let s be State of ; :: thesis: for I being parahalting Program of
for J being Program of st Initialized (stop I) c= s holds
for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
let I be parahalting Program of ; :: thesis: for J being Program of st Initialized (stop I) c= s holds
for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
let J be Program of ; :: thesis: ( Initialized (stop I) c= s implies for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT )
set SI = stop I;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan s implies Computation s,$1, Computation (s +* (I ';' J)),$1 equal_outside NAT );
A1: Computation (s +* (I ';' J)),0 = s +* (I ';' J) by AMI_1:13;
assume A2: Initialized (stop I) c= s ; :: thesis: for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT
then A3: ProgramPart s halts_on s by SCMPDS_4:63;
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom I) \/ (dom (Shift J,(card I))) by FUNCT_4:def 1;
then A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan s implies Computation s,m, Computation (s +* (I ';' J)),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
assume A7: m + 1 <= LifeSpan s ; :: thesis: Computation s,(m + 1), Computation (s +* (I ';' J)),(m + 1) equal_outside NAT
then A8: IC (Computation s,m) = IC (Computation (s +* (I ';' J)),m) by A6, AMI_1:121, NAT_1:13;
A9: Computation (s +* (I ';' J)),(m + 1) = Following (Computation (s +* (I ';' J)),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (I ';' J)),m)),(Computation (s +* (I ';' J)),m) ;
A10: Computation s,(m + 1) = Following (Computation s,m) by AMI_1:14
.= Exec (CurInstr (Computation s,m)),(Computation s,m) ;
A11: I ';' J c= Computation (s +* (I ';' J)),m by AMI_1:81, FUNCT_4:26;
A12: IC (Computation s,m) in dom (stop I) by A2, SCMPDS_4:def 9;
dom (stop I) misses dom (Start-At (inspos 0 )) by SCMPDS_4:54;
then stop I c= (stop I) +* (Start-At (inspos 0 )) by FUNCT_4:33;
then stop I c= s by A2, XBOOLE_1:1;
then stop I c= Computation s,m by AMI_1:81;
then A13: CurInstr (Computation s,m) = (stop I) . (IC (Computation s,m)) by A12, GRFUNC_1:8;
m < LifeSpan s by A7, NAT_1:13;
then (stop I) . (IC (Computation s,m)) <> halt SCMPDS by A3, A13, AMI_1:def 46;
then A14: IC (Computation s,m) in dom I by A12, Th3;
then CurInstr (Computation s,m) = I . (IC (Computation s,m)) by A13, SCMPDS_4:37
.= (I ';' J) . (IC (Computation s,m)) by A14, SCMPDS_4:37
.= CurInstr (Computation (s +* (I ';' J)),m) by A8, A11, A14, A5, GRFUNC_1:8 ;
hence Computation s,(m + 1), Computation (s +* (I ';' J)),(m + 1) equal_outside NAT by A6, A7, A10, A9, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

Computation s,0 = s by AMI_1:13;
then A15: S₁[ 0 ] by A1, AMI_1:120;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A15, A4); :: thesis: verum