let s be State of ; :: thesis: for I being Program of st I is_closed_on s & I is_halting_on s holds
for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
let I be Program of ; :: thesis: ( I is_closed_on s & I is_halting_on s implies for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) )
set s1 = s +* (I +* (Start-At (insloc 0 )));
set s2 = s +* ((loop I) +* (Start-At (insloc 0 )));
assume that
A1: I is_closed_on s and
A2: I is_halting_on s ; :: thesis: for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
let m be Element of NAT ; :: thesis: ( m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) )
A3: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom I by A1, SCMFSA7B:def 7;
then A4: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom (loop I) by FUNCT_4:105;
A5: I c= I +* (Start-At (insloc 0 )) by SCMFSA8A:9;
I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then I c= s +* (I +* (Start-At (insloc 0 ))) by A5, XBOOLE_1:1;
then I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
then A6: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A3, GRFUNC_1:8;
A7: loop I c= (loop I) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
(loop I) +* (Start-At (insloc 0 )) c= s +* ((loop I) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then loop I c= s +* ((loop I) +* (Start-At (insloc 0 ))) by A7, XBOOLE_1:1;
then A8: loop I c= Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m by AMI_1:81;
assume A9: m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
ProgramPart (s +* (I +* (Start-At (insloc 0 )))) halts_on s +* (I +* (Start-At (insloc 0 ))) by A2, SCMFSA7B:def 8;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A9, A6, AMI_1:def 46;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) = (loop I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by FUNCT_4:111;
hence CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A8, A4, A6, GRFUNC_1:8
.= CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) by A1, A2, A9, Th109, AMI_1:121 ;
:: thesis: verum