:: Event-based Proof of the Mutual Exclusion Property of {P}eterson's
:: Algorithm
:: by Ievgen Ivanov , Mykola Nikitchenko and Uri Abraham
::
:: Received August 14, 2015
:: Copyright (c) 2015-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies PETERSON, ABCMIZ_0, WAYBEL_3, RPR_1, MEMSTR_0, SUBSET_1,
XBOOLE_0, RELAT_1, FUNCT_1, XXREAL_0, ARYTM_3, CARD_1, PARTFUN1, TARSKI,
ZFMISC_1, ORDERS_1, ORDERS_2, RELAT_2, STRUCT_0;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, RELAT_1, RELSET_1, FUNCT_1,
SETFAM_1, FUNCT_2, PARTFUN1, STRUCT_0, BINOP_1, ORDERS_1, ORDERS_2,
RELAT_2;
constructors BINOP_1, BINOP_2, STRUCT_0, ORDERS_1, ORDERS_2, DICKSON, RELAT_1,
RELAT_2;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, CARD_1, STRUCT_0,
NECKLA_3;
requirements NUMERALS, BOOLE, SUBSET;
definitions TARSKI, STRUCT_0, BINOP_1, ORDERS_1, ORDERS_2;
theorems ZFMISC_1, RELAT_2, SUBSET_1, ORDERS_1, ORDERS_2;
schemes FUNCT_2;
begin
:: --------------------- Preliminaries ------------------------
definition
struct (1-sorted) Values_with_TF
(#
carrier -> set,
True -> Element of the carrier,
False -> Element of the carrier
#);
end;
definition
let A be Values_with_TF;
attr
A is consistent
means
the True of A <> the False of A;
end;
registration
cluster consistent for Values_with_TF;
existence
proof
set A = {0,1};
0 in A by ZFMISC_1:32;
then consider a be Element of A such that C1: a = 0;
1 in A by ZFMISC_1:32;
then consider b be Element of A such that C2: b = 1;
take L = Values_with_TF(# A, a, b #);
thus thesis by C1,C2;
end;
end;
definition
mode
Values_with_Bool
is
consistent Values_with_TF;
end;
definition
let A be RelStr;
attr
A is strongly_connected
means
the InternalRel of A is_strongly_connected_in the carrier of A;
end;
registration
cluster non empty reflexive transitive strongly_connected for RelStr;
existence
proof
set R = the Order of {{}};
take L = RelStr(# {{}}, the Order of {{}} #);
A1: field the Order of {{}} = the carrier of L by ORDERS_1:12;
thus L is non empty;
thus the InternalRel of L is_reflexive_in the carrier of L
by A1,RELAT_2:def 9;
thus the InternalRel of L is_transitive_in the carrier of L
by A1, RELAT_2:def 16;
thus the InternalRel of L is_strongly_connected_in the carrier of L
by A1, RELAT_2:def 15;
end;
end;
definition
mode
LinearPreorder
is
reflexive transitive strongly_connected RelStr;
end;
:: ---------------------- Main structure -----------------------
definition
let Values be Values_with_Bool;
struct Events_structure over Values
(#
events -> non empty LinearPreorder,
processes -> non empty set,
locations -> non empty set,
traces -> non empty set,
procE -> Function of
the processes,
bool the carrier of the events,
traceE -> Function of
the traces,
bool the carrier of the events,
readE -> Function of
the locations,
bool the carrier of the events,
writeE -> Function of
the locations,
bool the carrier of the events,
val_of -> PartFunc of
the carrier of the events,
the carrier of Values
#);
end;
:: ---------------------- Modes ---------------------------
definition
let Values be Values_with_Bool,
S be Events_structure over Values;
mode
Process of S
is
Element of the processes of S;
mode
Event of S
is
Element of the carrier of the events of S;
mode
EventSet of S
is
Subset of the carrier of the events of S;
mode
Location of S
is
Element of the locations of S;
mode
trace of S
is
Element of the traces of S;
end;
:: ------------------ Reserve ----------------------------------
reserve Values for Values_with_Bool;
reserve a, a1, a2 for Element of the carrier of Values;
reserve S for Events_structure over Values;
reserve p, p1, p2 for Process of S;
reserve x, x1, x2 for Location of S;
reserve tr, tr1, tr2 for trace of S;
reserve e, e0, e1, e2, e3, e4, e5 for Event of S;
reserve E for EventSet of S;
:: ----------------Predicates, functors, notation ---------------
definition
let Values, S, e, x;
pred
e reads x
means
e in (the readE of S) . x;
pred
e writesto x
means
e in (the writeE of S) . x;
end;
definition
let Values, S, x, E;
pred
E reads x
means
ex e st e in E & e reads x;
pred
E writesto x
means
ex e st e in E & e writesto x;
end;
definition
let Values, S, e, tr;
pred
e in tr
means
e in (the traceE of S) . tr;
