SCM_HALT: Initialization Halting Concepts and Their Basic Properties of SCM+FSA

:: Initialization Halting Concepts and Their Basic Properties of SCM+FSA
:: by JingChao Chen and Yatsuka Nakamura
::
:: Received June 17, 1998
:: Copyright (c) 1998-2011 Association of Mizar Users

begin

set SA0 = Start-At (0,SCM+FSA);

set iS = Initialize ((intloc 0) .--> 1);

reconsider EP = {} as PartState of SCM+FSA by FUNCT_1:174, RELAT_1:206;

IC in dom (Start-At (0,SCM+FSA))
by COMPOS_1:52;

then ICiS: IC (Initialize ((intloc 0) .--> 1)) = IC (Start-At (0,SCM+FSA)) by FUNCT_4:14
.= 0 by COMPOS_1:64 ;

zSA0: Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1)
by FUNCT_4:26;

diS: dom (Initialize ((intloc 0) .--> 1)) = (dom ((intloc 0) .--> 1)) \/ (dom (Start-At (0,SCM+FSA))) by FUNCT_4:def 1
.= {(intloc 0)} \/ (dom (Start-At (0,SCM+FSA))) by FUNCOP_1:19
.= {(intloc 0)} \/ {(IC )} by FUNCOP_1:19 ;

then ddiS: dom (Initialize ((intloc 0) .--> 1)) = {(intloc 0),(IC )}
by ENUMSET1:41;

intloc 0 in {(intloc 0)}
by TARSKI:def 1;

then iniS: intloc 0 in dom (Initialize ((intloc 0) .--> 1))
by diS, XBOOLE_0:def 3;

AA: intloc 0 <> IC
by SCMFSA_2:81;

dom (Start-At (0,SCM+FSA)) = {(IC )}
by FUNCOP_1:19;

then not intloc 0 in dom (Start-At (0,SCM+FSA))
by AA, TARSKI:def 1;

then iSint: (Initialize ((intloc 0) .--> 1)) . (intloc 0) = ((intloc 0) .--> 1) . (intloc 0) by FUNCT_4:12
.= 1 by FUNCOP_1:87 ;

definition

let I be Program of SCM+FSA;
attr I is InitClosed means :Def1: :: SCM_HALT:def 1
for s being State of SCM+FSA
for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st I c= P holds
for n being Element of NAT st Initialize ((intloc 0) .--> 1) c= s holds
IC (Comput (P,s,n)) in dom I;
attr I is InitHalting means :Def2: :: SCM_HALT:def 2
for s being State of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s holds
for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st I c= P holds
P halts_on s;
attr I is keepInt0_1 means :Def3: :: SCM_HALT:def 3
for s being State of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s holds
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT st I c= p holds
for k being Element of NAT holds (Comput (p,s,k)) . (intloc 0) = 1;

end;

:: deftheorem Def1 defines InitClosed SCM_HALT:def 1 :

:: deftheorem Def2 defines InitHalting SCM_HALT:def 2 :

:: deftheorem Def3 defines keepInt0_1 SCM_HALT:def 3 :

theorem Th1: :: SCM_HALT:1

for x being set
for i, m, n being Element of NAT holds
( not x in dom (((intloc i) .--> m) +* (Start-At (n,SCM+FSA))) or x = intloc i or x = IC )

proof end;

theorem Th2: :: SCM_HALT:2

for I being Program of SCM+FSA
for i, m, n being Element of NAT holds dom I misses dom (((intloc i) .--> m) +* (Start-At (n,SCM+FSA)))

proof end;

theorem :: SCM_HALT:3

canceled;

theorem Th4: :: SCM_HALT:4

Macro (halt SCM+FSA) is InitHalting

proof end;

registration

cluster non empty Relation-like NAT -defined the carrier of SCM+FSA -defined the Instructions of SCM+FSA -valued Function-like the Object-Kind of SCM+FSA -compatible V27() V63() V161() InitHalting set ;
existence by Th4;

end;

theorem Th5: :: SCM_HALT:5

theorem :: SCM_HALT:6

canceled;

theorem Th7: :: SCM_HALT:7

for I being Program of SCM+FSA
for s being State of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s holds
s . (intloc 0) = 1

