let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being 0 -started State of SCM+FSA
for I being paraclosed Program of SCM+FSA st I c= P & P halts_on s holds
for m being Element of NAT st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (loop I)),s,m)
let s be 0 -started State of SCM+FSA; :: thesis: for I being paraclosed Program of SCM+FSA st I c= P & P halts_on s holds
for m being Element of NAT st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (loop I)),s,m)
let I be paraclosed Program of SCM+FSA; :: thesis: ( I c= P & P halts_on s implies for m being Element of NAT st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (loop I)),s,m) )
assume A2: I c= P ; :: thesis: ( not P halts_on s or for m being Element of NAT st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (loop I)),s,m) )
defpred S₁[ Nat] means ( $1 <= LifeSpan (P,s) implies Comput (P,s,$1) = Comput ((P +* (loop I)),s,$1) );
assume A3: P halts_on s ; :: thesis: for m being Element of NAT st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (loop I)),s,m)
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

set sI = s;
set PI = P +* (loop I);
A5: loop I c= P +* (loop I) by FUNCT_4:25;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan (P,s) implies Comput (P,s,m) = Comput ((P +* (loop I)),s,m) ) ; :: thesis: S₁[m + 1]
A7: IC (Comput (P,s,m)) in dom I by A2, AMISTD_1:def 10;
then A8: IC (Comput (P,s,m)) in dom (loop I) by FUNCT_4:99;
A9: P /. (IC (Comput (P,s,m))) = P . (IC (Comput (P,s,m))) by PBOOLE:143;
A10: CurInstr (P,(Comput (P,s,m))) = I . (IC (Comput (P,s,m))) by A7, A9, A2, GRFUNC_1:2;
A11: Comput ((P +* (loop I)),s,(m + 1)) = Following ((P +* (loop I)),(Comput ((P +* (loop I)),s,m))) by EXTPRO_1:3;
A12: Comput (P,s,(m + 1)) = Following (P,(Comput (P,s,m))) by EXTPRO_1:3;
A13: (P +* (loop I)) /. (IC (Comput ((P +* (loop I)),s,m))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),s,m))) by PBOOLE:143;
assume A14: m + 1 <= LifeSpan (P,s) ; :: thesis: Comput (P,s,(m + 1)) = Comput ((P +* (loop I)),s,(m + 1))
then m < LifeSpan (P,s) by NAT_1:13;
then I . (IC (Comput (P,s,m))) <> halt SCM+FSA by A3, A10, EXTPRO_1:def 15;
then CurInstr (P,(Comput (P,s,m))) = (loop I) . (IC (Comput (P,s,m))) by A10, FUNCT_4:105
.= (P +* (loop I)) . (IC (Comput (P,s,m))) by A8, A5, GRFUNC_1:2
.= CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),s,m))) by A6, A14, A13, NAT_1:13 ;
hence Comput (P,s,(m + 1)) = Comput ((P +* (loop I)),s,(m + 1)) by A6, A14, A12, A11, NAT_1:13; :: thesis: verum

end;

A15: Comput ((P +* (loop I)),s,0) = s by EXTPRO_1:2;
Comput (P,s,0) = s by EXTPRO_1:2;
then A16: S₁[ 0 ] by A15;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A16, A4); :: thesis: verum