SCMFSA8C: The {\bf loop} and {\bf Times} Macroinstruction for {\SCMFSA}

:: The {\bf loop} and {\bf Times} Macroinstruction for {\SCMFSA}
:: by Noriko Asamoto
::
:: Received October 29, 1997
:: Copyright (c) 1997-2012 Association of Mizar Users

begin

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

set Q = (intloc 0) .--> 1;

theorem Th1: :: SCMFSA8C:1

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_pseudo-closed_on s,P holds
for k being Element of NAT st ( for n being Element of NAT st n <= k holds
IC (Comput ((P +* I),(Initialize s),n)) in dom I ) holds
k < pseudo-LifeSpan (s,P,I)

proof end;

theorem Th2: :: SCMFSA8C:2

for I, J being Program of SCM+FSA
for k being Element of NAT st card I <= k & k < (card I) + (card J) holds
for i being Instruction of SCM+FSA st i = J . (k -' (card I)) holds
(I ";" J) . k = IncAddr (i,(card I))

proof end;

theorem :: SCMFSA8C:3

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA holds IExec (I,P,s) = IExec (I,P,(Initialized s)) ;

theorem :: SCMFSA8C:4

canceled;

theorem Th5: :: SCMFSA8C:5

for I being Program of SCM+FSA st ( for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on Initialized s,P ) holds
Initialize ((intloc 0) .--> 1) is I -halted

proof end;

theorem Th6: :: SCMFSA8C:6

for I being Program of SCM+FSA st ( for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA holds I is_halting_on Initialized s,P ) holds
Initialize ((intloc 0) .--> 1) is I -halted by Th5;

theorem :: SCMFSA8C:7

canceled;

theorem :: SCMFSA8C:8

canceled;

theorem :: SCMFSA8C:9

canceled;

theorem :: SCMFSA8C:10

canceled;

theorem :: SCMFSA8C:11

canceled;

theorem Th12: :: SCMFSA8C:12

for s being State of SCM+FSA
for i being Instruction of SCM+FSA st InsCode i in {0,6,7,8} holds
DataPart (Exec (i,s)) = DataPart s

proof end;

theorem :: SCMFSA8C:13

canceled;

theorem Th14: :: SCMFSA8C:14

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA holds IExec ((Stop SCM+FSA),P,s) = Initialized s

proof end;

theorem Th15: :: SCMFSA8C:15

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_on s,P holds
0 in dom I

proof end;

theorem Th16: :: SCMFSA8C:16

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 holds
for n being Element of NAT st Reloc (I,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 Th17: :: SCMFSA8C:17

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being 0 -started State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I c= P1 & I c= P2 & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( IC (Comput (P1,s1,i)) = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) )

proof end;

theorem Th18: :: SCMFSA8C:18

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being 0 -started State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & I c= P1 & I c= P2 & DataPart s1 = DataPart s2 holds
LifeSpan (P1,s1) = LifeSpan (P2,s2)

proof end;

theorem :: SCMFSA8C:19

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st s1 . (intloc 0) = 1 & I is_closed_on s1,P1 & I is_halting_on s1,P1 & ( for a being read-write Int-Location holds s1 . a = s2 . a ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) holds
DataPart (IExec (I,P1,s1)) = DataPart (IExec (I,P2,s2))

proof end;

theorem Th20: :: SCMFSA8C:20

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st s1 . (intloc 0) = 1 & I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
DataPart (IExec (I,P1,s1)) = DataPart (IExec (I,P2,s2))

proof end;

theorem Th21: :: SCMFSA8C:21

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_pseudo-closed_on s,P holds
( I is_pseudo-closed_on Initialize s,P +* I & pseudo-LifeSpan (s,P,I) = pseudo-LifeSpan ((Initialize s),(P +* I),I) )

proof end;

theorem Th22: :: SCMFSA8C:22

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1 being 0 -started State of SCM+FSA
for s2 being State of SCM+FSA
for I being Program of SCM+FSA st I c= P1 & I is_pseudo-closed_on s1,P1 holds
for n being Element of NAT st Reloc (I,n) c= P2 & IC s2 = n & DataPart s1 = DataPart s2 holds
( ( for i being Element of NAT st i < pseudo-LifeSpan (s1,P1,I) holds
IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),n) = CurInstr (P2,(Comput (P2,s2,i))) ) & ( for i being Element of NAT st i <= pseudo-LifeSpan (s1,P1,I) holds
( (IC (Comput (P1,s1,i))) + n = IC (Comput (P2,s2,i)) & DataPart (Comput (P1,s1,i)) = DataPart (Comput (P2,s2,i)) ) ) )

