let s be State of SCMPDS ; :: thesis: for I, J being Program of SCMPDS
for k being Element of NAT st I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
let I, J be Program of SCMPDS ; :: thesis: for k being Element of NAT st I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) holds
Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
let k be Element of NAT ; :: thesis: ( I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) implies Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT )
set SA0 = Start-At (inspos 0 );
set spI = stop I;
set s1 = s +* (Initialized (stop I));
set s2 = s +* ((I ';' J) +* (Start-At (inspos 0 )));
set n = LifeSpan (s +* (Initialized (stop I)));
assume A1:
( I is_closed_on s & I is_halting_on s & k <= LifeSpan (s +* (Initialized (stop I))) )
; :: thesis: Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
A2: s +* (Initialized (stop I)) =
(s +* (stop I)) +* (Start-At (inspos 0 ))
by FUNCT_4:15
.=
(s +* (Start-At (inspos 0 ))) +* (stop I)
by SCMPDS_4:62
;
A3: s +* ((I ';' J) +* (Start-At (inspos 0 ))) =
(s +* (I ';' J)) +* (Start-At (inspos 0 ))
by FUNCT_4:15
.=
(s +* (Start-At (inspos 0 ))) +* (I ';' J)
by SCMPDS_4:62
;
set IL = NAT ;
A4:
(s +* (Start-At (inspos 0 ))) +* (stop I),s +* (Start-At (inspos 0 )) equal_outside NAT
by AMI_1:120, FUNCT_7:28;
A5:
s +* (Start-At (inspos 0 )),(s +* (Start-At (inspos 0 ))) +* (I ';' J) equal_outside NAT
by AMI_1:120;
defpred S1[ Element of NAT ] means ( $1 <= LifeSpan (s +* (Initialized (stop I))) implies Computation (s +* (Initialized (stop I))),$1, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),$1 equal_outside NAT );
( Computation (s +* (Initialized (stop I))),0 = s +* (Initialized (stop I)) & Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),0 = s +* ((I ';' J) +* (Start-At (inspos 0 ))) )
by AMI_1:13;
then A6:
S1[ 0 ]
by A2, A3, A4, A5, FUNCT_7:29;
A7:
for m being Element of NAT st S1[m] holds
S1[m + 1]
proof
let m be
Element of
NAT ;
:: thesis: ( S1[m] implies S1[m + 1] )
assume A8:
(
m <= LifeSpan (s +* (Initialized (stop I))) implies
Computation (s +* (Initialized (stop I))),
m,
Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),
m equal_outside NAT )
;
:: thesis: S1[m + 1]
assume A9:
m + 1
<= LifeSpan (s +* (Initialized (stop I)))
;
:: thesis: Computation (s +* (Initialized (stop I))),(m + 1), Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),(m + 1) equal_outside NAT
then A10:
m < LifeSpan (s +* (Initialized (stop I)))
by NAT_1:13;
A11:
Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),
(m + 1) =
Following (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m)
by AMI_1:14
.=
Exec (CurInstr (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m)),
(Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m)
;
A12:
Computation (s +* (Initialized (stop I))),
(m + 1) =
Following (Computation (s +* (Initialized (stop I))),m)
by AMI_1:14
.=
Exec (CurInstr (Computation (s +* (Initialized (stop I))),m)),
(Computation (s +* (Initialized (stop I))),m)
;
A13:
IC (Computation (s +* (Initialized (stop I))),m) in dom (stop I)
by A1, SCMPDS_6:def 2;
A14:
IC (Computation (s +* (Initialized (stop I))),m) in dom I
by A1, A10, SCMPDS_6:40;
then A15:
IC (Computation (s +* (Initialized (stop I))),m) in dom (I ';' J)
by FUNCT_4:13;
CurInstr (Computation (s +* (Initialized (stop I))),m) =
(s +* (Initialized (stop I))) . (IC (Computation (s +* (Initialized (stop I))),m))
by AMI_1:54
.=
(stop I) . (IC (Computation (s +* (Initialized (stop I))),m))
by A2, A13, FUNCT_4:14
.=
I . (IC (Computation (s +* (Initialized (stop I))),m))
by A14, SCMPDS_4:37
.=
(I ';' J) . (IC (Computation (s +* (Initialized (stop I))),m))
by A14, SCMPDS_4:37
.=
(s +* ((I ';' J) +* (Start-At (inspos 0 )))) . (IC (Computation (s +* (Initialized (stop I))),m))
by A3, A15, FUNCT_4:14
.=
(Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m) . (IC (Computation (s +* (Initialized (stop I))),m))
by AMI_1:54
.=
CurInstr (Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),m)
by A8, A9, AMI_1:121, NAT_1:13
;
hence
Computation (s +* (Initialized (stop I))),
(m + 1),
Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),
(m + 1) equal_outside NAT
by A8, A9, A11, A12, NAT_1:13, SCMPDS_4:15;
:: thesis: verum
end;
for k being Element of NAT holds S1[k]
from NAT_1:sch 1(A6, A7);
hence
Computation (s +* (Initialized (stop I))),k, Computation (s +* ((I ';' J) +* (Start-At (inspos 0 )))),k equal_outside NAT
by A1; :: thesis: verum