proof end;

registration

proof end;

end;

registration

proof end;

end;

registration

let I be InitHalting Program of SCM+FSA;
cluster Initialized I -> halting ;
coherence

proof end;

end;

registration

proof end;

end;

theorem :: SCM_HALT:8

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA
for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec (I,p,s)) . a = s . a

proof end;

theorem :: SCM_HALT:9

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec (I,p,s)) . f = s . f

proof end;

registration

end;

theorem :: SCM_HALT:10

canceled;

theorem :: SCM_HALT:11

canceled;

theorem Th12: :: SCM_HALT:12

for s1, s2 being State of SCM+FSA
for p1, p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for J being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & J c= p1 holds
for n being Element of NAT st Reloc (J,n) c= p2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (p1,s1,i))) + n = IC (Comput (p2,s2,i)) & IncAddr ((CurInstr (p1,(Comput (p1,s1,i)))),n) = CurInstr (p2,(Comput (p2,s2,i))) & DataPart (Comput (p1,s1,i)) = DataPart (Comput (p2,s2,i)) )

proof end;

theorem :: SCM_HALT:13

for I being Program of SCM+FSA
for s being State of SCM+FSA st Initialized I c= s holds
I c= s

proof end;

theorem Th14: :: SCM_HALT:14

for s1, s2 being State of SCM+FSA
for p1, p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 & NPP s1 = NPP s2 holds
for k being Element of NAT holds
( NPP (Comput (p1,s1,k)) = NPP (Comput (p2,s2,k)) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )

proof end;

theorem Th15: :: SCM_HALT:15

for s1, s2 being State of SCM+FSA
for p1, p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s1 & Initialize ((intloc 0) .--> 1) c= s2 & I c= p1 & I c= p2 & NPP s1 = NPP s2 holds
( LifeSpan (p1,s1) = LifeSpan (p2,s2) & NPP (Result (p1,s1)) = NPP (Result (p2,s2)) )

proof end;

registration

proof end;

end;

registration

proof end;

end;

theorem :: SCM_HALT:16

canceled;

theorem Th17: :: SCM_HALT:17

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA holds (IExec (I,p,s)) . (intloc 0) = 1

proof end;

registration

proof end;

end;

theorem Th18: :: SCM_HALT:18

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitClosed Program of SCM+FSA
for J being Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= p & p halts_on s holds
for m being Element of NAT st m <= LifeSpan (p,s) holds
NPP (Comput (p,s,m)) = NPP (Comput ((p +* (I ';' J)),s,m))

proof end;

theorem :: SCM_HALT:19

for I being Program of SCM+FSA
for s being State of SCM+FSA
for i, m, n being Element of NAT holds (s +* I) +* (((intloc i) .--> m) +* (Start-At (n,SCM+FSA))) = (s +* (((intloc i) .--> m) +* (Start-At (n,SCM+FSA)))) +* I

proof end;

theorem :: SCM_HALT:20

canceled;

theorem Th21: :: SCM_HALT:21

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitClosed Program of SCM+FSA st p +* I halts_on s & Directed I c= p & Initialize ((intloc 0) .--> 1) c= s holds
IC (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = card I

proof end;

theorem Th22: :: SCM_HALT:22

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitClosed Program of SCM+FSA st p +* I halts_on s & Directed I c= p & Initialize ((intloc 0) .--> 1) c= s holds
DataPart (Comput (p,s,(LifeSpan ((p +* I),s)))) = DataPart (Comput (p,s,((LifeSpan ((p +* I),s)) + 1)))

proof end;

theorem Th23: :: SCM_HALT:23

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA

proof end;

theorem Th24: :: SCM_HALT:24

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitClosed Program of SCM+FSA st p +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((p +* (I ';' J)),(s +* (Initialize ((intloc 0) .--> 1))),k))

proof end;

