let p be 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)))) )
let s be 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)))) )
let I be 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)))) )
let a be read-write Int-Location; ( not I destroys a & s . a > 0 implies ( (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)))) ) )
assume A1:
not I destroys a
; ( not s . a > 0 or ( (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)))) ) )
reconsider J3 = Macro (SubFrom (a,(intloc 0))) as good Program of SCM+FSA ;
set I1 = I ";" (SubFrom (a,(intloc 0)));
set ss = IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s);
set pp = p;
I ";" (SubFrom (a,(intloc 0))) is_closed_onInit s,p
by Th25;
then A2:
I ";" (SubFrom (a,(intloc 0))) is_closed_on Initialized s,p
by Th30;
I ";" (SubFrom (a,(intloc 0))) is_halting_onInit s,p
by Th26;
then A3:
( I ";" (SubFrom (a,(intloc 0))) = I ";" J3 & I ";" (SubFrom (a,(intloc 0))) is_halting_on Initialized s,p )
by Th31;
then A4:
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . (intloc 0) = 1
by A2, SCMFSA8C:67;
set P = if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))));
set s0 = Initialized s;
set p0 = p;
assume A5:
s . a > 0
; ( (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)))) )
then A6:
(Initialized s) . a > 0
by SCMFSA_M:37;
then consider s2 being State of SCM+FSA, p2 being Instruction-Sequence of SCM+FSA, k being Element of NAT such that
A7:
s2 = Initialized (Initialized s)
and
A8:
p2 = p +* (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
and
k = (LifeSpan ((p +* (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))),((Initialized s) +* (Initialize ((intloc 0) .--> 1))))) + 1
and
A9:
(Comput (p2,s2,k)) . a = ((Initialized s) . a) - 1
and
A10:
(Comput (p2,s2,k)) . (intloc 0) = 1
and
A11:
for b being read-write Int-Location st b <> a holds
(Comput (p2,s2,k)) . b = (IExec (I,p,(Initialized s))) . b
and
A12:
for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec (I,p,(Initialized s))) . f
and
A13:
IC (Comput (p2,s2,k)) = 0
and
A14:
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)))))))
by A1, Th64;
A15:
loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) c= p2
by A8, FUNCT_4:25;
A16:
now for f being FinSeq-Location holds (Comput (p2,s2,k)) . f = (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . flet f be
FinSeq-Location ;
(Comput (p2,s2,k)) . f = (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . fthus (Comput (p2,s2,k)) . f =
(IExec (I,p,(Initialized s))) . f
by A12
.=
(Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,(Initialized s))))) . f
by SCMFSA_2:65
.=
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . f
by Th24
.=
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . f
by SCMFSA8C:3
;
verum end;
A17: Initialize (Initialized s) =
((Initialized s) +* (Initialize ((intloc 0) .--> 1))) +* (Start-At (0,SCM+FSA))
by FUNCT_4:93
.=
(Initialized s) +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:14
.=
s2
by A7, FUNCT_4:93
;
thus A18: (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a =
(Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,s)))) . a
by Th23
.=
((IExec (I,p,s)) . a) - ((IExec (I,p,s)) . (intloc 0))
by SCMFSA_2:65
.=
((IExec (I,p,s)) . a) - 1
by Th9
.=
((Initialized s) . a) - 1
by A1, Th53
.=
(s . a) - 1
by SCMFSA_M:37
; DataPart (IExec ((Times (a,I)),p,s)) = DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
now for b being Int-Location holds (Comput (p2,s2,k)) . b = (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . blet b be
Int-Location;
(Comput (p2,s2,k)) . b1 = (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . b1per cases
( b = intloc 0 or b = a or ( b <> a & b <> intloc 0 ) )
;
suppose A19:
(
b <> a &
b <> intloc 0 )
;
(Comput (p2,s2,k)) . b1 = (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . b1then reconsider bb =
b as
read-write Int-Location by SCMFSA_M:def 2;
thus (Comput (p2,s2,k)) . b =
(IExec (I,p,(Initialized s))) . bb
by A11, A19
.=
(Exec ((SubFrom (a,(intloc 0))),(IExec (I,p,(Initialized s))))) . b
by A19, SCMFSA_2:65
.=
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . b
by Th23
.=
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . b
by SCMFSA8C:3
;
verum end; end; end;
then A20:
DataPart (Comput (p2,s2,k)) = DataPart (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))
by A16, SCMFSA_M:2;
set s21 = Initialize (Initialized s);
set p21 = p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA));
set ss0 = Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s));
set s31 = Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)));
set p31 = p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA));
0 in dom (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))
by SCMFSA8C:25;
then A21:
0 in dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by FUNCT_4:99;
A22:
(Initialized s) . (intloc 0) = 1
by SCMFSA_M:9;
then A23:
loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) is_pseudo-closed_on Initialized s,p
by A1, A6, Th60;
A24:
Directed (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) = loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))
by SCMFSA6A:22;
then A25:
Directed (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialized s,p
by A1, A22, A6, Th60;
A26:
DataPart (IExec ((Times (a,I)),p,(Initialized s))) = DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)),p,(Initialized s)))
by A1, A22, A6, A24, Th60, SCMFSA8C:40;