proof end;

theorem Th23: :: SCMFSA8C:23

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st DataPart s1 = DataPart s2 & I is_pseudo-closed_on s1,P1 holds
I is_pseudo-closed_on s2,P2

proof end;

theorem Th24: :: SCMFSA8C:24

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA st s . (intloc 0) = 1 holds
( I is_pseudo-closed_on s,P iff I is_pseudo-closed_on Initialized s,P )

proof end;

theorem Th25: :: SCMFSA8C:25

for a being Int-Location
for I, J being Program of SCM+FSA holds
( 0 in dom (if=0 (a,I,J)) & 1 in dom (if=0 (a,I,J)) & 0 in dom (if>0 (a,I,J)) & 1 in dom (if>0 (a,I,J)) )

proof end;

theorem Th26: :: SCMFSA8C:26

for a being Int-Location
for I, J being Program of SCM+FSA holds
( (if=0 (a,I,J)) . 0 = a =0_goto ((card J) + 3) & (if=0 (a,I,J)) . 1 = goto 2 & (if>0 (a,I,J)) . 0 = a >0_goto ((card J) + 3) & (if>0 (a,I,J)) . 1 = goto 2 )

proof end;

theorem Th27: :: SCMFSA8C:27

for a being Int-Location
for I, J being Program of SCM+FSA
for n being Element of NAT st n < ((card I) + (card J)) + 3 holds
( n in dom (if=0 (a,I,J)) & (if=0 (a,I,J)) . n <> halt SCM+FSA )

proof end;

theorem Th28: :: SCMFSA8C:28

for a being Int-Location
for I, J being Program of SCM+FSA
for n being Element of NAT st n < ((card I) + (card J)) + 3 holds
( n in dom (if>0 (a,I,J)) & (if>0 (a,I,J)) . n <> halt SCM+FSA )

proof end;

theorem Th29: :: SCMFSA8C:29

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA st Directed I is_pseudo-closed_on s,P holds
( I ";" (Stop SCM+FSA) is_closed_on s,P & I ";" (Stop SCM+FSA) is_halting_on s,P & LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s)) = pseudo-LifeSpan (s,P,(Directed I)) & ( for n being Element of NAT st n < pseudo-LifeSpan (s,P,(Directed I)) holds
IC (Comput ((P +* I),(Initialize s),n)) = IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),n)) ) & ( for n being Element of NAT st n <= pseudo-LifeSpan (s,P,(Directed I)) holds
DataPart (Comput ((P +* I),(Initialize s),n)) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),n)) ) )

proof end;

theorem Th30: :: SCMFSA8C:30

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA st Directed I is_pseudo-closed_on s,P holds
DataPart (Result ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) = DataPart (Comput ((P +* I),(Initialize s),(pseudo-LifeSpan (s,P,(Directed I)))))

proof end;

theorem :: SCMFSA8C:31

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of SCM+FSA st s . (intloc 0) = 1 & Directed I is_pseudo-closed_on s,P holds
DataPart (IExec ((I ";" (Stop SCM+FSA)),P,s)) = DataPart (Comput ((P +* I),(Initialize s),(pseudo-LifeSpan (s,P,(Directed I)))))

proof end;

theorem Th32: :: SCMFSA8C:32

for I, J being Program of SCM+FSA
for a being Int-Location holds (if=0 (a,I,J)) . (((card I) + (card J)) + 3) = halt SCM+FSA

proof end;

theorem Th33: :: SCMFSA8C:33

for I, J being Program of SCM+FSA
for a being Int-Location holds (if>0 (a,I,J)) . (((card I) + (card J)) + 3) = halt SCM+FSA

proof end;

theorem Th34: :: SCMFSA8C:34

for I, J being Program of SCM+FSA
for a being Int-Location holds (if=0 (a,I,J)) . ((card J) + 2) = goto (((card I) + (card J)) + 3)

proof end;