theorem Th25: :: SCM_HALT:25

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA
for s being State of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I ';' J c= p holds
( IC (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = card I & DataPart (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) = DataPart ((Comput ((p +* I),s,(LifeSpan ((p +* I),s)))) +* (Initialize ((intloc 0) .--> 1))) & Reloc (J,(card I)) c= p & (Comput (p,s,((LifeSpan ((p +* I),s)) + 1))) . (intloc 0) = 1 & p halts_on s & LifeSpan (p,s) = ((LifeSpan ((p +* I),s)) + 1) + (LifeSpan (((p +* I) +* J),((Result ((p +* I),s)) +* (Initialize ((intloc 0) .--> 1))))) & ( J is keeping_0 implies (Result (p,s)) . (intloc 0) = 1 ) )

proof end;

registration

let I be InitHalting keepInt0_1 Program of SCM+FSA;
let J be InitHalting Program of SCM+FSA;
cluster I ';' J -> InitHalting ;
coherence

proof end;

end;

theorem Th26: :: SCM_HALT:26

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being keepInt0_1 Program of SCM+FSA st p +* I halts_on s holds
for J being InitClosed Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I ';' J c= p holds
for k being Element of NAT holds NPP ((Comput (((p +* I) +* J),((Result ((p +* I),s)) +* (Initialize ((intloc 0) .--> 1))),k)) +* (Start-At (((IC (Comput (((p +* I) +* J),((Result ((p +* I),s)) +* (Initialize ((intloc 0) .--> 1))),k))) + (card I)),SCM+FSA))) = NPP (Comput ((p +* (I ';' J)),s,(((LifeSpan ((p +* I),s)) + 1) + k)))

proof end;

theorem Th27: :: SCM_HALT:27

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being keepInt0_1 Program of SCM+FSA st not p +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) holds
for J being Program of SCM+FSA
for k being Element of NAT holds NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) = NPP (Comput ((p +* (I ';' J)),(s +* (Initialize ((intloc 0) .--> 1))),k))

proof end;

theorem Th28: :: SCM_HALT:28

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA holds LifeSpan ((p +* (I ';' J)),(s +* (Initialize ((intloc 0) .--> 1)))) = ((LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1) + (LifeSpan (((p +* I) +* J),((Result ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))))) +* (Initialize ((intloc 0) .--> 1)))))

proof end;

theorem Th29: :: SCM_HALT:29

proof end;

registration

let i be parahalting Instruction of SCM+FSA;
cluster Macro i -> InitHalting ;
coherence ;

end;

registration

let i be parahalting Instruction of SCM+FSA;
let J be parahalting Program of SCM+FSA;
cluster i ';' J -> InitHalting ;
coherence ;

end;

registration

let i be parahalting keeping_0 Instruction of SCM+FSA;
let J be InitHalting Program of SCM+FSA;
cluster i ';' J -> InitHalting ;
coherence ;

end;

registration

let I, J be keepInt0_1 Program of SCM+FSA;
cluster I ';' J -> keepInt0_1 ;
coherence

proof end;

end;

registration

let j be parahalting keeping_0 Instruction of SCM+FSA;
let I be InitHalting keepInt0_1 Program of SCM+FSA;
cluster I ';' j -> InitHalting keepInt0_1 ;
coherence ;

end;

registration

let i be parahalting keeping_0 Instruction of SCM+FSA;
let J be InitHalting keepInt0_1 Program of SCM+FSA;
cluster i ';' J -> InitHalting keepInt0_1 ;
coherence ;

end;

registration

let j be parahalting Instruction of SCM+FSA;
let I be parahalting Program of SCM+FSA;
cluster I ';' j -> InitHalting ;
coherence ;

end;

registration

let i, j be parahalting Instruction of SCM+FSA;
cluster i ';' j -> InitHalting ;
coherence ;

end;

theorem Th30: :: SCM_HALT:30

for s being State of SCM+FSA
for a being Int-Location
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA holds (IExec ((I ';' J),p,s)) . a = (IExec (J,p,(IExec (I,p,s)))) . a

proof end;

theorem Th31: :: SCM_HALT:31

for s being State of SCM+FSA
for f being FinSeq-Location
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA holds (IExec ((I ';' J),p,s)) . f = (IExec (J,p,(IExec (I,p,s)))) . f

proof end;

theorem Th32: :: SCM_HALT:32

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for s being State of SCM+FSA holds DataPart (Initialized (IExec (I,p,s))) = DataPart (IExec (I,p,s))