per cases
( (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a = 0 or (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a <> 0 )
;
suppose A27:
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a = 0
;
DataPart (IExec ((Times (a,I)),p,s)) = DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))A28:
(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)
by SCMFSA8C:36;
A29:
(card (I ";" (SubFrom (a,(intloc 0))))) + (3 + 2) =
((card (I ";" (SubFrom (a,(intloc 0))))) + 1) + 4
.=
((card (Goto 2)) + (card (I ";" (SubFrom (a,(intloc 0)))))) + 4
by SCMFSA8A:15
.=
card (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))
by SCMFSA8B:11
.=
card (dom (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by CARD_1:62
.=
card (dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))))
by FUNCT_4:99
.=
card (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by CARD_1:62
;
then
((card (I ";" (SubFrom (a,(intloc 0))))) + 3) + 0 < card (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by XREAL_1:6;
then A30:
(card (I ";" (SubFrom (a,(intloc 0))))) + 3
in dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by AFINSQ_1:66;
p2 . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3) = (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)
by A30, A15, GRFUNC_1:2;
then A31:
p2 . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3) = goto ((card (I ";" (SubFrom (a,(intloc 0))))) + 5)
by A28, FUNCT_4:105;
A32:
p2 . 0 = (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) . 0
by A21, A15, GRFUNC_1:2;
A33:
Comput (
p2,
s2,
(k + 1)) =
Following (
p2,
(Comput (p2,s2,k)))
by EXTPRO_1:3
.=
Exec (
(p2 . 0),
(Comput (p2,s2,k)))
by A13, PBOOLE:143
;
A34:
(if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) . 0 = a =0_goto ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)
by SCMFSA8C:26;
A35:
p2 . 0 = a =0_goto ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)
by A32, A34, FUNCT_4:105;
then
InsCode (p2 . 0) = 7
by SCMFSA_2:24;
then
InsCode (p2 . 0) in {0,6,7,8}
by ENUMSET1:def 2;
then A36:
DataPart (Comput (p2,s2,k)) = DataPart (Comput (p2,s2,(k + 1)))
by A33, SCMFSA8C:12;
A37:
(Comput (p2,s2,k)) . a = 0
by A9, A18, A27, SCMFSA_M:37;
then A38:
IC (Comput (p2,s2,(k + 1))) = (card (I ";" (SubFrom (a,(intloc 0))))) + 3
by A33, A35, SCMFSA_2:70;
A39:
now for n being Element of NAT st not IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) holds
k + (1 + 1) <= nlet n be
Element of
NAT ;
( not IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) implies k + (1 + 1) <= n )assume A40:
not
IC (Comput (p2,s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
;
k + (1 + 1) <= nthen
k < n
by A14;
then
k + 1
<= n
by INT_1:7;
then
k + 1
< n
by A38, A30, A40, XXREAL_0:1;
then
(k + 1) + 1
<= n
by INT_1:7;
hence
k + (1 + 1) <= n
;
verum end; A41:
p2 /. (IC (Comput (p2,s2,(k + 1)))) = p2 . (IC (Comput (p2,s2,(k + 1))))
by PBOOLE:143;
A42:
Comput (
p2,
s2,
(k + (1 + 1))) =
Comput (
p2,
s2,
((k + 1) + 1))
.=
Following (
p2,
(Comput (p2,s2,(k + 1))))
by EXTPRO_1:3
.=
Exec (
(p2 . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)),
(Comput (p2,s2,(k + 1))))
by A37, A33, A35, A41, SCMFSA_2:70
;
then A43:
IC (Comput ((p +* (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))),(Initialize (Initialized s)),(k + 2))) = card (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))
by A8, A17, A29, A31, SCMFSA_2:69;
A44:
k + 2
= pseudo-LifeSpan (
(Initialized s),
p,
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))))
by A43, A8, A17, A23, A39, SCMFSA8A:def 4;
InsCode (p2 . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)) = 6
by A31, SCMFSA_2:23;
then
InsCode (p2 . ((card (I ";" (SubFrom (a,(intloc 0))))) + 3)) in {0,6,7,8}
by ENUMSET1:def 2;
then A45:
DataPart (Comput (p2,s2,k)) = DataPart (Comput (p2,s2,(k + 2)))
by A36, A42, SCMFSA8C:12;
A46:
Initialize (Initialized s) =
s +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:14
.=
Initialized s
by FUNCT_4:93
;
A47:
s2 =
(Initialized s) +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
by A7, FUNCT_4:93
.=
((Initialized s) +* (Initialize ((intloc 0) .--> 1))) +* (Start-At (0,SCM+FSA))
by FUNCT_4:14
.=
(Initialized s) +* (Start-At (0,SCM+FSA))
by FUNCT_4:93
;
thus DataPart (IExec ((Times (a,I)),p,s)) =
DataPart (IExec ((Times (a,I)),p,(Initialized s)))
by SCMFSA8C:3
.=
DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)),p,s))
by A26, SCMFSA8C:3
.=
DataPart (Result ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s))))
by A46, SCMFSA6B:def 1
.=
DataPart (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))
by A20, A45, A47, A24, A1, A22, A6, Th60, A8, A44, SCMFSA8C:30
.=
DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by A4, A27, Th65
;
verum end; suppose A48:
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a <> 0
;
DataPart (IExec ((Times (a,I)),p,s)) = DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
s . a >= 0 + 1
by A5, INT_1:7;
then A49:
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a > 0
by A18, A48, XREAL_1:19;
A50:
k < pseudo-LifeSpan (
(Initialized s),
p,
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))))
by A14, A8, A22, A17, A1, A6, Th60, SCMFSA8C:1;
then A51:
DataPart (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) = DataPart (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))
by A24, A25, A20, A17, A8, SCMFSA8C:29;
A52:
