let s be State of ; :: thesis: for I being paraclosed Program of st I +* (Start-At (insloc 0 )) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (loop I)),m equal_outside NAT
let I be paraclosed Program of ; :: thesis: ( I +* (Start-At (insloc 0 )) c= s & ProgramPart s halts_on s implies for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (loop I)),m equal_outside NAT )
assume A1: I +* (Start-At (insloc 0 )) c= s ; :: thesis: ( not ProgramPart s halts_on s or for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (loop I)),m equal_outside NAT )
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan s implies Computation s,$1, Computation (s +* (loop I)),$1 equal_outside NAT );
assume A2: ProgramPart s halts_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan s holds
Computation s,m, Computation (s +* (loop I)),m equal_outside NAT
A3: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

set sI = s +* (loop I);
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A4: ( m <= LifeSpan s implies Computation s,m, Computation (s +* (loop I)),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
A5: IC (Computation s,m) in dom I by A1, SCMFSA6B:def 2;
then A6: IC (Computation s,m) in dom (loop I) by FUNCT_4:105;
dom I misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then I c= I +* (Start-At (insloc 0 )) by FUNCT_4:33;
then I c= s by A1, XBOOLE_1:1;
then I c= Computation s,m by AMI_1:81;
then A7: CurInstr (Computation s,m) = I . (IC (Computation s,m)) by A5, GRFUNC_1:8;
A8: loop I c= Computation (s +* (loop I)),m by AMI_1:81, FUNCT_4:26;
A9: Computation (s +* (loop I)),(m + 1) = Following (Computation (s +* (loop I)),m) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (loop I)),m)),(Computation (s +* (loop I)),m) ;
A10: Computation s,(m + 1) = Following (Computation s,m) by AMI_1:14
.= Exec (CurInstr (Computation s,m)),(Computation s,m) ;
assume A11: m + 1 <= LifeSpan s ; :: thesis: Computation s,(m + 1), Computation (s +* (loop I)),(m + 1) equal_outside NAT
then m < LifeSpan s by NAT_1:13;
then I . (IC (Computation s,m)) <> halt SCM+FSA by A2, A7, AMI_1:def 46;
then CurInstr (Computation s,m) = (loop I) . (IC (Computation s,m)) by A7, FUNCT_4:111
.= (Computation (s +* (loop I)),m) . (IC (Computation s,m)) by A8, A6, GRFUNC_1:8
.= CurInstr (Computation (s +* (loop I)),m) by A4, A11, AMI_1:121, NAT_1:13 ;
hence Computation s,(m + 1), Computation (s +* (loop I)),(m + 1) equal_outside NAT by A4, A11, A10, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A12: Computation (s +* (loop I)),0 = s +* (loop I) by AMI_1:13;
Computation s,0 = s by AMI_1:13;
then A13: S₁[ 0 ] by A12, AMI_1:120;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A13, A3); :: thesis: verum