From 4449b68846db1ca9f46be97fffbaa27657803d4d Mon Sep 17 00:00:00 2001 From: merz Date: Fri, 25 May 2018 11:23:25 +0200 Subject: [PATCH] proofs updated to TLAPM 1.4.4 --- specifications/Bakery-Boulangerie/Bakery.pdf | Bin 209122 -> 197858 bytes specifications/Bakery-Boulangerie/Bakery.tla | 203 +++++++++++++++++- .../Bakery-Boulangerie/Boulanger.pdf | Bin 214068 -> 179425 bytes .../Bakery-Boulangerie/Boulanger.tla | 161 +++++++++++--- 4 files changed, 320 insertions(+), 44 deletions(-) diff --git a/specifications/Bakery-Boulangerie/Bakery.pdf b/specifications/Bakery-Boulangerie/Bakery.pdf index 4b4131474aacf63f86b2feb40d3a1f3aac7d69a3..ec18fa80b0775019be559c6b303a14dc6a09caa6 100644 GIT binary patch literal 197858 zcmb6AQ;;sO)&}afZQHhu*|u$dZQHhO+dbR1ZQC|xyZ1V$_T5?+|ISrXmCIDUPew+_ z6-CAAnHkt&$mf?w)?rwQn279+tzh{0U>IcqcIGY?L@Zn!Y()S60mCS6Y2yNLB4QM` zF>(Qj0!-{p0WbmrFwQPc03%x%kIflv9p|lfbicWJ!+e?+3=H?6*;C@|R^A=QC6~iY z0goMg?IHjL+C`FbPP2dS8PL9(q77x@5+p7dGy$H^nI9ESEh*0TSIo!5-W*m8n=$ec 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zz-uxN7%Q*MC2%M=#=*e_+cAE3=|R*hRqr`*v_)s04^c|djuJJ&27{xeHX=p^{#ZdnT3bemU diff --git a/specifications/Bakery-Boulangerie/Bakery.tla b/specifications/Bakery-Boulangerie/Bakery.tla index d078219d..0ae5e909 100644 --- a/specifications/Bakery-Boulangerie/Bakery.tla +++ b/specifications/Bakery-Boulangerie/Bakery.tla @@ -40,7 +40,7 @@ EXTENDS Naturals, TLAPS (* We first declare N to be the number of processes, and we assume that N *) (* is a natural number. *) (***************************************************************************) -CONSTANT N +CONSTANT N ASSUME N \in Nat (***************************************************************************) @@ -58,7 +58,7 @@ a \prec b == \/ a[1] < b[1] (*** this is a comment containing the PlusCal code * --algorithm Bakery -{ variable num = [i \in Procs |-> 0], flag = [i \in Procs |-> FALSE]; +{ variables num = [i \in Procs |-> 0], flag = [i \in Procs |-> FALSE]; fair process (p \in Procs) variables unchecked = {}, max = 0, nxt = 1 ; { ncs:- while (TRUE) @@ -249,11 +249,11 @@ Before(i,j) == /\ num[i] > 0 (***************************************************************************) Inv == /\ TypeOK /\ \A i \in Procs : - /\ (pc[i] \in {"ncs", "e1", "e2"}) => (num[i] = 0) +\* /\ (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 {(*"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") @@ -273,13 +273,156 @@ Inv == /\ TypeOK (* temporal reasoning. *) (***************************************************************************) THEOREM Spec => []MutualExclusion -<1> USE N \in Nat DEFS Procs, Inv, TypeOK, Before, \prec, ProcSet +<1> USE N \in Nat DEFS Procs, TypeOK, Before, \prec, ProcSet <1>1. Init => Inv - BY SMT DEF Init + BY DEF Init, Inv <1>2. Inv /\ [Next]_vars => Inv' - BY Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <2> SUFFICES ASSUME Inv, + [Next]_vars + PROVE Inv' + OBVIOUS + <2>1. ASSUME NEW self \in Procs, + ncs(self) + PROVE Inv' + BY <2>1 DEF ncs, Inv + <2>2. ASSUME NEW self \in Procs, + e1(self) + PROVE Inv' + <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 + <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 + <3>. QED BY <3>1, <3>2, <2>2 DEF e1 + <2>3. ASSUME NEW self \in Procs, + e2(self) + PROVE Inv' + <3>. /\ pc[self] = "e2" + /\ UNCHANGED << num, flag, nxt >> + BY <2>3 DEF e2 + <3>1. ASSUME NEW i \in unchecked[self], + unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {i}], + num[i] > max[self], + max' = [max EXCEPT ![self] = num[i]], + pc' = [pc