proof end;

theorem Th33: :: SCM_HALT:33

for s being State of SCM+FSA
for a being Int-Location
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),p,s)) . a = (Exec (j,(IExec (I,p,s)))) . a

proof end;

theorem Th34: :: SCM_HALT:34

for s being State of SCM+FSA
for f being FinSeq-Location
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being InitHalting keepInt0_1 Program of SCM+FSA
for j being parahalting Instruction of SCM+FSA holds (IExec ((I ';' j),p,s)) . f = (Exec (j,(IExec (I,p,s)))) . f

proof end;

definition

let I be Program of SCM+FSA;
let s be State of SCM+FSA;
let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ;
pred I is_closed_onInit s,p means :Def4: :: SCM_HALT:def 4
for k being Element of NAT holds IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) in dom I;
pred I is_halting_onInit s,p means :Def5: :: SCM_HALT:def 5
p +* I halts_on s +* (Initialize ((intloc 0) .--> 1));

end;

:: deftheorem Def4 defines is_closed_onInit SCM_HALT:def 4 :

:: deftheorem Def5 defines is_halting_onInit SCM_HALT:def 5 :

theorem Th35: :: SCM_HALT:35

for I being Program of SCM+FSA holds
( I is InitClosed iff for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT holds I is_closed_onInit s,p )

proof end;

theorem Th36: :: SCM_HALT:36

for I being Program of SCM+FSA holds
( I is InitHalting iff for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT holds I is_halting_onInit s,p )

proof end;

theorem Th37: :: SCM_HALT:37

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA
for a being Int-Location st not I destroys a & I is_closed_onInit s,p & Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT holds (Comput (p,s,k)) . a = s . a

proof end;

registration

proof end;

end;

registration

proof end;

end;

registration

cluster Stop SCM+FSA -> good InitHalting ;
coherence ;

end;

LmX: for s1, s2 being State of SCM+FSA
for J being InitHalting Program of SCM+FSA
for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT st NPP s1 = NPP s2 holds
NPP (IExec (J,P,s1)) = NPP (IExec (J,P,s2))

proof end;

theorem :: SCM_HALT:38

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being InitHalting Program of SCM+FSA
for a being Int-Location holds (IExec ((i ';' J),p,s)) . a = (IExec (J,p,(Exec (i,(Initialized s))))) . a

proof end;

theorem :: SCM_HALT:39

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for i being parahalting keeping_0 Instruction of SCM+FSA
for J being InitHalting Program of SCM+FSA
for f being FinSeq-Location holds (IExec ((i ';' J),p,s)) . f = (IExec (J,p,(Exec (i,(Initialized s))))) . f

proof end;

theorem Th40: :: SCM_HALT:40

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA holds
( I is_closed_onInit s,p iff I is_closed_on Initialized s,p )

proof end;

theorem Th41: :: SCM_HALT:41

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA holds
( I is_halting_onInit s,p iff I is_halting_on Initialized s,p )

proof end;

theorem :: SCM_HALT:42

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being Program of SCM+FSA
for s being State of SCM+FSA holds IExec (I,p,s) = IExec (I,p,(Initialized s))

proof end;

theorem Th43: :: SCM_HALT:43

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_onInit s,p & I is_halting_onInit s,p holds
( if=0 (a,I,J) is_closed_onInit s,p & if=0 (a,I,J) is_halting_onInit s,p )

proof end;

theorem Th44: :: SCM_HALT:44

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & I is_closed_onInit s,p & I is_halting_onInit s,p holds
IExec ((if=0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th45: :: SCM_HALT:45

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <> 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
( if=0 (a,I,J) is_closed_onInit s,p & if=0 (a,I,J) is_halting_onInit s,p )

proof end;

theorem Th46: :: SCM_HALT:46

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <> 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
IExec ((if=0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th47: :: SCM_HALT:47

