let s be State of SCM+FSA ; for I being Program of SCM+FSA
for a being read-write Int-Location st I is_closed_onInit s & I is_halting_onInit s & s . a > 0 holds
( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) )
let I be Program of SCM+FSA ; for a being read-write Int-Location st I is_closed_onInit s & I is_halting_onInit s & s . a > 0 holds
( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) )
let a be read-write Int-Location ; ( I is_closed_onInit s & I is_halting_onInit s & s . a > 0 implies ( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) ) )
set D = Int-Locations \/ FinSeq-Locations ;
set s0 = Initialized s;
set IA = I +* (Start-At 0 ,SCM+FSA );
set s1 = (Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ));
set s2 = Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),1;
set i = a >0_goto 4;
set sI = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ));
defpred S1[ Nat] means ( $1 <= LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))) implies ( IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + $1)) = (IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),$1)) + 4 & DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + $1)) = DataPart (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),$1) ) );
set loc4 = (card I) + 4;
set s3 = Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1);
A1:
(card I) + 4 in dom (while>0 a,I)
by SCMFSA_9:38;
assume A2:
I is_closed_onInit s
; ( not I is_halting_onInit s or not s . a > 0 or ( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) ) )
now let k be
Element of
NAT ;
IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom IX:
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))
by SCMFSA8A:13;
IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),k) in dom I
by A2, SCM_HALT:def 4;
hence
IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),k) in dom I
by X;
verum end;
then A3:
I is_closed_on Initialized s
by SCMFSA7B:def 7;
assume
I is_halting_onInit s
; ( not s . a > 0 or ( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) ) )
then X:
ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I)
by SCM_HALT:def 5;
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))
by SCMFSA8A:13;
then
ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))
by X;
then A4:
I is_halting_on Initialized s
by SCMFSA7B:def 8;
A5:
now let k be
Element of
NAT ;
( S1[k] implies S1[k + 1] )assume A6:
S1[
k]
;
S1[k + 1]now A7:
k + 0 < k + 1
by XREAL_1:8;
assume
k + 1
<= LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),
((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))
;
( IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = (IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1))) + 4 & DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = DataPart (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) )then
k < LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),
((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))
by A7, XXREAL_0:2;
hence
(
IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = (IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1))) + 4 &
DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = DataPart (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) )
by A3, A4, A6, SCMFSA_9:44;
verum end; hence
S1[
k + 1]
;
verum end;
while>0 a,I c= (while>0 a,I) +* (Start-At 0 ,SCM+FSA )
by SCMFSA8A:9;
then A8:
dom (while>0 a,I) c= dom ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))
by GRFUNC_1:8;
A9:
0 in dom (while>0 a,I)
by SCMFSA_9:10;
then A10: ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . 0 =
((while>0 a,I) +* (Start-At 0 ,SCM+FSA )) . 0
by A8, FUNCT_4:14
.=
(while>0 a,I) . 0
by A9, SCMFSA6B:7
.=
a >0_goto 4
by SCMFSA_9:11
;
IC SCM+FSA in dom ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))
by SF_MASTR:65;
then A11: IC ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) =
((while>0 a,I) +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA )
by FUNCT_4:14
.=
0
by SF_MASTR:66
;
Y:
(ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))) /. (IC ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))) = ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . (IC ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))))
by COMPOS_1:38;
A12: Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(0 + 1) =
Following (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),0 )
by AMI_1:14
.=
Following (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))
by AMI_1:13
.=
Exec (a >0_goto 4),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))
by A11, A10, Y
;
then A13:
for f being FinSeq-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) . f = ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . f
by SCMFSA_2:97;
for c being Int-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) . c = ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . c
by A12, SCMFSA_2:97;
then A14: DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) =
DataPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))
by A13, SCMFSA6A:38
.=
DataPart (Initialized s)
by SCMFSA8A:11
.=
DataPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))
by SCMFSA8A:11
;
set s4 = Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1);
set s2 = Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))));
not a in dom ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))
by SCMFSA6B:12;
then A16:
((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . a = (Initialized s) . a
by FUNCT_4:12;
assume
s . a > 0
; ( IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0 & DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)))) )
then A17:
(Initialized s) . a > 0
by SCMFSA6C:3;
A18:
S1[ 0 ]
proof
assume
0 <= LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),
((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))
;
( IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = (IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),0 )) + 4 & DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = DataPart (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) )
A19:
IC SCM+FSA in dom (I +* (Start-At 0 ,SCM+FSA ))
by SF_MASTR:65;
IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) =
((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC SCM+FSA )
by AMI_1:13
.=
(I +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA )
by A19, FUNCT_4:14
.=
0
by SF_MASTR:66
;
hence
(
IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = (IC (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),0 )) + 4 &
DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = DataPart (Comput (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) )
by A17, A12, A16, A14, AMI_1:13, SCMFSA_2:97;
verum
end;
for k being Element of NAT holds S1[k]
from NAT_1:sch 1(A18, A5);
then A20:
S1[ LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))]
;
T:
ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))))
by AMI_1:123;
A21: Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) =
Following (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))))
by AMI_1:14
.=
Exec (goto ((card I) + 4)),(Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))))
by A3, A4, A20, T, SCMFSA_9:45
;
then A22:
for f being FinSeq-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . f = (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))))) . f
by SCMFSA_2:95;
A23: (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . ((card I) + 4) =
((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) . ((card I) + 4)
by AMI_1:54
.=
((while>0 a,I) +* (Start-At 0 ,SCM+FSA )) . ((card I) + 4)
by A8, A1, FUNCT_4:14
.=
(while>0 a,I) . ((card I) + 4)
by A1, SCMFSA6B:7
.=
goto 0
by SCMFSA_9:46
;
Y:
(ProgramPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))) /. (IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))) = (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . (IC (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)))
by COMPOS_1:38;
T:
ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))
by AMI_1:123;
A24: Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1) =
Following (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))
by AMI_1:14
.=
Exec (goto 0 ),(Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))
by A21, A23, Y, T, SCMFSA_2:95
;
then A25:
for f being FinSeq-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)) . f = (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . f
by SCMFSA_2:95;
for c being Int-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . c = (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))))) . c
by A21, SCMFSA_2:95;
then A26:
DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) = DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))))
by A22, SCMFSA6A:38;
X:
s +* (Initialized (while>0 a,I)) = (Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))
by SCMFSA8A:13;
XX:
(Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) = s +* (Initialized I)
by SCMFSA8A:13;
A27: Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3) =
Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)
by X
.=
Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)
by XX
;
hence
IC (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = 0
by A24, SCMFSA_2:95; DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) = DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))
X:
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))
by SCMFSA8A:13;
XX:
(Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )) = s +* (Initialized I)
by SCMFSA8A:13;
for c being Int-Location holds (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)) . c = (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . c
by A24, SCMFSA_2:95;
then
DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)) = DataPart (Comput (ProgramPart ((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* ((while>0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA ))))))
by A26, A25, SCMFSA6A:38;
hence DataPart (Comput (ProgramPart (s +* (Initialized (while>0 a,I)))),(s +* (Initialized (while>0 a,I))),((LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) + 3)) =
DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart ((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))),((Initialized s) +* (I +* (Start-At 0 ,SCM+FSA )))))
by A20, A27, X
.=
DataPart (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),(LifeSpan (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))))
by XX
;
verum