end;
definition
let Values, S, e, p;
pred
e in p
means
e in (the procE of S) . p;
end;
definition
let Values, S, e;
func
val e
equals
(the val_of of S) . e;
correctness;
end;
definition
let Values, S, e, p, tr;
pred
e in p,tr
means
e in p & e in tr;
end;
definition
let Values, S, e, x, a;
pred
e writesto x,a
means
e writesto x & val e = a;
pred
e reads x,a
means
e reads x & val e = a;
end;
::------------------ Attributes -----------------------
definition
let Values, S;
attr
S is pr-complete
means
for tr, e st
e in tr
holds
ex p st e in p;
attr
S is pr-ordered
means
for p, e1, e2 st
e1 in p &
e2 in p
holds
(e1 <= e2 & e2 <= e1 implies e1 = e2);
attr
S is rw-ordered
means
for x, e1, e2 st
(e1 reads x or e1 writesto x) &
(e2 reads x or e2 writesto x)
holds
(e1 <= e2 & e2 <= e1 implies e1 = e2);
attr
S is rw-consistent
means
for tr, x, e, a st
e in tr & e reads x & val e = a
holds
ex e0 st
e0 in tr & e0 < e & e0 writesto x & val e0 = a &
for e1 st
e1 in tr & e1 <= e & e1 writesto x
holds
e1 <= e0;
attr
S is rw-exclusive
means
for e, x1, x2 holds
not (e reads x1 & e writesto x2);
end;
definition
let Values, S;
attr
S is consistent
means
S is pr-complete pr-ordered rw-ordered
rw-consistent rw-exclusive;
end;
registration
let Values;
cluster consistent for Events_structure over Values;
existence
proof
{} is Element of bool the carrier of the non empty LinearPreorder
by SUBSET_1:1;
then consider E0 being
Element of bool the carrier of the non empty LinearPreorder
such that E0def: E0={};
deffunc ProcE0(Element of the non empty set) = E0;
consider procE0 be Function of the non empty set,
bool the carrier of the non empty LinearPreorder
such that procE0def:
for p being Element of the non empty set holds
procE0 . p = ProcE0(p) from FUNCT_2:sch 4;
deffunc TraceE0(Element of the non empty set) = E0;
consider traceE0 be Function of the non empty set,
bool the carrier of the non empty LinearPreorder
such that traceE0def:
for tr being Element of the non empty set holds
traceE0 . tr = TraceE0(tr) from FUNCT_2:sch 4;
deffunc ReadE0(Element of the non empty set) = E0;
consider readE0 be Function of the non empty set,
bool the carrier of the non empty LinearPreorder
such that readE0def:
for x being Element of the non empty set holds
readE0 . x = ReadE0(x) from FUNCT_2:sch 4;
deffunc WriteE0(Element of the non empty set) = E0;
consider writeE0 be Function of the non empty set,
bool the carrier of the non empty LinearPreorder
such that writeE0def:
for x being Element of the non empty set holds
writeE0 . x = WriteE0(x) from FUNCT_2:sch 4;
take Se = Events_structure(#
the non empty LinearPreorder,
the non empty set,
the non empty set,
the non empty set,
procE0,
traceE0,
readE0,
writeE0,
the PartFunc of the carrier of the non empty LinearPreorder,
the carrier of Values
#);
C1: Se is pr-complete
proof
let tr be trace of Se;
let e be Event of Se such that U1: e in tr;
take the Process of Se;
thus e in the Process of Se by U1, E0def, traceE0def;
end;
C2: Se is pr-ordered
proof
let p be Process of Se;
let e1, e2 be Event of Se such that U1: e1 in p & e2 in p;
thus thesis by U1, E0def, procE0def;
end;
C3: Se is rw-ordered
proof
let x be Location of Se;
let e1, e2 be Event of Se such that
U1: (e1 reads x or e1 writesto x) & (e2 reads x or e2 writesto x);
thus thesis by U1, E0def, readE0def, writeE0def;
end;
C4: Se is rw-consistent
proof
let tr be trace of Se;
let x be Location of Se;
let e be Event of Se;
let a be Element of the carrier of Values such that
U1: e in tr & e reads x & val e = a;
thus thesis by U1, E0def, traceE0def;
end;
Se is rw-exclusive
proof
let e be Event of Se;
let x1, x2 be Location of Se;
thus thesis by E0def, readE0def;
end;
hence thesis by C1,C2,C3,C4;
end;
end;
definition
let Values;
mode
DistributedSysWithSharedMem of Values
is
consistent Events_structure over Values;
end;
:: -------------------------- Peterson's algorithm ---------------------------
reserve DS for DistributedSysWithSharedMem of Values;
reserve p, p1, p2 for Process of DS;
reserve x, x1, x2, flag1, flag2, turn for Location of DS;
reserve tr, tr1, tr2 for trace of DS;
reserve e, e0, e1, e2, e3, e4, e5 for Event of DS;
reserve E for EventSet of DS;
