let s be State of SCM+FSA; :: thesis: for I being parahalting good 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)))),s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) )
let I be parahalting good Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not I destroys a & s . a > 0 holds
( (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) )
let a be read-write Int-Location ; :: thesis: ( not I destroys a & s . a > 0 implies ( (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) ) )
set I1 = I ';' (SubFrom (a,(intloc 0)));
set ss = IExec ((I ';' (SubFrom (a,(intloc 0)))),s);
set s0 = Initialized s;
set ss0 = Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s));
set P = if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))));
set s21 = (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)));
set s31 = (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)));
A1: I is_halting_on Initialized s by SCMFSA7B:25;
A2: I ';' (SubFrom (a,(intloc 0))) is_halting_on Initialized s by SCMFSA7B:25;
reconsider J3 = Macro (SubFrom (a,(intloc 0))) as good Program of SCM+FSA by Th99, SCMFSA7B:14;
set D = Int-Locations \/ FinSeq-Locations;
set A = NAT ;
assume A3: not I destroys a ; :: thesis: ( not s . a > 0 or ( (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) ) )
0 in dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by Th54;
then A4: 0 in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by FUNCT_4:105;
A5: I ';' (SubFrom (a,(intloc 0))) is_closed_on Initialized s by SCMFSA7B:24;
A6: I ';' (SubFrom (a,(intloc 0))) = I ';' J3 ;
then A7: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . (intloc 0) = 1 by A2, A5, Th96;
assume A8: s . a > 0 ; :: thesis: ( (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = (s . a) - 1 & DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) )
then A9: (Initialized s) . a > 0 by SCMFSA6C:3;
A10: (Initialized s) . (intloc 0) = 1 by SCMFSA6C:3;
then A11: loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) is_pseudo-closed_on Initialized s by A3, A9, Th117;
A12: 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:63;
then A13: Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on Initialized s by A3, A10, A9, Th117;
consider s2 being State of SCM+FSA, k being Element of NAT such that
A14: s2 = (Initialized s) +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) and
k = (LifeSpan ((ProgramPart ((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* ((if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) +* (Start-At (0,SCM+FSA)))))) + 1 and
A15: (Comput ((ProgramPart s2),s2,k)) . a = ((Initialized s) . a) - 1 and
A16: (Comput ((ProgramPart s2),s2,k)) . (intloc 0) = 1 and
A17: for b being read-write Int-Location st b <> a holds
(Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,(Initialized s))) . b and
A18: for f being FinSeq-Location holds (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,(Initialized s))) . f and
A19: IC (Comput ((ProgramPart s2),s2,k)) = 0 and
A20: for n being Element of NAT st n <= k holds
IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by A3, A10, A9, Th122;

A21: now

let f be FinSeq-Location ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . f = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . f
thus (Comput ((ProgramPart s2),s2,k)) . f = (IExec (I,(Initialized s))) . f by A18
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,(Initialized s))))) . f by SCMFSA_2:91
.= (IExec ((I ';' (SubFrom (a,(intloc 0)))),(Initialized s))) . f by SCMFSA6C:8
.= (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . f by Th17 ; :: thesis: verum

end;

thus A22: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = (Exec ((SubFrom (a,(intloc 0))),(IExec (I,s)))) . a by SCMFSA6C:7
.= ((IExec (I,s)) . a) - ((IExec (I,s)) . (intloc 0)) by SCMFSA_2:91
.= ((IExec (I,s)) . a) - 1 by A1, Th92
.= ((Initialized s) . a) - 1 by A3, Th91
.= (s . a) - 1 by SCMFSA6C:3 ; :: thesis: DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s))))

now

let b be Int-Location ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b₁ = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b₁

per cases ( b = intloc 0 or b = a or ( b <> a & b <> intloc 0 ) ) ;

suppose b = intloc 0 ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b₁ = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b₁

hence (Comput ((ProgramPart s2),s2,k)) . b = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b by A16, A6, A2, A5, Th96; :: thesis: verum

end;

suppose b = a ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b₁ = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b₁

hence (Comput ((ProgramPart s2),s2,k)) . b = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b by A15, A22, SCMFSA6C:3; :: thesis: verum

end;

suppose A23: ( b <> a & b <> intloc 0 ) ; :: thesis: (Comput ((ProgramPart s2),s2,k)) . b₁ = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b₁

then reconsider bb = b as read-write Int-Location by SF_MASTR:def 5;
thus (Comput ((ProgramPart s2),s2,k)) . b = (IExec (I,(Initialized s))) . bb by A17, A23
.= (Exec ((SubFrom (a,(intloc 0))),(IExec (I,(Initialized s))))) . b by A23, SCMFSA_2:91
.= (IExec ((I ';' (SubFrom (a,(intloc 0)))),(Initialized s))) . b by SCMFSA6C:7
.= (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . b by Th17 ; :: thesis: verum

end;

end;

end;