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( if=0 (a,I,J) is InitHalting & ( s . a = 0 implies IExec ((if=0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <> 0 implies IExec ((if=0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )

proof end;

theorem :: SCM_HALT:48

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if=0 (a,I,J)),p,s)) = ((card I) + (card J)) + 3 & ( s . a = 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),p,s)) . d = (IExec (I,p,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),p,s)) . f = (IExec (I,p,s)) . f ) ) ) & ( s . a <> 0 implies ( ( for d being Int-Location holds (IExec ((if=0 (a,I,J)),p,s)) . d = (IExec (J,p,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if=0 (a,I,J)),p,s)) . f = (IExec (J,p,s)) . f ) ) ) )

proof end;

theorem Th49: :: SCM_HALT:49

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_onInit s,p & I is_halting_onInit s,p holds
( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p )

proof end;

theorem Th50: :: SCM_HALT:50

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & I is_closed_onInit s,p & I is_halting_onInit s,p holds
IExec ((if>0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th51: :: SCM_HALT:51

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a <= 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
( if>0 (a,I,J) is_closed_onInit s,p & if>0 (a,I,J) is_halting_onInit s,p )

proof end;

theorem Th52: :: SCM_HALT:52

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I, J being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st s . a <= 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
IExec ((if>0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA))

proof end;

theorem Th53: :: SCM_HALT:53

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( if>0 (a,I,J) is InitHalting & ( s . a > 0 implies IExec ((if>0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) & ( s . a <= 0 implies IExec ((if>0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At ((((card I) + (card J)) + 3),SCM+FSA)) ) )

proof end;

theorem :: SCM_HALT:54

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( IC (IExec ((if>0 (a,I,J)),p,s)) = ((card I) + (card J)) + 3 & ( s . a > 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),p,s)) . d = (IExec (I,p,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),p,s)) . f = (IExec (I,p,s)) . f ) ) ) & ( s . a <= 0 implies ( ( for d being Int-Location holds (IExec ((if>0 (a,I,J)),p,s)) . d = (IExec (J,p,s)) . d ) & ( for f being FinSeq-Location holds (IExec ((if>0 (a,I,J)),p,s)) . f = (IExec (J,p,s)) . f ) ) ) )

proof end;

theorem Th55: :: SCM_HALT:55

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a < 0 & I is_closed_onInit s,p & I is_halting_onInit s,p holds
IExec ((if<0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))

proof end;

theorem Th56: :: SCM_HALT:56

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a = 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
IExec ((if<0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))

proof end;

theorem Th57: :: SCM_HALT:57

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being Program of SCM+FSA
for a being read-write Int-Location st s . a > 0 & J is_closed_onInit s,p & J is_halting_onInit s,p holds
IExec ((if<0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))

proof end;

theorem Th58: :: SCM_HALT:58

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is InitHalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),p,s) = (IExec (I,p,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),p,s) = (IExec (J,p,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )

proof end;

registration

let I, J be InitHalting Program of SCM+FSA;
let a be read-write Int-Location ;
cluster if=0 (a,I,J) -> InitHalting ;
correctness by Th47;
cluster if>0 (a,I,J) -> InitHalting ;
correctness by Th53;
cluster if<0 (a,I,J) -> InitHalting ;
correctness by Th58;

end;

theorem Th59: :: SCM_HALT:59

for I being Program of SCM+FSA holds
( I is InitHalting iff for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT holds I is_halting_on Initialized s,p )

proof end;

theorem Th60: :: SCM_HALT:60

for I being Program of SCM+FSA holds
( I is InitClosed iff for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT holds I is_closed_on Initialized s,p )

proof end;

theorem Th61: :: SCM_HALT:61

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds (IExec (I,p,s)) . a = (Comput ((p +* I),(Initialize (Initialized s)),(LifeSpan ((p +* I),(Initialize (Initialized s)))))) . a

proof end;

theorem Th62: :: SCM_HALT:62

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitHalting Program of SCM+FSA
for a being Int-Location
for k being Element of NAT st not I destroys a holds
(IExec (I,p,s)) . a = (Comput ((p +* I),(Initialize (Initialized s)),k)) . a

proof end;

set A = NAT ;

set D = Data-Locations SCM+FSA;

theorem Th63: :: SCM_HALT:63

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitHalting Program of SCM+FSA
for a being Int-Location st not I destroys a holds
(IExec (I,p,s)) . a = (Initialized s) . a

