalp : - /\ \/ /\ b > maxBal[self] - /\ maxBalp = [maxBal EXCEPT ![self] = b] - /\ votesp = votes - \/ \E v \in Value : - /\ maxBal[self] \leq b - /\ DidNotVoteIn(self, b) - /\ \A p \in Acceptor \ {self} : - \A w \in Value : VotedFor(p, b, w) => (w = v) - /\ SafeAt(b, v) - /\ votesp = [votes EXCEPT ![self] = votes[self] + <6>1. SUFFICES + ASSUME P + PROVE \E maxBalp, votesp: + /\ \E v \in Value : + /\ maxBal[self] \leq b + /\ DidNotVoteIn(self, b) + /\ \A p \in Acceptor \ {self} : + \A w \in Value : VotedFor(p, b, w) => (w = v) + /\ SafeAt(b, v) + /\ votesp = [votes EXCEPT ![self] = votes[self] \cup {<>}] - /\ maxBalp = [maxBal EXCEPT ![self] = b] - /\ <> # <> -\* BY DEF BallotAction, IncreaseMaxBal, VoteFor, vars, SafeAt, -\* DidNotVoteIn, VotedFor - PROOF OMITTED - <6> SUFFICES - \E votesp, maxBalp: - /\ \E v \in Value : - /\ maxBal[self] \leq b - /\ DidNotVoteIn(self, b) - /\ \A p \in Acceptor \ {self} : - \A w \in Value : VotedFor(p, b, w) => (w = v) - /\ SafeAt(b, v) - /\ votesp = [votes EXCEPT ![self] = votes[self] - \cup {<>}] - /\ maxBalp = [maxBal EXCEPT ![self] = b] - /\ <> # <> - BY <6>2 + /\ maxBalp = [maxBal EXCEPT ![self] = b] + /\ <> # <> + BY ExpandENABLED + DEF BallotAction, IncreaseMaxBal, VoteFor, vars <6> DEFINE someVoted == \E p \in Acceptor \ {self} : \E w \in Value : VotedFor(p, b, w) vp == CHOOSE p \in Acceptor \ {self} : @@ -1412,13 +1414,14 @@ THEOREM Liveness == LiveSpec => C!LiveSpec BY <6>1, <6>3, VT4 DEF VInv, VInv2, Ballot <6> DEFINE votesp == [votes EXCEPT ![self] = votes[self] \cup {<>}] maxBalp == [maxBal EXCEPT ![self] = b] - <6> WITNESS votesp, maxBalp + <6> WITNESS maxBalp, votesp <6> SUFFICES /\ maxBal[self] \leq b /\ DidNotVoteIn(self, b) /\ \A p \in Acceptor \ {self} : \A w \in Value : VotedFor(p, b, w) => (w = v) /\ votesp # votes - BY <6>4, Zenon + <7>. HIDE DEF v + <7>. QED BY <6>4 <6>5. maxBal[self] \leq b BY <6>1 DEF Ballot <6>6. DidNotVoteIn(self, b) @@ -1473,9 +1476,35 @@ THEOREM Liveness == LiveSpec => C!LiveSpec BY <4>1, VT2, PTL DEF Spec <3>4. QED BY <3>2, <3>3, PTL - <2>4. (\A a \in Q : <>[]Voted(a)) => <>[](\A a \in Q : Voted(a)) -\* BY <1>a, EventuallyAlwaysForall \* doesn't check - PROOF OMITTED + <2>4. (\A self \in Q : <>[]Voted(self)) => <>[](\A self \in Q : Voted(self)) + \* again, we need to redo the proof instead of using lemma EventuallyAlwaysForall + <3>. DEFINE A(x) == <>[]Voted(x) + L(T) == \A self \in T : A(self) + R(T) == \A self \in T : Voted(self) + I(T) == L(T) => <>[]R(T) + <3>1. I({}) + <4>1. R({}) OBVIOUS + <4>. QED BY <4>1, PTL + <3>2. ASSUME NEW T, NEW x + PROVE I(T) => I(T \cup {x}) + <4>1. L(T \cup {x}) => A(x) + <5>. HIDE DEF A + <5>. QED OBVIOUS + <4>2. L(T \cup {x}) /\ I(T) => <>[]R(T) + OBVIOUS + <4>3. <>[]R(T) /\ A(x) => <>[](R(T) /\ Voted(x)) + BY PTL + <4>4. R(T) /\ Voted(x) => R(T \cup {x}) + OBVIOUS + <4>5. <>[](R(T) /\ Voted(x)) => <>[]R(T \cup {x}) + BY <4>4, PTL + <4>. QED BY <4>1, <4>2, <4>3, <4>5 + <3>. HIDE DEF I + <3>3. \A T : IsFiniteSet(T) => I(T) + BY <3>1, <3>2, FS_Induction, IsaM("blast") + <3>4. I(Q) + BY <1>a, <3>3 + <3>. QED BY <3>4 DEF I <2>. QED BY <2>1, VT2, <2>2, <2>3, <2>4, PTL @@ -1488,9 +1517,3 @@ THEOREM Liveness == LiveSpec => C!LiveSpec BY <2>2, <1>1, Isa =============================================================================== -\* Modification History -\* Last modified Fri Jul 24 18:20:31 CEST 2020 by merz -\* Last