then A24: DataPart (Comput ((ProgramPart s2),s2,k)) = DataPart (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) by A21, SCMFSA6A:38;
A25: DataPart (IExec ((Times (a,I)),(Initialized s))) = DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)),(Initialized s))) by A3, A10, A9, A12, Th69, Th117;

per cases ( (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = 0 or (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a <> 0 ) ;

suppose A26: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a = 0 ; :: thesis: DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s))))

loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) c= (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A27: dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) c= dom ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
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 Th65;
A30: (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:29
.= card (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) by SCMFSA8B:14
.= card (dom (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))
.= card (dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))) by FUNCT_4:105
.= card (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ;
then ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) + 0 < card (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by XREAL_1:8;
then A31: (card (I ';' (SubFrom (a,(intloc 0))))) + 3 in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) by AFINSQ_1:70;
(Comput ((ProgramPart s2),s2,(k + 1))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) = s2 . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) by AMI_1:54
.= ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) by A14, A31, A27, FUNCT_4:14
.= (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) by A31, SCMFSA6B:7 ;
then A32: (Comput ((ProgramPart s2),s2,(k + 1))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) = goto ((card (I ';' (SubFrom (a,(intloc 0))))) + 5) by A28, FUNCT_4:111;
then InsCode ((Comput ((ProgramPart s2),s2,(k + 1))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3)) = 6 by SCMFSA_2:47;
then A33: InsCode ((Comput ((ProgramPart s2),s2,(k + 1))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3)) in {0,6,7,8} by ENUMSET1:def 2;
Y: (ProgramPart (Comput ((ProgramPart s2),s2,k))) /. (IC (Comput ((ProgramPart s2),s2,k))) = (Comput ((ProgramPart s2),s2,k)) . (IC (Comput ((ProgramPart s2),s2,k))) by COMPOS_1:38;
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,k)) by AMI_1:123;
A34: Comput ((ProgramPart s2),s2,(k + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) by EXTPRO_1:4
.= Exec (((Comput ((ProgramPart s2),s2,k)) . 0),(Comput ((ProgramPart s2),s2,k))) by A19, Y, T ;
loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) c= (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
then A35: dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) c= dom ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) by GRFUNC_1:8;
A36: (Comput ((ProgramPart s2),s2,k)) . 0 = s2 . 0 by AMI_1:54
.= ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) +* (Start-At (0,SCM+FSA))) . 0 by A14, A4, A35, FUNCT_4:14
.= (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) . 0 by A4, SCMFSA6B:7 ;
A37: (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))) . 0 = a =0_goto ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) by Th55;
A38: (Comput ((ProgramPart s2),s2,k)) . 0 = a =0_goto ((card (I ';' (SubFrom (a,(intloc 0))))) + 3) by A36, A37, FUNCT_4:111;
A39: (Comput ((ProgramPart s2),s2,k)) . a = 0 by A15, A22, A26, SCMFSA6C:3;
then A40: IC (Comput ((ProgramPart s2),s2,(k + 1))) = (card (I ';' (SubFrom (a,(intloc 0))))) + 3 by A34, A38, SCMFSA_2:96;

A41: now

let n be Element of NAT ; :: thesis: ( not IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) implies k + (1 + 1) <= n )
assume A42: not IC (Comput ((ProgramPart s2),s2,n)) in dom (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ; :: thesis: k + (1 + 1) <= n
then k < n by A20;
then k + 1 <= n by INT_1:20;
then k + 1 < n by A40, A31, A42, XXREAL_0:1;
then (k + 1) + 1 <= n by INT_1:20;
hence k + (1 + 1) <= n ; :: thesis: verum

end;

Y: (ProgramPart (Comput ((ProgramPart s2),s2,(k + 1)))) /. (IC (Comput ((ProgramPart s2),s2,(k + 1)))) = (Comput ((ProgramPart s2),s2,(k + 1))) . (IC (Comput ((ProgramPart s2),s2,(k + 1)))) by COMPOS_1:38;
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,(k + 1))) by AMI_1:123;
A43: Comput ((ProgramPart s2),s2,(k + (1 + 1))) = Comput ((ProgramPart s2),s2,((k + 1) + 1))
.= Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,(k + 1)))) by EXTPRO_1:4
.= Exec (((Comput ((ProgramPart s2),s2,(k + 1))) . ((card (I ';' (SubFrom (a,(intloc 0))))) + 3)),(Comput ((ProgramPart s2),s2,(k + 1)))) by A39, A34, A38, Y, T, SCMFSA_2:96 ;
then IC (Comput ((ProgramPart s2),s2,(k + 2))) = (card (I ';' (SubFrom (a,(intloc 0))))) + 5 by A32, SCMFSA_2:95
.= card (ProgramPart (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))) by A30, RELAT_1:209 ;
then A44: k + 2 = pseudo-LifeSpan ((Initialized s),(loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))) by A14, A11, A41, SCMFSA8A:def 5;
InsCode ((Comput ((ProgramPart s2),s2,k)) . 0) = 7 by A38, SCMFSA_2:48;
then InsCode ((Comput ((ProgramPart s2),s2,k)) . 0) in {0,6,7,8} by ENUMSET1:def 2;
then DataPart (Comput ((ProgramPart s2),s2,k)) = DataPart (Comput ((ProgramPart s2),s2,(k + 1))) by A34, Th32;
then A45: DataPart (Comput ((ProgramPart s2),s2,k)) = DataPart (Comput ((ProgramPart s2),s2,(k + 2))) by A43, A33, Th32;
X: (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) = s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))) by SCMFSA8A:13;
thus DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(Initialized s))) by Th17
.= DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)),s)) by A25, Th17
.= DataPart ((Result ((ProgramPart (s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))))),(s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)))))) +* (s | NAT)) by SCMFSA6B:def 1
.= DataPart ((Result ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) +* (s | NAT)) by X
.= DataPart (Result ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) by Th35
.= DataPart (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) by A3, A10, A9, A14, A12, A24, A44, A45, Th59, Th117
.= DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) by A7, A26, Th123 ; :: thesis: verum