theorem Th35: :: SCMFSA8C:35

for I, J being Program of SCM+FSA
for a being Int-Location holds (if>0 (a,I,J)) . ((card J) + 2) = goto (((card I) + (card J)) + 3)

proof end;

theorem Th36: :: SCMFSA8C:36

for J being Program of SCM+FSA
for a being Int-Location holds (if=0 (a,(Goto 2),J)) . ((card J) + 3) = goto ((card J) + 5)

proof end;

theorem Th37: :: SCMFSA8C:37

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 & Directed I is_pseudo-closed_on s,P holds
( if=0 (a,I,J) is_halting_on s,P & if=0 (a,I,J) is_closed_on s,P & LifeSpan ((P +* (if=0 (a,I,J))),(Initialize s)) = (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) + 1 )

proof end;

theorem :: SCMFSA8C:38

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 . (intloc 0) = 1 & s . a = 0 & Directed I is_pseudo-closed_on s,P holds
DataPart (IExec ((if=0 (a,I,J)),P,s)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,s))

proof end;

theorem Th39: :: SCMFSA8C: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 & Directed I is_pseudo-closed_on s,P holds
( if>0 (a,I,J) is_halting_on s,P & if>0 (a,I,J) is_closed_on s,P & LifeSpan ((P +* (if>0 (a,I,J))),(Initialize s)) = (LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s))) + 1 )

proof end;

theorem Th40: :: SCMFSA8C: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 . (intloc 0) = 1 & s . a > 0 & Directed I is_pseudo-closed_on s,P holds
DataPart (IExec ((if>0 (a,I,J)),P,s)) = DataPart (IExec ((I ";" (Stop SCM+FSA)),P,s))

proof end;

theorem Th41: :: SCMFSA8C: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 & Directed J is_pseudo-closed_on s,P holds
( if=0 (a,I,J) is_halting_on s,P & if=0 (a,I,J) is_closed_on s,P & LifeSpan ((P +* (if=0 (a,I,J))),(Initialize s)) = (LifeSpan ((P +* (J ";" (Stop SCM+FSA))),(Initialize s))) + 3 )

proof end;

theorem :: SCMFSA8C:42

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 . (intloc 0) = 1 & s . a <> 0 & Directed J is_pseudo-closed_on s,P holds
DataPart (IExec ((if=0 (a,I,J)),P,s)) = DataPart (IExec ((J ";" (Stop SCM+FSA)),P,s))

proof end;

theorem Th43: :: SCMFSA8C:43

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 & Directed J is_pseudo-closed_on s,P holds
( if>0 (a,I,J) is_halting_on s,P & if>0 (a,I,J) is_closed_on s,P & LifeSpan ((P +* (if>0 (a,I,J))),(s +* (Start-At (0,SCM+FSA)))) = (LifeSpan ((P +* (J ";" (Stop SCM+FSA))),(Initialize s))) + 3 )

proof end;

theorem Th44: :: SCMFSA8C:44

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 . (intloc 0) = 1 & s . a <= 0 & Directed J is_pseudo-closed_on s,P holds
DataPart (IExec ((if>0 (a,I,J)),P,s)) = DataPart (IExec ((J ";" (Stop SCM+FSA)),P,s))

proof end;

theorem :: SCMFSA8C: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 Directed I is_pseudo-closed_on s,P & Directed J is_pseudo-closed_on s,P holds
( if=0 (a,I,J) is_closed_on s,P & if=0 (a,I,J) is_halting_on s,P )

proof end;

theorem :: SCMFSA8C: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 Directed I is_pseudo-closed_on s,P & Directed J is_pseudo-closed_on s,P holds
( if>0 (a,I,J) is_closed_on s,P & if>0 (a,I,J) is_halting_on s,P )

proof end;

theorem :: SCMFSA8C:47

for I being Program of SCM+FSA
for a being Int-Location st not I destroys a holds
not Directed I destroys a by SCMFSA8A:13;

theorem Th48: :: SCMFSA8C:48

for i being Instruction of SCM+FSA
for a being Int-Location st not i destroys a holds
not Macro i destroys a

proof end;

theorem Th49: :: SCMFSA8C:49

for a being Int-Location holds not halt SCM+FSA refers a

proof end;

theorem :: SCMFSA8C:50

for a, b, c being Int-Location st a <> b holds
not AddTo (c,b) refers a

proof end;

