:: Conditional branch macro instructions of SCM+FSA, Part I (preliminary)
:: by Noriko Asamoto
::
:: Received August 27, 1996
:: Copyright (c) 1996-2012 Association of Mizar Users


begin

set A = NAT ;

set D = Data-Locations ;

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

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

theorem :: SCMFSA8A:1
canceled;

theorem :: SCMFSA8A:2
canceled;

theorem :: SCMFSA8A:3
canceled;

theorem :: SCMFSA8A:4
canceled;

theorem :: SCMFSA8A:5
canceled;

theorem :: SCMFSA8A:6
canceled;

theorem :: SCMFSA8A:7
for P being preProgram of SCM+FSA
for l being Element of NAT
for x being set st x in dom P holds
( ( P . x = halt SCM+FSA implies (Directed (P,l)) . x = goto l ) & ( P . x <> halt SCM+FSA implies (Directed (P,l)) . x = P . x ) ) by FUNCT_4:105, FUNCT_4:106;

theorem Th8: :: SCMFSA8A:8
for i being Instruction of SCM+FSA
for a being Int-Location
for n being Element of NAT st not i destroys a holds
not IncAddr (i,n) destroys a
proof end;

theorem Th9: :: SCMFSA8A:9
for P being preProgram of SCM+FSA
for n being Element of NAT
for a being Int-Location st not P destroys a holds
not Reloc (P,n) destroys a
proof end;

theorem Th10: :: SCMFSA8A:10
for P being good preProgram of SCM+FSA
for n being Element of NAT holds Reloc (P,n) is good
proof end;

theorem Th11: :: SCMFSA8A:11
for I, J being preProgram 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 Th12: :: SCMFSA8A:12
for I, J being good preProgram of SCM+FSA holds I +* J is good
proof end;

theorem Th13: :: SCMFSA8A:13
for I being preProgram of SCM+FSA
for l being Element of NAT
for a being Int-Location st not I destroys a holds
not Directed (I,l) destroys a
proof end;

registration
let I be good preProgram of SCM+FSA;
let l be Element of NAT ;
cluster Directed (I,l) -> good ;
correctness
coherence
Directed (I,l) is good ;
proof end;
end;

registration
let I be good Program of SCM+FSA;
cluster Directed I -> good ;
correctness
coherence
Directed I is good ;
;
end;

registration
let I be Program of SCM+FSA;
let l be Element of NAT ;
cluster Directed (I,l) -> initial ;
correctness
coherence
Directed (I,l) is initial ;
proof end;
end;

registration
let I, J be good Program of SCM+FSA;
cluster I ";" J -> good ;
coherence
I ";" J is good
proof end;
end;

Lm1: for l being Element of NAT holds
( dom (0 .--> (goto l)) = {0} & 0 in dom (0 .--> (goto l)) & (0 .--> (goto l)) . 0 = goto l & card (0 .--> (goto l)) = 1 & not halt SCM+FSA in rng (0 .--> (goto l)) )
proof end;

definition
let l be Element of NAT ;
func Goto l -> Program of SCM+FSA equals :: SCMFSA8A:def 1
0 .--> (goto l);
coherence
0 .--> (goto l) is Program of SCM+FSA
proof end;
end;

:: deftheorem defines Goto SCMFSA8A:def 1 :
for l being Element of NAT holds Goto l = 0 .--> (goto l);

registration
let l be Element of NAT ;
cluster Goto l -> halt-free good ;
coherence
( Goto l is halt-free & Goto l is good )
proof end;
end;

registration
cluster non empty Relation-like NAT -defined the InstructionsF of SCM+FSA -valued Function-like T-Sequence-like V28() initial halt-free good for set ;
existence
ex b1 being Program of SCM+FSA st
( b1 is halt-free & b1 is good )
proof end;
end;

definition
let s be State of SCM+FSA;
let P be Instruction-Sequence of SCM+FSA;
let I be Program of SCM+FSA;
pred I is_pseudo-closed_on s,P means :Def2: :: SCMFSA8A:def 2
 ex k being Element of NAT st
( IC (Comput ((P +* I),(Initialize s),k)) = card I & ( for n being Element of NAT st n < k holds
IC (Comput ((P +* I),(Initialize s),n)) in dom I ) );
end;

:: deftheorem Def2 defines is_pseudo-closed_on SCMFSA8A:def 2 :
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA holds
( I is_pseudo-closed_on s,P iff ex k being Element of NAT st
( IC (Comput ((P +* I),(Initialize s),k)) = card I & ( for n being Element of NAT st n < k holds
IC (Comput ((P +* I),(Initialize s),n)) in dom I ) ) );

registration
cluster with_explicit_jumps IC-relocable sequential for Element of the InstructionsF of SCM+FSA;
existence
ex b1 being Instruction of SCM+FSA st b1 is sequential
proof end;
end;