modified Wed Apr 29 12:24:23 CEST 2020 by merz -\* Last modified Mon May 28 08:53:38 PDT 2012 by lamport - - From 762cf476cf45e77539e86e0427708f88c225863c Mon Sep 17 00:00:00 2001 From: Stephan Merz Date: Fri, 27 Jun 2025 16:46:05 +0200 Subject: [PATCH 05/46] more fixed proofs of examples Signed-off-by: Stephan Merz --- README.md | 2 +- .../LoopInvariance/BinarySearch.pdf | Bin 187951 -> 187326 bytes .../LoopInvariance/BinarySearch.tla | 7 +- specifications/LoopInvariance/Quicksort.pdf | Bin 226974 -> 223252 bytes specifications/LoopInvariance/Quicksort.tla | 423 ++++++++++++------ specifications/LoopInvariance/SumSequence.pdf | Bin 202006 -> 199276 bytes specifications/LoopInvariance/SumSequence.tla | 62 ++- specifications/Majority/MajorityProof.tla | 12 +- .../MisraReachability/ParReachProofs.pdf | Bin 144878 -> 148204 bytes .../MisraReachability/ReachabilityProofs.pdf | Bin 133549 -> 132732 bytes .../MisraReachability/ReachabilityProofs.tla | 4 +- .../MisraReachability/ReachableProofs.pdf | Bin 161285 -> 162666 bytes specifications/TeachingConcurrency/Simple.pdf | Bin 193454 -> 196822 bytes specifications/TeachingConcurrency/Simple.tla | 18 +- .../TeachingConcurrency/SimpleRegular.pdf | Bin 192157 -> 195582 bytes .../TeachingConcurrency/SimpleRegular.tla | 18 +- specifications/byzpaxos/PConProof.tla | 1 - 17 files changed, 364 insertions(+), 183 deletions(-) diff --git a/README.md b/README.md index 52bc0fe1..35294519 100644 --- a/README.md +++ b/README.md @@ -19,7 +19,7 @@ Here is a list of specs included in this repository, with links to the relevant | [Loop Invariance](specifications/LoopInvariance) | Leslie Lamport | ✔ | ✔ | ✔ | ✔ | | | [Learn TLA⁺ Proofs](specifications/LearnProofs) | Andrew Helwer | ✔ | ✔ | ✔ | ✔ | | | [Boyer-Moore Majority Vote](specifications/Majority) | Stephan Merz | ✔ | ✔ | | ✔ | | -| [Proof x+x is Even](specifications/sums_even) | Stephan Merz | ✔ | ✔ | | ✔ | | +| [Proof x+x is Even](specifications/sums_even) | Martin Riener | ✔ | ✔ | | ✔ | | | [The N-Queens Puzzle](specifications/N-Queens) | Stephan Merz | ✔ | | ✔ | ✔ | | | [The Dining Philosophers Problem](specifications/DiningPhilosophers) | Jeff Hemphill | ✔ | | ✔ | ✔ | | | [The Car Talk Puzzle](specifications/CarTalkPuzzle) | Leslie Lamport | ✔ | | | ✔ | | diff --git a/specifications/LoopInvariance/BinarySearch.pdf b/specifications/LoopInvariance/BinarySearch.pdf index 54a5ca328ece5352cc2acbebe9873b028424b8a8..44160ad0ee81395ec5a7890d69dd57681d2e5d86 100644 GIT binary patch literal 187326 zcma&MW2`8_x^_El+qP}nSkpGwv~AnAZQHhO+s3yp&Y!z?a!)7Kc`NCplS)-Rqq_#F zf`}L`BON;w>DAg_n?UjLK{+}(m>5_?xos?KS|@I{ApYFw^`DbdMe83V`(gXqah0Dk;jLb8 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zk6ZLJP?)@!TNt6&n!Uz diff --git a/specifications/LoopInvariance/BinarySearch.tla b/specifications/LoopInvariance/BinarySearch.tla index 77871204..48383673 100644 --- a/specifications/LoopInvariance/BinarySearch.tla +++ b/specifications/LoopInvariance/BinarySearch.tla @@ -130,7 +130,7 @@ Inv == /\ TypeOK (***************************************************************************) THEOREM Spec => []resultCorrect <1>1. Init => Inv - BY DEF Init, Inv, TypeOK + BY DEF Init, Inv, TypeOK, SortedSeqs <1>2. Inv /\ [Next]_vars => Inv' <2> SUFFICES ASSUME Inv, [Next]_vars @@ -252,8 +252,3 @@ THEOREM Spec => []resultCorrect <1>4. 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