definition
let Values, DS, e1, e2;
pred e1 << e2 means
e1 <= e2 & not (e2 <= e1);
end;
definition
let Values, DS, e1, e2;
func
(e1,e2) inter -> EventSet of DS
equals
{ e : e1 < e & e < e2 };
correctness
proof
defpred P[Event of DS] means e1 < $1 & $1 < e2;
{e: P[e]} c= the carrier of the events of DS
proof
let o be object such that O1: o in {e: P[e]};
consider e such that O2: e = o & P[e] by O1;
thus thesis by O2;
end;
hence thesis;
end;
end;
definition
let Values, DS, e1, e2, p, tr;
func
(e1,e2) interval_in (p,tr) -> EventSet of DS
equals
{ e : e1 < e & e < e2 & e in p,tr };
correctness
proof
defpred P[Event of DS] means e1 < $1 & $1 < e2 & $1 in p,tr;
{e: P[e]} c= the carrier of the events of DS
proof
let o be object such that O1: o in {e: P[e]};
consider e such that O2: e = o & P[e] by O1;
thus thesis by O2;
end;
hence thesis;
end;
end;
theorem
(e1,e2) interval_in (p,tr) c= (e1,e2) inter
proof
let o be object such that O1: o in (e1,e2) interval_in (p,tr);
consider e such that O2: e = o & e1 < e & e < e2 & e in p,tr by O1;
thus thesis by O2;
end;
theorem thLinPreordEvents:
e1 <= e2 or e2 <= e1
proof
the events of DS is strongly_connected;
then [e1,e2] in the InternalRel of the events of DS or
[e2,e1] in the InternalRel of the events of DS by RELAT_2:def 7;
hence e1 <= e2 or e2 <= e1 by ORDERS_2:def 5;
end;
theorem
e in p,tr & e1 < e & e < e2 implies e in (e1,e2) interval_in (p,tr);
theorem
e1 < e2 implies e1 <= e2 by ORDERS_2:def 6;
theorem thEvStrictPrec:
e1 in p & e2 in p & e1 < e2 implies e1 << e2
proof
assume A1: e1 in p & e2 in p;
assume A2: e1 < e2;
DS is consistent;
then DS is pr-ordered;
then not e1 <= e2 or not e2 <= e1 by A1, A2;
hence thesis by A2, ORDERS_2:def 6;
end;
theorem
e1 in p,tr & e2 in p,tr & e1 < e2 implies e1 << e2 by thEvStrictPrec;
theorem
e1 << e2 implies e1 < e2 by ORDERS_2:def 6;
theorem
e1 in p & e2 in p implies e1 = e2 or e1 << e2 or e2 << e1
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
theorem thEvTrans:
e1 <= e2 & e2 <= e3 implies e1 <= e3
proof
[e1,e2] in the InternalRel of the events of DS &
[e2,e3] in the InternalRel of the events of DS
implies
[e1,e3] in the InternalRel of the events of DS
by ORDERS_2:def 3, RELAT_2:def 8;
hence thesis by ORDERS_2:def 5;
end;
theorem thEvStrictTrans1:
e1 <= e2 & e2 << e3 implies e1 << e3 by thEvTrans;
theorem thEvStrictTrans2:
e1 << e2 & e2 <= e3 implies e1 << e3 by thEvTrans;
theorem
e1 << e2 & e2 << e3 implies e1 << e3 by thEvTrans;
definition
let Values, DS, e1, e2;
:: simultaneous events
pred simultev e1, e2 means e1 <= e2 & e2 <= e1;
end;
theorem
not simultev e1, e2 implies e1 << e2 or e2 << e1 by thLinPreordEvents;
definition
let Values, DS, p, tr, e, x1, x2, turn, a1, a2;
pred e has_Peterson_mutex_in p,x1,x2,turn,a1,a2,tr means
ex e1, e2, e3 st
e1 in p,tr & e2 in p,tr & e3 in p,tr &
e1 < e2 & e2 < e3 & e3 < e &
e1 writesto x1,the True of Values &
not (e1,e) interval_in (p,tr) writesto x1 &
e2 writesto turn,a2 &
not (e2,e) interval_in (p,tr) writesto turn &
(e3 reads x2,the False of Values or e3 reads turn,a1);
end;
definition
let Values, DS, tr;
let Es be set;
pred Es has_Peterson_mutex_in tr means
ex p1, p2 st
((for p being Process of DS holds p = p1 or p = p2) &
ex flag1, flag2, turn st
(for e st e in p1,tr holds
not e writesto flag2 &
not e writesto turn,the False of Values) &
(for e st e in p2,tr holds
not e writesto flag1 &
not e writesto turn, the True of Values) &
for e st e in Es holds
e has_Peterson_mutex_in p1, flag1, flag2, turn,
the False of Values, the True of Values, tr &
e has_Peterson_mutex_in p2, flag2, flag1, turn,
the True of Values, the False of Values, tr);
end;
theorem lemwbefr:
e1 in tr & e2 in tr &
e1 reads x,a1 & e2 reads x,a2 & e1<=e2 & a1<>a2
implies
ex e st
e in tr & e1 << e & e << e2 & e writesto x,a2
proof
assume A0: e1 in tr & e2 in tr;
assume A1: e1 reads x,a1 & e2 reads x,a2 & e1<=e2 & a1<>a2;
DS is consistent;
then DS is rw-consistent;
then consider e2w being Event of DS such that
B1: e2w in tr & e2w < e2 & e2w writesto x & val e2w = a2 &
for e st e in tr & e <= e2 & e writesto x
holds e <= e2w by A0,A1;
DS is consistent;
then JJ0: DS is rw-ordered;
C1: not e2w << e1
proof