definition
canceled;
let s be State of SCM+FSA;
let P be Instruction-Sequence of SCM+FSA;
let I be Program of SCM+FSA;
assume X1: I is_pseudo-closed_on s,P ;
func pseudo-LifeSpan (s,P,I) -> Element of NAT means :Def4: :: SCMFSA8A:def 4
( IC (Comput ((P +* I),(Initialize s),it)) = card I & ( for n being Element of NAT st not IC (Comput ((P +* I),(Initialize s),n)) in dom I holds
it <= n ) );
existence
ex b1 being Element of NAT st
( IC (Comput ((P +* I),(Initialize s),b1)) = card I & ( for n being Element of NAT st not IC (Comput ((P +* I),(Initialize s),n)) in dom I holds
b1 <= n ) )
proof end;
uniqueness
for b1, b2 being Element of NAT st IC (Comput ((P +* I),(Initialize s),b1)) = card I & ( for n being Element of NAT st not IC (Comput ((P +* I),(Initialize s),n)) in dom I holds
b1 <= n ) & IC (Comput ((P +* I),(Initialize s),b2)) = card I & ( for n being Element of NAT st not IC (Comput ((P +* I),(Initialize s),n)) in dom I holds
b2 <= n ) holds
b1 = b2
proof end;
end;

:: deftheorem SCMFSA8A:def 3 :
canceled;

:: deftheorem Def4 defines pseudo-LifeSpan SCMFSA8A:def 4 :
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_pseudo-closed_on s,P holds
for b4 being Element of NAT holds
( b4 = pseudo-LifeSpan (s,P,I) iff ( IC (Comput ((P +* I),(Initialize s),b4)) = card I & ( for n being Element of NAT st not IC (Comput ((P +* I),(Initialize s),n)) in dom I holds
b4 <= n ) ) );

theorem Th14: :: SCMFSA8A:14
for I, J being Program of SCM+FSA
for x being set st x in dom I holds
(I ";" J) . x = (Directed I) . x
proof end;

theorem :: SCMFSA8A:15
for l being Element of NAT holds card (Goto l) = 1 by Lm1;

theorem :: SCMFSA8A:16
for P being preProgram of SCM+FSA
for x being set st x in dom P holds
( ( P . x = halt SCM+FSA implies (Directed P) . x = goto (card P) ) & ( P . x <> halt SCM+FSA implies (Directed P) . x = P . x ) ) by FUNCT_4:105, FUNCT_4:106;

theorem Th17: :: SCMFSA8A:17
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_pseudo-closed_on s,P holds
for n being Element of NAT st n < pseudo-LifeSpan (s,P,I) holds
( IC (Comput ((P +* I),(Initialize s),n)) in dom I & CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),n))) <> halt SCM+FSA )
proof end;

theorem Th18: :: SCMFSA8A:18
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA st I is_pseudo-closed_on s,P holds
for k being Element of NAT st k <= pseudo-LifeSpan (s,P,I) holds
Comput ((P +* I),(Initialize s),k) = Comput ((P +* (I ";" J)),(Initialize s),k)
proof end;

theorem Th19: :: SCMFSA8A:19
for I being preProgram of SCM+FSA
for l being Element of NAT holds card (Directed (I,l)) = card I
proof end;

theorem :: SCMFSA8A:20
for I being Program of SCM+FSA holds card (Directed I) = card I by Th19;

theorem Th21: :: SCMFSA8A:21
for s being State of SCM+FSA
for P being Instruction-Sequence 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 k being Element of NAT st k <= LifeSpan ((P +* I),(Initialize s)) holds
( Comput ((P +* I),(Initialize s),k) = Comput ((P +* (Directed I)),(Initialize s),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) <> halt SCM+FSA )
proof end;

theorem Th22: :: SCMFSA8A:22
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (Directed I)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) = card I & DataPart (Comput ((P +* I),(Initialize s),(LifeSpan ((P +* I),(Initialize s))))) = DataPart (Comput ((P +* (Directed I)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) )
proof end;

Lm2: for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( Directed I is_pseudo-closed_on s,P & pseudo-LifeSpan (s,P,(Directed I)) = (LifeSpan ((P +* I),(Initialize s))) + 1 )
proof end;

theorem :: SCMFSA8A:23
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
Directed I is_pseudo-closed_on s,P by Lm2;

theorem :: SCMFSA8A:24
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
pseudo-LifeSpan (s,P,(Directed I)) = (LifeSpan ((P +* I),(Initialize s))) + 1 by Lm2;

theorem Th25: :: SCMFSA8A:25
for I, J being Program of SCM+FSA holds (Directed I) ";" J = I ";" J
proof end;

theorem Th26: :: SCMFSA8A:26
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( ( for k being Element of NAT st k <= LifeSpan ((P +* I),(Initialize s)) holds
( IC (Comput ((P +* (Directed I)),(Initialize s),k)) = IC (Comput ((P +* (I ";" J)),(Initialize s),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(Initialize s),k))) = CurInstr ((P +* (I ";" J)),(Comput ((P +* (I ";" J)),(Initialize s),k))) ) ) & DataPart (Comput ((P +* (Directed I)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) = DataPart (Comput ((P +* (I ";" J)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) & IC (Comput ((P +* (Directed I)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) = IC (Comput ((P +* (I ";" J)),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) )
proof end;