end;

suppose A46: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a <> 0 ; :: thesis: DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s))))

s . a >= 0 + 1 by A8, INT_1:20;
then A47: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a > 0 by A22, A46, XREAL_1:21;
A48: DataPart (Initialized s) = DataPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) by SCMFSA8A:11;
A49: k < pseudo-LifeSpan ((Initialized s),(loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0)))))))) by A3, A10, A9, A14, A20, Th2, Th117;
then A50: DataPart (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) = DataPart (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) by A14, A12, A13, A24, Th58;

A51: now

A52: DataPart (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) = DataPart ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) by SCMFSA8A:11;

hereby :: thesis: for f being FinSeq-Location holds (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . f = ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . f

let a be Int-Location ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . b₁ = ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . b₁

per cases ( a = intloc 0 or a <> intloc 0 ) ;

suppose A53: a = intloc 0 ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . b₁ = ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . b₁

thus (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . a = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a by A50, SCMFSA6A:38
.= 1 by A53, SCMFSA6B:35
.= (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) . a by A53, SCMFSA6C:3
.= ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . a by A52, SCMFSA6A:38 ; :: thesis: verum

end;

suppose a <> intloc 0 ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . b₁ = ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . b₁

then A54: a is read-write Int-Location by SF_MASTR:def 5;
thus (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . a = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . a by A50, SCMFSA6A:38
.= (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) . a by A54, SCMFSA6C:3
.= ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . a by A52, SCMFSA6A:38 ; :: thesis: verum

end;

end;

end;

let f be FinSeq-Location ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . f = ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . f
thus (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) . f = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . f by A50, SCMFSA6A:38
.= (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) . f by SCMFSA6C:3
.= ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) . f by A52, SCMFSA6A:38 ; :: thesis: verum

end;

IC (Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart s2),s2,k)) by A14, A12, A13, A49, Th58
.= IC (((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))) +* (Start-At (0,SCM+FSA))) by A19, FUNCT_4:121
.= IC ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))) by FUNCT_4:15 ;
then A55: Comput ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))),k),(Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) equal_outside NAT by A51, SCMFSA10:91;
A56: ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)) c= (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
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:63;
then A57: Directed (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) is_pseudo-closed_on (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by A3, A10, A9, A48, Th52, Th117;
then A58: (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA) is_closed_on (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by Th58;
A59: (loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA) is_halting_on (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by A57, Th58;
A60: (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) . (intloc 0) = 1 by A6, A2, A5, Th96;
A61: ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)) c= (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
X: (Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) = s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))) by SCMFSA8A:13;
Y: (Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))) = (IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))) by SCMFSA8A:13;
DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)),(Initialized s))) = DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)),s)) by Th17
.= DataPart ((Result ((ProgramPart (s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))))),(s +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)))))) +* (s | NAT)) by SCMFSA6B:def 1
.= DataPart ((Result ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) +* (s | NAT)) by X
.= DataPart (Result ((ProgramPart ((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) by Th35
.= DataPart (Result ((ProgramPart ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) by A56, A61, A58, A59, A55, Th103, SCMFSA6A:39
.= DataPart ((Result ((ProgramPart ((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA))))),((Initialized (IExec ((I ';' (SubFrom (a,(intloc 0)))),s))) +* (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)) +* (Start-At (0,SCM+FSA)))))) +* ((IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) | NAT)) by Th35
.= DataPart ((Result ((ProgramPart ((IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA))))),((IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) +* (Initialized ((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)))))) +* ((IExec ((I ';' (SubFrom (a,(intloc 0)))),s)) | NAT)) by Y
.= DataPart (IExec (((loop (if=0 (a,(Goto 2),(I ';' (SubFrom (a,(intloc 0))))))) ';' (Stop SCM+FSA)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) by SCMFSA6B:def 1
.= DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) by A3, A12, A60, A47, Th69, Th117 ;
hence DataPart (IExec ((Times (a,I)),s)) = DataPart (IExec ((Times (a,I)),(IExec ((I ';' (SubFrom (a,(intloc 0)))),s)))) by A25, Th17; :: thesis: verum

end;

end;