proof end;

theorem Th64: :: SCM_HALT:64

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitHalting keepInt0_1 Program of SCM+FSA
for a being read-write Int-Location st not I destroys a holds
(Comput ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)),(LifeSpan ((p +* (I ';' (SubFrom (a,(intloc 0))))),(Initialize (Initialized s)))))) . a = (s . a) - 1

proof end;

theorem Th65: :: SCM_HALT:65

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitClosed Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= p & p halts_on s holds
for m being Element of NAT st m <= LifeSpan (p,s) holds
NPP (Comput (p,s,m)) = NPP (Comput ((p +* (loop I)),s,m))

proof end;

theorem :: SCM_HALT:66

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being InitHalting Program of SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= p holds
for k being Element of NAT st k <= LifeSpan (p,s) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),s,k))) <> halt SCM+FSA

proof end;

theorem :: SCM_HALT:67

canceled;

theorem Th68: :: SCM_HALT:68

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_onInit s,p & I is_halting_onInit s,p holds
for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))

proof end;

theorem Th69: :: SCM_HALT:69

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_onInit s,p & I is_halting_onInit s,p holds
for m being Element of NAT st m < LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) = CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)))

proof end;

theorem :: SCM_HALT:70

for l being Element of NAT holds not l in dom (Initialize ((intloc 0) .--> 1))

proof end;

theorem Th71: :: SCM_HALT:71

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_onInit s,p & I is_halting_onInit s,p holds
( CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) <> halt SCM+FSA ) )

proof end;

theorem :: SCM_HALT:72

canceled;

theorem Th73: :: SCM_HALT:73

for s being State of SCM+FSA
for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 & s . a > 0 holds
loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) is_pseudo-closed_on s,p

proof end;

theorem :: SCM_HALT:74

proof end;

theorem :: SCM_HALT:75

proof end;

theorem :: SCM_HALT:76

for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a holds
Initialized (Times (a,I)) is halting

proof end;

registration

let a be read-write Int-Location ;
let I be good Program of SCM+FSA;
cluster Times (a,I) -> good ;
coherence ;

end;

theorem Th77: :: SCM_HALT:77

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
ex s2 being State of SCM+FSA ex p2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT ex k being Element of NAT st
( s2 = s +* (Initialize ((intloc 0) .--> 1)) & p2 = p +* (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) & k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 & (Comput (p2,s2,k)) . a = (s . a) - 1 & (Comput (p2,s2,k)) . (intloc 0) = 1 & ( for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,s)) . b ) & ( for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,s)) . f ) & IC (Comput (p2,s2,k)) = 0 & ( for n being Element of NAT st n <= k holds
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ) )

proof end;

theorem Th78: :: SCM_HALT:78

proof end;

theorem Th79: :: SCM_HALT:79

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
( (IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),p,s)) = DataPart (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) )

proof end;

theorem :: SCM_HALT:80

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st s . a <= 0 holds
(IExec ((Times (a,I)),p,s)) . f = s . f

proof end;

theorem :: SCM_HALT:81

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for b being Int-Location
for a being read-write Int-Location st s . a <= 0 holds
(IExec ((Times (a,I)),p,s)) . b = (Initialized s) . b

proof end;

theorem :: SCM_HALT:82

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . f = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . f

proof end;

theorem :: SCM_HALT:83

for p being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for b being Int-Location
for a being read-write Int-Location st not I destroys a & s . a > 0 holds
(IExec ((Times (a,I)),p,s)) . b = (IExec ((Times (a,I)),p,(IExec ((I ';' (SubFrom (a,(intloc 0)))),p,s)))) . b

proof end;

definition

let i be Instruction of SCM+FSA;
redefine attr i is good means :Def6: :: SCM_HALT:def 6
not i destroys intloc 0;
compatibility

proof end;

end;

:: deftheorem Def6 defines good SCM_HALT:def 6 :

registration

cluster good parahalting with_explicit_jumps IC-relocable Exec-preserving Element of the Instructions of SCM+FSA;
existence

proof end;

end;

registration

let n be Element of NAT ;
cluster intloc (n + 1) -> read-write ;
coherence

proof end;

end;