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) 1998-2012 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_1:104, RELAT_1:171;

Lm1: IC (Initialize ((intloc 0) .--> 1)) = 0
by MEMSTR_0:def 11;

Lm2: Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1)
by FUNCT_4:25;

Lm3: 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_1:13
.= {(intloc 0)} \/ {(IC )} by FUNCOP_1:13 ;

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 Instruction-Sequence of SCM+FSA 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 Instruction-Sequence of SCM+FSA 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 Instruction-Sequence of SCM+FSA 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 :: SCM_HALT:1

canceled;

theorem Th2: :: SCM_HALT:2

Macro (halt SCM+FSA) is InitHalting

proof end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() V162() InitHalting for set ;

existence by Th2;

end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() paraclosed -> InitClosed for set ;

coherence

proof end;

end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() parahalting -> InitHalting for set ;

coherence

proof end;

end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() InitHalting -> InitClosed for set ;

coherence

proof end;

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() keepInt0_1 -> InitClosed for set ;

coherence

proof end;

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() keeping_0 -> keepInt0_1 for set ;

coherence

proof end;

end;

theorem :: SCM_HALT:3

canceled;

theorem :: SCM_HALT:4

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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:5

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() InitHalting -> for set ;

coherence ;

end;

theorem Th6: :: SCM_HALT:6

for s1, s2 being State of SCM+FSA
for p1, p2 being Instruction-Sequence of SCM+FSA
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 Th7: :: SCM_HALT:7

for s1, s2 being State of SCM+FSA
for p1, p2 being Instruction-Sequence of SCM+FSA
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 & s1 = s2 holds
for k being Element of NAT holds
( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )

proof end;

theorem Th8: :: SCM_HALT:8

for s1, s2 being State of SCM+FSA
for p1, p2 being Instruction-Sequence of SCM+FSA
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 & s1 = s2 holds
( LifeSpan (p1,s1) = LifeSpan (p2,s2) & Result (p1,s1) = Result (p2,s2) )

proof end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() keeping_0 V162() InitHalting for set ;

existence

proof end;

end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() V162() InitHalting keepInt0_1 for set ;

existence

proof end;

end;

theorem Th9: :: SCM_HALT:9

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being InitHalting keepInt0_1 Program of SCM+FSA holds (IExec (I,p,s)) . (intloc 0) = 1

proof end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() V162() InitClosed for set ;

existence

proof end;

end;

theorem Th10: :: SCM_HALT:10

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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
Comput (p,s,m) = Comput ((p +* (I ";" J)),s,m)

proof end;

theorem Th11: :: SCM_HALT:11

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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 Th12: :: SCM_HALT:12

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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 Th13: :: SCM_HALT:13

for s being State of SCM+FSA
for p being Instruction-Sequence 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 +* (Directed I)),(Comput ((p +* (Directed I)),s,k))) <> halt SCM+FSA

proof end;

theorem Th14: :: SCM_HALT:14

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being InitClosed Program of SCM+FSA st p +* I halts_on Initialized s holds
for J being Program of SCM+FSA
for k being Element of NAT st k <= LifeSpan ((p +* I),(Initialized s)) holds
Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

proof end;

theorem Th15: :: SCM_HALT:15

for p being Instruction-Sequence of SCM+FSA
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 Th16: :: SCM_HALT:16

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
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 (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)) = Comput ((p +* (I ";" J)),s,(((LifeSpan ((p +* I),s)) + 1) + k))

proof end;

theorem Th17: :: SCM_HALT:17

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being keepInt0_1 Program of SCM+FSA st not p +* I halts_on Initialized s holds
for J being Program of SCM+FSA
for k being Element of NAT holds Comput ((p +* I),(Initialized s),k) = Comput ((p +* (I ";" J)),(Initialized s),k)

proof end;

theorem Th18: :: SCM_HALT:18

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA holds LifeSpan ((p +* (I ";" J)),(Initialized s)) = ((LifeSpan ((p +* I),(Initialized s))) + 1) + (LifeSpan (((p +* I) +* J),((Result ((p +* I),(Initialized s))) +* (Initialize ((intloc 0) .--> 1)))))

proof end;

theorem Th19: :: SCM_HALT:19

for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA
for I being InitHalting keepInt0_1 Program of SCM+FSA
for J being InitHalting Program of SCM+FSA holds IExec ((I ";" J),p,s) = IncIC ((IExec (J,p,(IExec (I,p,s)))),(card I))