assume D1: e2w << e1;
DS is consistent;
then DS is rw-consistent;
then consider e1w being Event of DS such that
D2: e1w in tr & e1w < e1 & e1w writesto x & val e1w = a1 &
for e st e in tr & e <= e1 & e writesto x
holds e <= e1w by A0,A1;
D3: e2w <= e1w by D1,D2,B1;
e1w <= e1 & e1 <= e2 by A1, D2, ORDERS_2:def 6;
then e1w <= e2 by thEvTrans;
then D4: e1w <= e2w by B1,D2;
DS is consistent;
then DS is rw-ordered;
hence contradiction by B1,D2,A1,D3,D4;
end;
C2: not e2 << e2w by B1,ORDERS_2:def 6;
e1 << e2w & e2w << e2 & e2w writesto x,a2
by thLinPreordEvents,JJ0,A1,C1,C2,B1;
hence thesis by B1;
end;
::$N Main result: Mutual exclusion property of Peterson's algorithm
theorem
e1 in tr & e2 in tr & {e1, e2} has_Peterson_mutex_in tr
implies
e1 = e2 or e1 << e2 or e2 << e1
proof
assume U0: e1 in tr & e2 in tr;
assume {e1, e2} has_Peterson_mutex_in tr;
then consider p1, p2 such that
U1: for p being Process of DS holds p = p1 or p = p2
and
U2: ex flag1, flag2, turn st
(for e st e in p1,tr holds
not e writesto flag2 &
not e writesto turn,the False of Values) &
(for e st e in p2,tr holds
not e writesto flag1 &
not e writesto turn,the True of Values) &
for e st e in {e1,e2} holds
e has_Peterson_mutex_in p1, flag1, flag2, turn,
the False of Values, the True of Values, tr &
e has_Peterson_mutex_in p2, flag2, flag1, turn,
the True of Values, the False of Values, tr;
consider flag1, flag2, turn such that
U3nw: (for e st e in p1,tr holds
not e writesto flag2 & not e writesto turn,the False of Values)
& (for e st e in p2,tr holds
not e writesto flag1 & not e writesto turn,the True of Values)
and
U3: for e st e in {e1,e2} holds
e has_Peterson_mutex_in p1, flag1, flag2, turn,
the False of Values, the True of Values, tr &
e has_Peterson_mutex_in p2, flag2, flag1, turn,
the True of Values, the False of Values, tr by U2;
{e1} c= {e1,e2} & {e2} c= {e1,e2} by ZFMISC_1:36;
then e1 in {e1,e2} & e2 in {e1,e2} by ZFMISC_1:31;
then U4:
e1 has_Peterson_mutex_in p1, flag1, flag2, turn,
the False of Values, the True of Values, tr &
e1 has_Peterson_mutex_in p2, flag2, flag1, turn,
the True of Values, the False of Values, tr &
e2 has_Peterson_mutex_in p1, flag1, flag2, turn,
the False of Values, the True of Values, tr &
e2 has_Peterson_mutex_in p2, flag2, flag1, turn,
the True of Values, the False of Values, tr
by U3;
assume Aneq: not e1 = e2;
W1: (e1 in p1 & e2 in p1) or (e1 in p2 & e2 in p2)
implies
e1 << e2 or e2 << e1
by Aneq,thLinPreordEvents,thEvStrictPrec,ORDERS_2:def 6;
DS is consistent;
then DS is pr-complete;
then W2: (ex p being Process of DS st e1 in p) &
(ex p being Process of DS st e2 in p) by U0;
W3: not (e1 in p1 & e2 in p2 & simultev e1, e2)
proof
assume A0: e1 in p1 & e2 in p2;
assume A0s: simultev e1, e2;
consider u1 being Event of DS, w1 being Event of DS, r1 being Event of DS
such that
V1: u1 in p1,tr & w1 in p1,tr & r1 in p1,tr &
u1 < w1 & w1 < r1 & r1 < e1 &
u1 writesto flag1,the True of Values &
not (u1,e1) interval_in (p1,tr) writesto flag1 &
w1 writesto turn,the True of Values &
not (w1,e1) interval_in (p1,tr) writesto turn &
(r1 reads flag2,the False of Values or
r1 reads turn,the False of Values) by U4;
V1o: u1 << w1 & w1 << r1 & r1 << e1 by V1, A0, thEvStrictPrec;
consider u2 being Event of DS, w2 being Event of DS, r2 being Event of DS
such that
V2: u2 in p2,tr & w2 in p2,tr & r2 in p2,tr &
u2 < w2 & w2 < r2 & r2 < e2 &
u2 writesto flag2,the True of Values &
not (u2,e2) interval_in (p2,tr) writesto flag2 &
w2 writesto turn,the False of Values &
not (w2,e2) interval_in (p2,tr) writesto turn &
(r2 reads flag1,the False of Values or
r2 reads turn,the True of Values) by U4;
V2o: u2 << w2 & w2 << r2 & r2 << e2 by V2, A0, thEvStrictPrec;
RR1: not r1 <= r2
proof
assume D0: r1 <= r2;
D01: u1 << r2 & r2 << e1
by V1o,thEvTrans,D0,A0s,V2,A0,thEvStrictPrec,thEvStrictTrans2;
D1: r2 in tr & r2 reads turn,the True of Values
proof
not r2 reads flag1,the False of Values
proof
assume Y1: r2 reads flag1,the False of Values;
DS is consistent;
then DS is rw-consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2w < r2 & r2w writesto flag1 &
val r2w = the False of Values &
for e st e in tr & e <= r2 & e writesto flag1
holds e <= r2w by Y1,V2;