EXCEPT ![self] = "e2"] + PROVE Inv' + BY <3>1, Z3T(10) DEF Inv + <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>0. \A ii \in Procs : (pc'[ii] \in {"ncs", "e1", "e2"}) => (num'[ii] = 0) +\* 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 + <3>3. CASE /\ unchecked[self] = {} + /\ pc' = [pc EXCEPT ![self] = "e3"] + /\ UNCHANGED << unchecked, max >> + BY <3>3 DEF Inv + <3>. QED BY <3>1, <3>2, <3>3, <2>3 DEF e2 + <2>4. ASSUME NEW self \in Procs, + e3(self) + PROVE Inv' + <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 + <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, Z3 DEF Inv + <3>3. QED BY <3>1, <3>2, <2>4 DEF e3 + <2>5. ASSUME NEW self \in Procs, + e4(self) + PROVE Inv' + <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 + <3>2. CASE /\ flag' = [flag EXCEPT ![self] = FALSE] + /\ unchecked' = [unchecked EXCEPT ![self] = Procs \ {self}] + /\ pc' = [pc EXCEPT ![self] = "w1"] + BY <3>2 DEF Inv + <3>. QED BY <3>1, <3>2, <2>5 DEF e4 + <2>6. ASSUME NEW self \in Procs, + w1(self) + PROVE Inv' + <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>2. CASE /\ unchecked[self] = {} + /\ pc' = [pc EXCEPT ![self] = "cs"] + /\ nxt' = nxt + BY <3>2, Z3 DEF Inv + <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, Z3 DEF w2, Inv + <2>8. ASSUME NEW self \in Procs, + cs(self) + PROVE Inv' + BY <2>8, Z3 DEF cs, Inv + <2>9. ASSUME NEW self \in Procs, + exit(self) + PROVE Inv' + <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 + <3>2. CASE /\ num' = [num EXCEPT ![self] = 0] + /\ pc' = [pc EXCEPT ![self] = "ncs"] + BY <3>2 DEF Inv + <3>. QED BY <3>1, <3>2, <2>9 DEF exit + <2>10. CASE UNCHANGED vars + BY <2>10 DEF vars, Inv + <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 + BY SMT DEF MutualExclusion, Inv <1>4. QED BY <1>1, <1>2, <1>3, PTL DEF Spec ------------------------------------------------------------------------------ @@ -287,7 +430,47 @@ 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) + +----------------------------------------------------------------------------- +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 + ============================================================================= -\* Modification History -\* Last modified Tue Jul 21 17:55:24 PDT 2015 by lamport +\* Modification History +\* Last modified Fri May 25 11:18:47 CEST 2018 by merz +\* 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 + diff --git a/specifications/Bakery-Boulangerie/Boulanger.pdf b/specifications/Bakery-Boulangerie/Boulanger.pdf index af1238801b77e8e61f55f1545b776e11bea451c4..e3b23dd71044366676ed6f4c4d6dcf2c275e6936 100644 GIT binary patch literal 179425 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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(***************************************************************************) THEOREM Spec => []MutualExclusion -<1> USE N \in Nat DEFS Procs, Inv, TypeOK, Before, \prec, ProcSet +<1> USE N \in Nat DEFS Procs, TypeOK, Before, \prec, ProcSet <1>1. Init => Inv - BY SMT DEF Init + BY SMT DEF Init, Inv <1>2. Inv /\ [Next]_vars => Inv' <2> SUFFICES ASSUME Inv, @@ -369,66 +369,157 @@ THEOREM Spec => []MutualExclusion <2>1. ASSUME NEW self \in Procs, ncs(self) PROVE Inv' - BY <2>1, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + BY <2>1, Z3 DEF ncs, Inv <2>2. ASSUME NEW self \in Procs, e1(self) PROVE Inv' - BY <2>2, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <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 + <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 + <3>. QED BY <3>1, <3>2, <2>2 DEF e1 <2>3. ASSUME NEW self \in Procs, e2(self) PROVE Inv' - BY <2>3, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <3>. /\ pc[self] = "e2" + /\ UNCHANGED << num, flag, nxt, previous >> + BY <2>3 DEF e2 + <3>1. ASSUME NEW i \in unchecked[self], + unchecked' = [unchecked EXCEPT ![self] = unchecked[self] \ {i}], + num[i] > max[self], + max' = [max EXCEPT ![self] = num[i]], + pc' = [pc EXCEPT ![self] = "e2"] + PROVE Inv' + BY <3>1, Z3 DEF Inv + <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 + <3>3. CASE /\ unchecked[self] = {} + /\ pc' = [pc EXCEPT ![self] = "e3"] + /\ UNCHANGED << unchecked, max >> + BY <3>3, Z3 DEF Inv + <3>. QED BY <3>1, <3>2, <3>3, <2>3 DEF e2 <2>4. ASSUME NEW self \in Procs, e3(self) PROVE Inv' - BY <2>4, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <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 + <3>2. CASE /\ num' = [num EXCEPT ![self] = max[self] + 1] + /\ pc' = [pc EXCEPT ![self] = "e4"] + BY <3>2, Z3 DEF Inv + <3>. QED BY <3>1, <3>2, <2>4 DEF e3 <2>5. ASSUME NEW self \in Procs, e4(self) PROVE Inv' - <3> ASSUME NEW i \in Procs', (pc[i] \in {"w1", "w2"})' - PROVE (\A j \in (Procs \ unchecked[i]) \ {i} : Before(i, j))' - <4>1. CASE self = i - <5> SUFFICES ASSUME NEW j \in ((Procs \ unchecked[i]) \ {i})' - PROVE Before(i, j)' - OBVIOUS - <5>1. CASE i < j - BY <4>1, <5>1, <2>5, Z3 DEF ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars - <5>2. CASE j =< i - <6> unchecked'[i] = 1..(i-1) - BY <4>1, <2>5 DEF e4 - <6> QED - BY <4>1, <2>5, Z3 DEF ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars - <5>3. QED - BY <5>1, <5>2 - <4>2. CASE self # i - BY <4>2, <2>5, Z3 DEF ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars - <4>3. QED - BY <4>1, <4>2 - <3> QED - BY <2>5, Z3 DEF ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <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 + <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 + <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 + <3>. QED BY <3>1, <3>2, <3>3, <2>5 DEF e4 <2>6. ASSUME NEW self \in Procs, w1(self) PROVE Inv' - BY <2>6, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <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>2. CASE /\ unchecked[self] = {} + /\ pc' = [pc EXCEPT ![self] = "cs"] + /\ UNCHANGED << nxt, previous >> + BY <3>2, Z3 DEF Inv + <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, Z3 DEF ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <3>. /\ pc[self] = "w2" + /\ UNCHANGED << num, flag, max, nxt >> + BY <2>7 DEF w2 + <3>1. 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] = "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 + \/ <> \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 <2>8. ASSUME NEW self \in Procs, cs(self) PROVE Inv' - BY <2>8, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + BY <2>8, Z3 DEF cs, Inv <2>9. ASSUME NEW self \in Procs, exit(self) PROVE Inv' - BY <2>9, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + <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 <2>10. CASE UNCHANGED vars - BY <2>10, Z3 DEF Next, ncs, p, e1, e2, e3, e4, w1, w2, cs, exit, vars + BY <2>10, Z3 DEF vars, Inv <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 + BY SMT DEF Inv, MutualExclusion <1>4. QED BY <1>1, <1>2, <1>3, PTL DEF Spec @@ -438,6 +529,8 @@ 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 +\* Modification History +\* 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