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 Th20: :: SCM_HALT:20

for s being State of SCM+FSA
for a being Int-Location
for p being Instruction-Sequence of SCM+FSA
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 Th21: :: SCM_HALT:21

for s being State of SCM+FSA
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA
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 Th22: :: SCM_HALT:22

for p being Instruction-Sequence of SCM+FSA
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 Th23: :: SCM_HALT:23

for s being State of SCM+FSA
for a being Int-Location
for p being Instruction-Sequence of SCM+FSA
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 Th24: :: SCM_HALT:24

for s being State of SCM+FSA
for f being FinSeq-Location
for p being Instruction-Sequence of SCM+FSA
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 Instruction-Sequence of SCM+FSA;

pred I is_closed_onInit s,p means :Def4: :: SCM_HALT:def 4
for k being Element of NAT holds IC (Comput ((p +* I),(Initialized s),k)) in dom I;

pred I is_halting_onInit s,p means :Def5: :: SCM_HALT:def 5
p +* I halts_on Initialized s;

end;

:: deftheorem Def4 defines is_closed_onInit SCM_HALT:def 4 :

:: deftheorem Def5 defines is_halting_onInit SCM_HALT:def 5 :

theorem Th25: :: SCM_HALT:25

for I being Program of SCM+FSA holds
( I is InitClosed iff for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA holds I is_closed_onInit s,p )

proof end;

theorem Th26: :: SCM_HALT:26

for I being Program of SCM+FSA holds
( I is InitHalting iff for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA holds I is_halting_onInit s,p )

proof end;

theorem Th27: :: SCM_HALT:27

for p being Instruction-Sequence of SCM+FSA
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

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() good V162() InitHalting for set ;

existence

proof end;

end;

registration

cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like V30() V69() good InitClosed -> keepInt0_1 for set ;

correctness

proof end;

end;

registration

cluster Stop SCM+FSA -> good InitHalting ;

coherence ;

end;

theorem :: SCM_HALT:28

for p being Instruction-Sequence of SCM+FSA
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:29

for p being Instruction-Sequence of SCM+FSA
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 Th30: :: SCM_HALT:30

for p being Instruction-Sequence of SCM+FSA
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 Th31: :: SCM_HALT:31

for p being Instruction-Sequence of SCM+FSA
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:32

for p being Instruction-Sequence of SCM+FSA
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 Th33: :: SCM_HALT:33

for p being Instruction-Sequence of SCM+FSA
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 Th34: :: SCM_HALT:34

for p being Instruction-Sequence of SCM+FSA
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 Th35: :: SCM_HALT:35

for p being Instruction-Sequence of SCM+FSA
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 Th36: :: SCM_HALT:36

for p being Instruction-Sequence of SCM+FSA
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 Th37: :: SCM_HALT:37

for p being Instruction-Sequence of SCM+FSA
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:38

for p being Instruction-Sequence of SCM+FSA
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 Th39: :: SCM_HALT:39

for p being Instruction-Sequence of SCM+FSA
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 Th40: :: SCM_HALT:40

for p being Instruction-Sequence of SCM+FSA
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 Th41: :: SCM_HALT:41

for p being Instruction-Sequence of SCM+FSA
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 Th42: :: SCM_HALT:42

for p being Instruction-Sequence of SCM+FSA
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 Th43: :: SCM_HALT:43

for p being Instruction-Sequence of SCM+FSA
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:44