u1 = r2w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then TA01: r2w in p1,tr or r2w in p2,tr by U1,Y2;
TA02: u1 < r2w & r2w < e1
implies
r2w in (u1,e1) interval_in (p1,tr) by U3nw,Y2,TA01;
TB01: r2w << e1 by D01,thEvStrictTrans1,ORDERS_2:def 6,Y2;
u1 in p1,tr & r2w in p1,tr & not (u1 << r2w)
by TB01,Y2,TA02,TA01,U3nw,V1,ORDERS_2:def 6;
then u1 = r2w or r2w << u1
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V1,D01;
end;
then val r2w = the True of Values & Values is consistent by V1;
hence contradiction by Y2;
end;
hence thesis by V2;
end;
D2: r1 reads flag2,the False of Values
proof
not r1 reads turn,the False of Values
proof
assume Y1: r1 reads turn,the False of Values;
Values is consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r1 << r2w & r2w << r2 &
r2w writesto turn,the True of Values by lemwbefr,V1,D1,D0,Y1;
Y3: r2w in p1,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then (r2w in p1,tr or r2w in p2,tr) &
r2w writesto turn,the True of Values &
Values is consistent by U1,Y2;
hence thesis by U3nw;
end;
r2w << r2 & r2 <= e2 by ORDERS_2:def 6,Y2,V2;
then IA0: r2w << e2 by thEvTrans;
IB0: w1 <= r1 & r1 << r2w by Y2, V1, ORDERS_2:def 6;
w1 < r2w & r2w < e1 by thEvTrans,A0s,IA0,IB0,ORDERS_2:def 6;
then r2w in (w1,e1) interval_in (p1,tr) by Y3;
hence contradiction by V1,Y2;
end;
hence thesis by V1;
end;
DS is consistent;
then DS is rw-consistent;
then consider wr2 be Event of DS such that
H1: wr2 in tr & wr2 < r2 & wr2 writesto turn &
val wr2 = the True of Values &
for e1 st
e1 in tr & e1 <= r2 & e1 writesto turn
holds
e1 <= wr2 by D1;
H2: wr2 in p1,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st wr2 in p by H1;
then (wr2 in p1,tr or wr2 in p2,tr) &
wr2 writesto turn,the True of Values &
Values is consistent by U1,H1;
hence thesis by U3nw;
end;
M1: wr2 << r1
proof
r2 << e1 by V2,A0,thEvStrictPrec,A0s,thEvStrictTrans2;
then J1: wr2 << e1 by ORDERS_2:def 6,H1,thEvStrictTrans1;
J2: not r1 << wr2
proof
assume r1 << wr2;
then w1 << wr2 & wr2 < e1 by ORDERS_2:def 6,J1,V1,thEvStrictTrans1;
then w1 < wr2 & wr2 < e1 by ORDERS_2:def 6;
then wr2 in (w1,e1) interval_in (p1,tr) by H2;
hence contradiction by V1,H1;
end;
DS is consistent;
then J30: DS is pr-ordered;
DS is consistent;
then DS is rw-exclusive;
then not r1 = wr2 by D2,H1;
hence wr2 << r1 by J2,H2,V1,thLinPreordEvents,J30;
end;
Q0: not (u2 <= r1 & r1 <= e2)
proof
assume Q0a: u2 <= r1 & r1 <= e2;
DS is consistent;
then DS is rw-consistent;
then consider r1w being Event of DS such that
Y2: r1w in tr & r1w < r1 & r1w writesto flag2 &
val r1w = the False of Values &
for e st e in tr & e <= r1 & e writesto flag2
holds e <= r1w by V1,D2;
u2 = r1w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r1w in p by Y2;
then TA01: r1w in p1,tr or r1w in p2,tr by U1,Y2;
TA02: u2 < r1w & r1w < e2
implies
r1w in (u2,e2) interval_in (p2,tr) by TA01,U3nw,Y2;
r1w <= r1 & r1 << e1 by V1,A0,thEvStrictPrec,Y2,ORDERS_2:def 6;
then r1w <= r1 & r1 << e2 by A0s, thEvTrans;
then u2 in p2,tr & r1w in p2,tr & not (u2 << r1w)
by Y2,TA01,U3nw,V2,TA02,thEvTrans,ORDERS_2:def 6;
then u2 = r1w or r1w << u2
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V2,Q0a;
end;
then val r1w = the True of Values & Values is consistent by V2;
hence contradiction by Y2;
end;
M20: r1 << e1 by V1,A0,thEvStrictPrec;
u2 << w2 & w2 <= wr2 & wr2 << r1
by M1, V2, thEvStrictPrec, H1,ORDERS_2:def 6;
then u2 << wr2 & wr2 << r1 by thEvTrans;
hence contradiction by M20,Q0,A0s,thEvTrans;
end;
not r2 <= r1
proof
assume D0: r2 <= r1;
D01: u2 << r1 & r1 << e2
by V2o,thEvTrans,D0,A0s,V1,A0,thEvStrictPrec,thEvStrictTrans2;
D1: r1 in tr & r1 reads turn,the False of Values
proof
not r1 reads flag2,the False of Values
proof
assume Y1: r1 reads flag2,the False of Values;
DS is consistent;
then DS is rw-consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2w < r1 & r2w writesto flag2 &
val r2w = the False of Values &
for e st e in tr & e <= r1 & e writesto flag2
holds e <= r2w by Y1,V1;
u2 = r2w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then TA01: r2w in p2,tr or r2w in p1,tr by U1,Y2;
TA02: u2 < r2w & r2w < e2
implies
r2w in (u2,e2) interval_in (p2,tr) by U3nw,Y2,TA01;