theorem :: SCMFSA8C:51

for i being Instruction of SCM+FSA
for a being Int-Location st not i refers a holds
not Macro i refers a

proof end;

theorem Th52: :: SCMFSA8C:52

for I, J being Program of SCM+FSA
for a being Int-Location st not I destroys a & not J destroys a holds
not I ";" J destroys a

proof end;

theorem Th53: :: SCMFSA8C:53

for J being Program of SCM+FSA
for i being Instruction of SCM+FSA
for a being Int-Location st not i destroys a & not J destroys a holds
not i ";" J destroys a

proof end;

theorem :: SCMFSA8C:54

for I being Program of SCM+FSA
for j being Instruction of SCM+FSA
for a being Int-Location st not I destroys a & not j destroys a holds
not I ";" j destroys a

proof end;

theorem :: SCMFSA8C:55

for i, j being Instruction of SCM+FSA
for a being Int-Location st not i destroys a & not j destroys a holds
not i ";" j destroys a

proof end;

theorem Th56: :: SCMFSA8C:56

for a being Int-Location holds not Stop SCM+FSA destroys a

proof end;

theorem Th57: :: SCMFSA8C:57

for a being Int-Location
for l being Element of NAT holds not Goto l destroys a

proof end;

theorem Th58: :: SCMFSA8C: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_halting_on Initialized s,P holds
( ( 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 ) & ( for f being FinSeq-Location holds (IExec (I,P,s)) . f = (Comput ((P +* I),(Initialize (Initialized s)),(LifeSpan ((P +* I),(Initialize (Initialized s)))))) . f ) )

proof end;

theorem Th59: :: SCMFSA8C:59

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being parahalting 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 Th60: :: SCMFSA8C:60

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
for k being Element of NAT st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P & not I destroys a holds
(IExec (I,P,s)) . a = (Comput ((P +* I),(Initialize (Initialized s)),k)) . a

proof end;

theorem Th61: :: SCMFSA8C:61

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being parahalting 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;

theorem Th62: :: SCMFSA8C:62

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being parahalting 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 Th63: :: SCMFSA8C:63

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being keeping_0 Program of SCM+FSA st I is_halting_on Initialized s,P holds
( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) )

proof end;

theorem Th64: :: SCMFSA8C:64

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 holds
for k being Element of NAT st IC (Comput ((P +* I),(Initialize s),k)) in dom I holds
(Comput ((P +* I),(Initialize s),(k + 1))) . a = (Comput ((P +* I),(Initialize s),k)) . a

proof end;

theorem Th65: :: SCMFSA8C:65

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 holds
for m being Element of NAT st ( for n being Element of NAT st n < m holds
IC (Comput ((P +* I),(Initialize s),n)) in dom I ) holds
for n being Element of NAT st n <= m holds
(Comput ((P +* I),(Initialize s),n)) . a = s . a

proof end;

theorem Th66: :: SCMFSA8C:66

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good Program of SCM+FSA
for m being Element of NAT st ( for n being Element of NAT st n < m holds
IC (Comput ((P +* I),(Initialize s),n)) in dom I ) holds
for n being Element of NAT st n <= m holds
(Comput ((P +* I),(Initialize s),n)) . (intloc 0) = s . (intloc 0)

proof end;

theorem Th67: :: SCMFSA8C:67

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good Program of SCM+FSA st I is_halting_on Initialized s,P & I is_closed_on Initialized s,P holds
( (IExec (I,P,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((P +* I),(Initialize (Initialized s)),k)) . (intloc 0) = 1 ) )

proof end;

theorem :: SCMFSA8C:68

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good Program of SCM+FSA st I is_closed_on s,P holds
for k being Element of NAT holds (Comput ((P +* I),(Initialize s),k)) . (intloc 0) = s . (intloc 0)

proof end;

theorem Th69: :: SCMFSA8C:69

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being keeping_0 parahalting 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 Th70: :: SCMFSA8C:70

for i being Instruction of SCM+FSA st not i destroys intloc 0 holds
Macro i is good

proof end;

theorem Th71: :: SCMFSA8C:71

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
for k being Element of NAT holds
( Comput ((P1 +* I),(Initialize s1),k) = Comput ((P2 +* I),(Initialize s2),k) & CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),k))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),k))) )

proof end;