theorem Th27: :: SCMFSA8A:27
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
( ( for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k)) = IC (Comput ((P +* (I ";" J)),(s +* (Initialize ((intloc 0) .--> 1))),k)) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) = CurInstr ((P +* (I ";" J)),(Comput ((P +* (I ";" J)),(s +* (Initialize ((intloc 0) .--> 1))),k))) ) ) & DataPart (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = DataPart (Comput ((P +* (I ";" J)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) & IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = IC (Comput ((P +* (I ";" J)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) )
proof end;

theorem Th28: :: SCMFSA8A:28
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
( Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k) = Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k) & CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA )
proof end;

theorem Th29: :: SCMFSA8A:29
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
( IC (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = card I & DataPart (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))))) = DataPart (Comput ((P +* (Directed I)),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) )
proof end;

Lm3: for I being Program of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) = card I & DataPart (Comput ((P +* I),(Initialize s),(LifeSpan ((P +* I),(Initialize s))))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) & P +* (I ";" (Stop SCM+FSA)) halts_on Initialize s & LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(Initialize s)) = (LifeSpan ((P +* I),(Initialize s))) + 1 & I ";" (Stop SCM+FSA) is_closed_on s,P & I ";" (Stop SCM+FSA) is_halting_on s,P )
proof end;

theorem :: SCMFSA8A:30
for I being Program of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( I ";" (Stop SCM+FSA) is_closed_on s,P & I ";" (Stop SCM+FSA) is_halting_on s,P ) by Lm3;

theorem :: SCMFSA8A:31
for l being Element of NAT holds
( 0 in dom (Goto l) & (Goto l) . 0 = goto l ) by Lm1;

Lm4: for I being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
( IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = card I & DataPart (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) & P +* (I ";" (Stop SCM+FSA)) halts_on s +* (Initialize ((intloc 0) .--> 1)) & LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1)))) = (LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 )
proof end;

theorem :: SCMFSA8A:32
for I being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IC (Comput ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = card I by Lm4;

theorem :: SCMFSA8A:33
for I being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
DataPart (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))))) = DataPart (Comput ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) by Lm4;

theorem :: SCMFSA8A:34
for I being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
P +* (I ";" (Stop SCM+FSA)) halts_on s +* (Initialize ((intloc 0) .--> 1)) by Lm4;

theorem :: SCMFSA8A:35
for I being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
LifeSpan ((P +* (I ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1)))) = (LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1 by Lm4;

theorem :: SCMFSA8A:36
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((I ";" (Stop SCM+FSA)),P,s) = (IExec (I,P,s)) +* (Start-At ((card I),SCM+FSA))
proof end;

Lm5: for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* I),(Initialize s),(LifeSpan ((P +* I),(Initialize s))))) = DataPart (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 2))) & ( for k being Element of NAT st k < (LifeSpan ((P +* I),(Initialize s))) + 2 holds
CurInstr ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s),k))) <> halt SCM+FSA ) & ( for k being Element of NAT st k <= LifeSpan ((P +* I),(Initialize s)) holds
IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s),k)) = IC (Comput ((P +* I),(Initialize s),k)) ) & IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s),((LifeSpan ((P +* I),(Initialize s))) + 1))) = card I & P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)) halts_on Initialize s & LifeSpan ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Initialize s)) = (LifeSpan ((P +* I),(Initialize s))) + 2 )
proof end;

theorem :: SCMFSA8A:37
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
( ((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA) is_closed_on s,P & ((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA) is_halting_on s,P )
proof end;

theorem :: SCMFSA8A:38
for I, J being Program of SCM+FSA
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)) halts_on Initialize s by Lm5;

Lm6: for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
( IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 2))) = ((card I) + (card J)) + 1 & DataPart (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),(LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))))) = DataPart (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 2))) & ( for k being Element of NAT st k < (LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 2 holds
CurInstr ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),k))) <> halt SCM+FSA ) & ( for k being Element of NAT st k <= LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),k)) = IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),k)) ) & IC (Comput ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1))),((LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 1))) = card I & P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)) halts_on s +* (Initialize ((intloc 0) .--> 1)) & LifeSpan ((P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA))),(s +* (Initialize ((intloc 0) .--> 1)))) = (LifeSpan ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))))) + 2 )
proof end;

theorem :: SCMFSA8A:39
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
P +* (((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)) halts_on s +* (Initialize ((intloc 0) .--> 1)) by Lm6;

theorem :: SCMFSA8A:40
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IC (IExec ((((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)),P,s)) = ((card I) + (card J)) + 1
proof end;

theorem :: SCMFSA8A:41
for P being Instruction-Sequence of SCM+FSA
for I, J being Program of SCM+FSA
for s being State of SCM+FSA st I is_closed_on Initialized s,P & I is_halting_on Initialized s,P holds
IExec ((((I ";" (Goto ((card J) + 1))) ";" J) ";" (Stop SCM+FSA)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 1),SCM+FSA))
proof end;