TB01: r2w << e2 by D01,thEvStrictTrans1,ORDERS_2:def 6,Y2;
u2 in p2,tr & r2w in p2,tr & not (u2 << r2w)
by TB01,Y2,TA02,TA01,U3nw,V2,ORDERS_2:def 6;
then u2 = r2w or r2w << u2
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V2,D01;
end;
then val r2w = the True of Values & Values is consistent by V2;
hence contradiction by Y2;
end;
hence thesis by V1;
end;
D2: r2 reads flag1,the False of Values
proof
not r2 reads turn,the True of Values
proof
assume Y1: r2 reads turn,the True of Values;
Values is consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2 << r2w & r2w << r1 &
r2w writesto turn,the False of Values by lemwbefr,V2,D1,D0,Y1;
Y3: r2w in p2,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then (r2w in p2,tr or r2w in p1,tr) &
r2w writesto turn,the False of Values &
Values is consistent by U1,Y2;
hence thesis by U3nw;
end;
r2w << r1 & r1 <= e1 by ORDERS_2:def 6,Y2,V1;
then IA0: r2w << e1 by thEvTrans;
IB0: w2 <= r2 & r2 << r2w by Y2, V2, ORDERS_2:def 6;
w2 < r2w & r2w < e2 by thEvTrans,A0s,IA0,IB0,ORDERS_2:def 6;
then r2w in (w2,e2) interval_in (p2,tr) by Y3;
hence contradiction by V2,Y2;
end;
hence thesis by V2;
end;
DS is consistent;
then DS is rw-consistent;
then consider wr2 be Event of DS such that
H1: wr2 in tr & wr2 < r1 & wr2 writesto turn &
val wr2 = the False of Values &
for e2 st
e2 in tr & e2 <= r1 & e2 writesto turn
holds
e2 <= wr2 by D1;
H2: wr2 in p2,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st wr2 in p by H1;
then (wr2 in p2,tr or wr2 in p1,tr) &
wr2 writesto turn,the False of Values &
Values is consistent by U1,H1;
hence thesis by U3nw;
end;
M1: wr2 << r2
proof
r1 << e2 by V1,A0,thEvStrictPrec,A0s,thEvStrictTrans2;
then J1: wr2 << e2 by ORDERS_2:def 6,H1,thEvStrictTrans1;
J2: not r2 << wr2
proof
assume r2 << wr2;
then w2 << wr2 & wr2 < e2 by ORDERS_2:def 6,J1,V2,thEvStrictTrans1;
then w2 < wr2 & wr2 < e2 by ORDERS_2:def 6;
then wr2 in (w2,e2) interval_in (p2,tr) by H2;
hence contradiction by V2,H1;
end;
DS is consistent;
then J30: DS is pr-ordered;
DS is consistent;
then DS is rw-exclusive;
then not r2 = wr2 by D2,H1;
hence wr2 << r2 by J2,H2,V2,thLinPreordEvents,J30;
end;
Q0: not (u1 <= r2 & r2 <= e1)
proof
assume Q0a: u1 <= r2 & r2 <= e1;
DS is consistent;
then DS is rw-consistent;
then consider r1w being Event of DS such that
Y2: r1w in tr & r1w < r2 & r1w writesto flag1 &
val r1w = the False of Values &
for e st e in tr & e <= r2 & e writesto flag1
holds e <= r1w by V2,D2;
u1 = r1w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r1w in p by Y2;
then TA01: r1w in p2,tr or r1w in p1,tr by U1,Y2;
TA02: u1 < r1w & r1w < e1 implies
r1w in (u1,e1) interval_in (p1,tr) by TA01,U3nw,Y2;
r1w <= r2 & r2 << e2 by V2,A0,thEvStrictPrec,Y2,ORDERS_2:def 6;
then r1w <= r2 & r2 << e1 by A0s, thEvTrans;
then u1 in p1,tr & r1w in p1,tr & not (u1 << r1w)
by Y2,TA01,U3nw,V1,TA02,thEvTrans,ORDERS_2:def 6;
then u1 = r1w or r1w << u1
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V1,Q0a;
end;
then val r1w = the True of Values & Values is consistent by V1;
hence contradiction by Y2;
end;
M20: r2 << e1 by V2, A0s, A0, thEvStrictPrec, thEvStrictTrans2;
u1 << w1 & w1 <= wr2 & wr2 << r2
by M1,thEvStrictPrec,V1,H1,ORDERS_2:def 6;
then u1 << wr2 & wr2 << r2 by thEvTrans;
hence contradiction by M20,Q0,thEvTrans;
end;
hence contradiction by thLinPreordEvents, RR1;
end;
not (e1 in p2 & e2 in p1 & simultev e1, e2)
proof
assume A0: e1 in p2 & e2 in p1;
assume A0s: simultev e1, e2;
consider u1 being Event of DS, w1 being Event of DS, r1 being Event of DS
such that
V1: u1 in p2,tr & w1 in p2,tr & r1 in p2,tr &
u1 < w1 & w1 < r1 & r1 < e1 &
u1 writesto flag2,the True of Values &
not (u1,e1) interval_in (p2,tr) writesto flag2 &
w1 writesto turn,the False of Values &
not (w1,e1) interval_in (p2,tr) writesto turn &
(r1 reads flag1,the False of Values or
r1 reads turn,the True of Values) by U4;
V1o: u1 << w1 & w1 << r1 & r1 << e1 by V1, A0, thEvStrictPrec;
consider u2 being Event of DS, w2 being Event of DS, r2 being Event of DS
such that
V2: u2 in p1,tr & w2 in p1,tr & r2 in p1,tr &
u2 < w2 & w2 < r2 & r2 < e2 &
u2 writesto flag1,the True of Values &
not (u2,e2) interval_in (p1,tr) writesto flag1 &