for p being Instruction-Sequence of SCM+FSA
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 Th45: :: SCM_HALT:45

for p being Instruction-Sequence of SCM+FSA
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 Th46: :: SCM_HALT:46

for p being Instruction-Sequence of SCM+FSA
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 Th47: :: SCM_HALT:47

for p being Instruction-Sequence of SCM+FSA
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 Th48: :: SCM_HALT:48

for p being Instruction-Sequence of SCM+FSA
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 Th37;

cluster if>0 (a,I,J) -> InitHalting ;

correctness by Th43;

cluster if<0 (a,I,J) -> InitHalting ;

correctness by Th48;

end;

theorem Th49: :: SCM_HALT:49

for I being Program of SCM+FSA holds
( I is InitHalting iff for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA holds I is_halting_on Initialized s,p )

proof end;

theorem Th50: :: SCM_HALT:50

for I being Program of SCM+FSA holds
( I is InitClosed iff for s being State of SCM+FSA
for p being Instruction-Sequence of SCM+FSA holds I is_closed_on Initialized s,p )

proof end;

theorem Th51: :: SCM_HALT:51

for p being Instruction-Sequence of SCM+FSA
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 Th52: :: SCM_HALT:52

for p being Instruction-Sequence of SCM+FSA
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 ;

theorem Th53: :: SCM_HALT:53

for p being Instruction-Sequence of SCM+FSA
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 Th54: :: SCM_HALT:54

for p being Instruction-Sequence of SCM+FSA
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 Th55: :: SCM_HALT:55

for p being Instruction-Sequence of SCM+FSA
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
Comput (p,s,m) = Comput ((p +* (loop I)),s,m)

proof end;

theorem :: SCM_HALT:56

for p being Instruction-Sequence of SCM+FSA
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 Th57: :: SCM_HALT:57

for p being Instruction-Sequence of SCM+FSA
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),(Initialized s)) holds
Comput ((p +* I),(Initialized s),m) = Comput ((p +* (loop I)),(Initialized s),m)

proof end;

theorem Th58: :: SCM_HALT:58

for p being Instruction-Sequence of SCM+FSA
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),(Initialized s)) holds
CurInstr ((p +* I),(Comput ((p +* I),(Initialized s),m))) = CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(Initialized s),m)))

proof end;

theorem Th59: :: SCM_HALT:59

for p being Instruction-Sequence of SCM+FSA
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)),(Initialized s),(LifeSpan ((p +* I),(Initialized s)))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((p +* I),(Initialized s)) holds
CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(Initialized s),m))) <> halt SCM+FSA ) )

proof end;

theorem Th60: :: SCM_HALT:60

for s being State of SCM+FSA
for p being Instruction-Sequence 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 . (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:61

proof end;

theorem :: SCM_HALT:62

proof end;

theorem :: SCM_HALT:63

for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a holds
Initialize ((intloc 0) .--> 1) is Times (a,I) -halted

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 Th64: :: SCM_HALT:64

for p being Instruction-Sequence of SCM+FSA
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 Instruction-Sequence of SCM+FSA ex k being Element of NAT st
( s2 = Initialized s & 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))))))),(Initialized s))) + 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 Th65: :: SCM_HALT:65

for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good InitHalting Program of SCM+FSA
for a being read-write Int-Location st s . (intloc 0) = 1 & s . a <= 0 holds
DataPart (IExec ((Times (a,I)),p,s)) = DataPart s

proof end;

theorem Th66: :: SCM_HALT:66

for p being Instruction-Sequence of SCM+FSA
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:67

for p being Instruction-Sequence of SCM+FSA
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:68

for p being Instruction-Sequence of SCM+FSA
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:69

for p being Instruction-Sequence of SCM+FSA
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:70

for p being Instruction-Sequence of SCM+FSA
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 :: SCM_HALT:def 6
not i destroys intloc 0;

compatibility

proof end;

end;

:: deftheorem defines good SCM_HALT:def 6 :