let s be State of SCM+FSA ; :: thesis: for I being paraclosed Program of SCM+FSA st I +* (Start-At 0 ,SCM+FSA ) c= s & ProgramPart s halts_on s holds
for m being Element of NAT st m <= LifeSpan (ProgramPart s),s holds
Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m equal_outside NAT
let I be paraclosed Program of SCM+FSA ; :: thesis: ( I +* (Start-At 0 ,SCM+FSA ) c= s & ProgramPart s halts_on s implies for m being Element of NAT st m <= LifeSpan (ProgramPart s),s holds
Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m equal_outside NAT )
assume A1: I +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: ( not ProgramPart s halts_on s or for m being Element of NAT st m <= LifeSpan (ProgramPart s),s holds
Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m equal_outside NAT )
defpred S₁[ Nat] means ( $1 <= LifeSpan (ProgramPart s),s implies Comput (ProgramPart s),s,$1, Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),$1 equal_outside NAT );
assume A2: ProgramPart s halts_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan (ProgramPart s),s holds
Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (loop I))),(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 (ProgramPart s),s implies Comput (ProgramPart s),s,m, Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m equal_outside NAT ) ; :: thesis: S₁[m + 1]
A5: IC (Comput (ProgramPart s),s,m) in dom I by A1, SCMFSA6B:def 2;
then A6: IC (Comput (ProgramPart s),s,m) in dom (loop I) by FUNCT_4:105;
Y: (ProgramPart (Comput (ProgramPart s),s,m)) /. (IC (Comput (ProgramPart s),s,m)) = (Comput (ProgramPart s),s,m) . (IC (Comput (ProgramPart s),s,m)) by COMPOS_1:38;
TX: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,m) by AMI_1:123;
dom I misses dom (Start-At 0 ,SCM+FSA ) by SF_MASTR:64;
then I c= I +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:33;
then I c= s by A1, XBOOLE_1:1;
then I c= Comput (ProgramPart s),s,m by AMI_1:81;
then A7: CurInstr (ProgramPart s),(Comput (ProgramPart s),s,m) = I . (IC (Comput (ProgramPart s),s,m)) by A5, Y, TX, GRFUNC_1:8;
A8: loop I c= Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m by AMI_1:81, FUNCT_4:26;
T: ProgramPart (s +* (loop I)) = ProgramPart (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) by AMI_1:123;
A9: Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1) = Following (ProgramPart (s +* (loop I))),(Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)),(Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)),(Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) by T ;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,m) by AMI_1:123;
A10: Comput (ProgramPart s),s,(m + 1) = Following (ProgramPart s),(Comput (ProgramPart s),s,m) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s),s,m)),(Comput (ProgramPart s),s,m)),(Comput (ProgramPart s),s,m) by T ;
Y: (ProgramPart (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)) /. (IC (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)) = (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) . (IC (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)) by COMPOS_1:38;
assume A11: m + 1 <= LifeSpan (ProgramPart s),s ; :: thesis: Comput (ProgramPart s),s,(m + 1), Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1) equal_outside NAT
then m < LifeSpan (ProgramPart s),s by NAT_1:13;
then I . (IC (Comput (ProgramPart s),s,m)) <> halt SCM+FSA by A2, A7, AMI_1:def 46;
then CurInstr (ProgramPart (Comput (ProgramPart s),s,m)),(Comput (ProgramPart s),s,m) = (loop I) . (IC (Comput (ProgramPart s),s,m)) by A7, TX, FUNCT_4:111
.= (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) . (IC (Comput (ProgramPart s),s,m)) by A8, A6, GRFUNC_1:8
.= CurInstr (ProgramPart (Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m)),(Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),m) by A4, A11, Y, COMPOS_1:24, NAT_1:13 ;
hence Comput (ProgramPart s),s,(m + 1), Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),(m + 1) equal_outside NAT by A4, A11, A10, A9, NAT_1:13, SCMFSA6A:32; :: thesis: verum

end;

A12: Comput (ProgramPart (s +* (loop I))),(s +* (loop I)),0 = s +* (loop I) by AMI_1:13;
Comput (ProgramPart s),s,0 = s by AMI_1:13;
then A13: S₁[ 0 ] by A12, FUNCT_7:132;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A13, A3); :: thesis: verum