w2 writesto turn,the True of Values &
not (w2,e2) interval_in (p1,tr) writesto turn &
(r2 reads flag2,the False of Values or
r2 reads turn,the False of Values)
by U4;
V2o: u2 << w2 & w2 << r2 & r2 << e2 by V2, A0, thEvStrictPrec;
RR1: not r1 <= r2
proof
assume D0: r1 <= r2;
D01: u1 << r2 & r2 << e1
by V1o,thEvTrans,D0,A0s,V2,A0,thEvStrictPrec,thEvStrictTrans2;
D1: r2 in tr & r2 reads turn,the False of Values
proof
not r2 reads flag2,the False of Values
proof
assume Y1: r2 reads flag2,the False of Values;
DS is consistent;
then DS is rw-consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2w < r2 & r2w writesto flag2 &
val r2w = the False of Values &
for e st e in tr & e <= r2 & e writesto flag2
holds e <= r2w by Y1,V2;
u1 = r2w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then TA01: r2w in p2,tr or r2w in p1,tr by U1,Y2;
TA02: u1 < r2w & r2w < e1
implies
r2w in (u1,e1) interval_in (p2,tr) by U3nw,Y2,TA01;
TB01: r2w << e1 by D01,thEvStrictTrans1,ORDERS_2:def 6,Y2;
u1 in p2,tr & r2w in p2,tr & not (u1 << r2w)
by TB01,Y2,TA02,TA01,U3nw,V1,ORDERS_2:def 6;
then u1 = r2w or r2w << u1
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V1,D01;
end;
then val r2w = the True of Values & Values is consistent by V1;
hence contradiction by Y2;
end;
hence thesis by V2;
end;
D2: r1 reads flag1,the False of Values
proof
not r1 reads turn,the True of Values
proof
assume Y1: r1 reads turn,the True of Values;
Values is consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r1 << r2w & r2w << r2 &
r2w writesto turn,the False of Values by lemwbefr,V1,D1,D0,Y1;
Y3: r2w in p2,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then (r2w in p2,tr or r2w in p1,tr) &
r2w writesto turn,the False of Values &
Values is consistent by U1,Y2;
hence thesis by U3nw;
end;
r2w << r2 & r2 <= e2 by ORDERS_2:def 6,Y2,V2;
then IA0: r2w << e2 by thEvTrans;
IB0: w1 <= r1 & r1 << r2w by Y2, V1, ORDERS_2:def 6;
w1 < r2w & r2w < e1 by thEvTrans,A0s,IA0,IB0,ORDERS_2:def 6;
then r2w in (w1,e1) interval_in (p2,tr) by Y3;
hence contradiction by V1,Y2;
end;
hence thesis by V1;
end;
DS is consistent;
then DS is rw-consistent;
then consider wr2 be Event of DS such that
H1: wr2 in tr & wr2 < r2 & wr2 writesto turn &
val wr2 = the False of Values &
for e1 st
e1 in tr & e1 <= r2 & e1 writesto turn
holds
e1 <= wr2 by D1;
H2: wr2 in p2,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st wr2 in p by H1;
then (wr2 in p2,tr or wr2 in p1,tr) &
wr2 writesto turn,the False of Values &
Values is consistent by U1,H1;
hence thesis by U3nw;
end;
M1: wr2 << r1
proof
r2 << e1 by V2,A0,thEvStrictPrec,A0s,thEvStrictTrans2;
then J1: wr2 << e1 by ORDERS_2:def 6,H1,thEvStrictTrans1;
J2: not r1 << wr2
proof
assume r1 << wr2;
then w1 << wr2 & wr2 < e1 by ORDERS_2:def 6,J1,V1,thEvStrictTrans1;
then w1 < wr2 & wr2 < e1 by ORDERS_2:def 6;
then wr2 in (w1,e1) interval_in (p2,tr) by H2;
hence contradiction by V1,H1;
end;
DS is consistent;
then J30: DS is pr-ordered;
DS is consistent;
then DS is rw-exclusive;
then not r1 = wr2 by D2,H1;
hence wr2 << r1 by J2,H2,V1,thLinPreordEvents,J30;
end;
Q0: not (u2 <= r1 & r1 <= e2)
proof
assume Q0a: u2 <= r1 & r1 <= e2;
DS is consistent;
then DS is rw-consistent;
then consider r1w being Event of DS such that
Y2: r1w in tr & r1w < r1 & r1w writesto flag1 &
val r1w = the False of Values &
for e st e in tr & e <= r1 & e writesto flag1
holds e <= r1w by V1,D2;
u2 = r1w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r1w in p by Y2;
then TA01: r1w in p2,tr or r1w in p1,tr by U1,Y2;
TA02: u2 < r1w & r1w < e2
implies
r1w in (u2,e2) interval_in (p1,tr) by TA01,U3nw,Y2;
r1w <= r1 & r1 << e1 by V1,A0,thEvStrictPrec,Y2,ORDERS_2:def 6;
then r1w <= r1 & r1 << e2 by A0s, thEvTrans;
then u2 in p1,tr & r1w in p1,tr & not (u2 << r1w)
by Y2,TA01,U3nw,V2,TA02,thEvTrans,ORDERS_2:def 6;
then u2 = r1w or r1w << u2
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V2,Q0a;
end;
then val r1w = the True of Values & Values is consistent by V2;
hence contradiction by Y2;
end;
M20: r1 << e1 by V1,A0,thEvStrictPrec;
u2 << w2 & w2 <= wr2 & wr2 << r1
by M1, V2, thEvStrictPrec, H1,ORDERS_2:def 6;
then u2 << wr2 & wr2 << r1 by thEvTrans;
hence contradiction by M20,Q0,A0s,thEvTrans;