theorem Th72: :: SCMFSA8C:72

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & Result ((P1 +* I),(Initialize s1)) = Result ((P2 +* I),(Initialize s2)) )

proof end;

theorem Th73: :: SCMFSA8C:73

for P1, P2 being Instruction-Sequence of SCM+FSA
for s1, s2 being 0 -started State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & I c= P1 & I c= P2 & ex k being Element of NAT st Comput (P1,s1,k) = s2 holds
Result (P1,s1) = Result (P2,s2)

proof end;

begin

definition

let I be Program of SCM+FSA;

func loop I -> halt-free Program of SCM+FSA equals :: SCMFSA8C:def 1
Directed (I,0);

coherence

proof end;

end;

:: deftheorem defines loop SCMFSA8C:def 1 :

theorem :: SCMFSA8C:74

for I being Program of SCM+FSA
for a being Int-Location st loop I destroys a holds
I destroys a by SCMFSA8A:13;

registration

let I be good Program of SCM+FSA;

cluster loop I -> halt-free good ;

correctness ;

end;

theorem Th75: :: SCMFSA8C:75

for I being Program of SCM+FSA holds not halt SCM+FSA in rng (loop I) by FUNCT_4:100;

theorem Th76: :: SCMFSA8C:76

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_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
Comput ((P +* I),(Initialize s),m) = Comput ((P +* (loop I)),(Initialize s),m)

proof end;

theorem Th77: :: SCMFSA8C:77

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_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m < LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))

proof end;

Lm1: 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_on s,P & I is_halting_on s,P holds
( CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),(LifeSpan ((P +* I),(Initialize s)))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) <> halt SCM+FSA ) )

proof end;

theorem :: SCMFSA8C:78

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_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) <> halt SCM+FSA by Lm1;

theorem :: SCMFSA8C:79

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_on s,P & I is_halting_on s,P holds
CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),(LifeSpan ((P +* I),(Initialize s)))))) = goto 0 by Lm1;

theorem Th80: :: SCMFSA8C:80

for P being Instruction-Sequence of SCM+FSA
for s being 0 -started State of SCM+FSA
for I being paraclosed Program of SCM+FSA st 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 :: SCMFSA8C:81

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being parahalting 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;

begin

definition

let a be Int-Location;
let I be Program of SCM+FSA;

func Times (a,I) -> Program of SCM+FSA equals :: SCMFSA8C:def 2
if>0 (a,(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))),(Stop SCM+FSA));

correctness ;

end;

:: deftheorem defines Times SCMFSA8C:def 2 :

registration

let a be Int-Location;
let I be Program of SCM+FSA;

cluster Times (a,I) -> non halt-free ;

coherence ;

end;

theorem Th82: :: SCMFSA8C:82

for I being good Program of SCM+FSA
for a being read-write Int-Location holds if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))) is good

proof end;

theorem Th83: :: SCMFSA8C:83

for I, J being Program of SCM+FSA
for a being Int-Location holds (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3) = goto ((card (I ";" (SubFrom (a,(intloc 0))))) + 5)

proof end;

theorem Th84: :: SCMFSA8C:84

proof end;

theorem :: SCMFSA8C:85

proof end;

theorem :: SCMFSA8C:86

proof end;

theorem :: SCMFSA8C:87

for I being good parahalting 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;

theorem :: SCMFSA8C:88

for I, J being Program of SCM+FSA
for a, c being Int-Location st not I destroys c & not J destroys c holds
( not if=0 (a,I,J) destroys c & not if>0 (a,I,J) destroys c )

proof end;

theorem Th89: :: SCMFSA8C:89

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good parahalting Program of SCM+FSA
for a being read-write Int-Location st not I destroys a & s . (intloc 0) = 1 & 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 = s +* (Start-At (0,SCM+FSA)) & 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 +* (Start-At (0,SCM+FSA))))) + 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 Th90: :: SCMFSA8C:90

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good parahalting 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 Th91: :: SCMFSA8C:91

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being good parahalting 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;

begin

theorem :: SCMFSA8C:92

for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for a, b, c being read-write Int-Location st a <> b & a <> c & b <> c & s . a >= 0 holds
(IExec ((Times (a,(Macro (AddTo (b,c))))),P,s)) . b = (s . b) + ((s . c) * (s . a))

proof end;