now ( ( for a being Int-Location holds (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . a = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . a ) & ( for f being FinSeq-Location holds (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . f = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . f ) )A53:
DataPart (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))) =
(DataPart (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) +* {}
.=
(DataPart (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) +* (DataPart (Start-At (0,SCM+FSA)))
by MEMSTR_0:20
.=
DataPart (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by FUNCT_4:71
;
hereby for f being FinSeq-Location holds (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . f = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . f
let a be
Int-Location;
(Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . b1 = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . b1per cases
( a = intloc 0 or a <> intloc 0 )
;
suppose A54:
a = intloc 0
;
(Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . b1 = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . b1thus (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . a =
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a
by A51, SCMFSA_M:2
.=
1
by A54, Th9
.=
(Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))) . a
by A54, SCMFSA_M:9
.=
(Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . a
by A53, SCMFSA_M:2
;
verum end; suppose
a <> intloc 0
;
(Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . b1 = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . b1then A55:
a is
read-write Int-Location
by SCMFSA_M:def 2;
thus (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . a =
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . a
by A51, SCMFSA_M:2
.=
(Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))) . a
by A55, SCMFSA_M:37
.=
(Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . a
by A53, SCMFSA_M:2
;
verum end; end;
end; let f be
FinSeq-Location ;
(Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . f = (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . fthus (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) . f =
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . f
by A51, SCMFSA_M:2
.=
(Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))) . f
by SCMFSA_M:37
.=
(Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))) . f
by A53, SCMFSA_M:2
;
verum end; A56:
IC (Comput ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s)),k)) =
IC (Comput (p2,s2,k))
by A24, A25, A50, A8, A17, SCMFSA8C:29
.=
IC (Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by A13, FUNCT_4:113
;
A57:
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) . (intloc 0) = 1
by A3, A2, SCMFSA8C:67;
A58:
DataPart (Initialized s) =
(DataPart (Initialized s)) +* {}
.=
(DataPart (Initialized s)) +* (DataPart (Start-At (0,SCM+FSA)))
by MEMSTR_0:20
.=
DataPart (Initialize (Initialized s))
by FUNCT_4:71
;
Directed (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) = loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))
by SCMFSA6A:22;
then
Directed (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialize (Initialized s),
p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))
by A1, A22, A6, Th60, A58, SCMFSA8C:23;
then A59:
(
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA) is_closed_on Initialize (Initialized s),
p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)) &
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA) is_halting_on Initialize (Initialized s),
p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)) )
by SCMFSA8C:29;
A60:
Initialize (Initialized s) =
s +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:14
.=
Initialized s
by FUNCT_4:93
;
A61:
Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))) =
(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)) +* ((Initialize ((intloc 0) .--> 1)) +* (Start-At (0,SCM+FSA)))
by FUNCT_4:14
.=
Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))
by FUNCT_4:93
;
A62:
(
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA) c= p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)) &
(loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA) c= p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)) )
by FUNCT_4:25;
A63:
Result (
(p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),
(Initialize (Initialized s)))
= Result (
(p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),
(Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s)))))
by A59, A56, A62, A52, SCMFSA8C:73, SCMFSA_2:61;
DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)),p,(Initialized s))) =
DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)),p,s))
by SCMFSA8C:3
.=
DataPart (Result ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized s))))
by A60, SCMFSA6B:def 1
.=
DataPart (Result ((p +* ((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA))),(Initialize (Initialized (IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))))
by A63
.=
DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) ";" (Stop SCM+FSA)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by A61, SCMFSA6B:def 1
.=
DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by A1, A24, A57, A49, Th60, SCMFSA8C:40
;
hence
DataPart (IExec ((Times (a,I)),p,s)) = DataPart (IExec ((Times (a,I)),p,(IExec ((I ";" (SubFrom (a,(intloc 0)))),p,s))))
by A26, SCMFSA8C:3;
verum end; end;