end;
not r2 <= r1
proof
assume D0: r2 <= r1;
D01: u2 << r1 & r1 << e2
by V2o,thEvTrans,D0,A0s,V1,A0,thEvStrictPrec,thEvStrictTrans2;
D1: r1 in tr & r1 reads turn,the True of Values
proof
not r1 reads flag1,the False of Values
proof
assume Y1: r1 reads flag1,the False of Values;
DS is consistent;
then DS is rw-consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2w < r1 & r2w writesto flag1 &
val r2w = the False of Values &
for e st e in tr & e <= r1 & e writesto flag1
holds e <= r2w by Y1,V1;
u2 = r2w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then TA01: r2w in p1,tr or r2w in p2,tr by U1,Y2;
TA02: u2 < r2w & r2w < e2
implies
r2w in (u2,e2) interval_in (p1,tr) by U3nw,Y2,TA01;
TB01: r2w << e2 by D01,thEvStrictTrans1,ORDERS_2:def 6,Y2;
u2 in p1,tr & r2w in p1,tr & not (u2 << r2w)
by TB01,Y2,TA02,TA01,U3nw,V2,ORDERS_2:def 6;
then u2 = r2w or r2w << u2
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V2,D01;
end;
then val r2w = the True of Values & Values is consistent by V2;
hence contradiction by Y2;
end;
hence thesis by V1;
end;
D2: r2 reads flag2,the False of Values
proof
not r2 reads turn,the False of Values
proof
assume Y1: r2 reads turn,the False of Values;
Values is consistent;
then consider r2w being Event of DS such that
Y2: r2w in tr & r2 << r2w & r2w << r1 &
r2w writesto turn,the True of Values by lemwbefr,V2,D1,D0,Y1;
Y3: r2w in p1,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st r2w in p by Y2;
then (r2w in p1,tr or r2w in p2,tr) &
r2w writesto turn,the True of Values &
Values is consistent by U1,Y2;
hence thesis by U3nw;
end;
r2w << r1 & r1 <= e1 by ORDERS_2:def 6,Y2,V1;
then IA0: r2w << e1 by thEvTrans;
IB0: w2 <= r2 & r2 << r2w by Y2, V2, ORDERS_2:def 6;
w2 < r2w & r2w < e2 by thEvTrans,A0s,IA0,IB0,ORDERS_2:def 6;
then r2w in (w2,e2) interval_in (p1,tr) by Y3;
hence contradiction by V2,Y2;
end;
hence thesis by V2;
end;
DS is consistent;
then DS is rw-consistent;
then consider wr2 be Event of DS such that
H1: wr2 in tr & wr2 < r1 & wr2 writesto turn &
val wr2 = the True of Values &
for e2 st
e2 in tr & e2 <= r1 & e2 writesto turn
holds
e2 <= wr2 by D1;
H2: wr2 in p1,tr
proof
DS is consistent;
then DS is pr-complete;
then ex p st wr2 in p by H1;
then (wr2 in p1,tr or wr2 in p2,tr) &
wr2 writesto turn,the True of Values &
Values is consistent by U1,H1;
hence thesis by U3nw;
end;
M1: wr2 << r2
proof
r1 << e2 by V1,A0,thEvStrictPrec,A0s,thEvStrictTrans2;
then J1: wr2 << e2 by ORDERS_2:def 6,H1,thEvStrictTrans1;
J2: not r2 << wr2
proof
assume r2 << wr2;
then w2 << wr2 & wr2 < e2 by ORDERS_2:def 6,J1,V2,thEvStrictTrans1;
then w2 < wr2 & wr2 < e2 by ORDERS_2:def 6;
then wr2 in (w2,e2) interval_in (p1,tr) by H2;
hence contradiction by V2,H1;
end;
DS is consistent;
then J30: DS is pr-ordered;
DS is consistent;
then DS is rw-exclusive;
then not r2 = wr2 by D2,H1;
hence wr2 << r2 by J2,H2,V2,thLinPreordEvents,J30;
end;
Q0: not (u1 <= r2 & r2 <= e1)
proof
assume Q0a: u1 <= r2 & r2 <= e1;
DS is consistent;
then DS is rw-consistent;
then consider r1w being Event of DS such that
Y2: r1w in tr & r1w < r2 & r1w writesto flag2 &
val r1w = the False of Values &
for e st e in tr & e <= r2 & e writesto flag2
holds e <= r1w by V2,D2;
u1 = r1w
proof
DS is consistent;
then DS is pr-complete;
then ex p st r1w in p by Y2;
then TA01: r1w in p1,tr or r1w in p2,tr by U1,Y2;
TA02: u1 < r1w & r1w < e1
implies
r1w in (u1,e1) interval_in (p2,tr) by TA01,U3nw,Y2;
r1w <= r2 & r2 << e2 by V2,A0,thEvStrictPrec,Y2,ORDERS_2:def 6;
then r1w <= r2 & r2 << e1 by A0s, thEvTrans;
then u1 in p2,tr & r1w in p2,tr & not (u1 << r1w)
by Y2,TA01,U3nw,V1,TA02,thEvTrans,ORDERS_2:def 6;
then u1 = r1w or r1w << u1
by thLinPreordEvents, thEvStrictPrec, ORDERS_2:def 6;
hence thesis by Y2,V1,Q0a;
end;
then val r1w = the True of Values & Values is consistent by V1;
hence contradiction by Y2;
end;
M20: r2 << e1 by V2, A0s, A0, thEvStrictPrec, thEvStrictTrans2;
u1 << w1 & w1 <= wr2 & wr2 << r2
by M1,thEvStrictPrec,V1,H1,ORDERS_2:def 6;
then u1 << wr2 & wr2 << r2 by thEvTrans;
hence contradiction by M20,Q0,thEvTrans;
end;
hence contradiction by thLinPreordEvents, RR1;
end;
hence thesis by thLinPreordEvents, W1, W2, U1, W3;
end;