let s be 0 -started State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being paraclosed Program of st P +* I halts_on s & Directed I c= P holds
IC (Comput (P,s,((LifeSpan ((P +* I),s)) + 1))) = card I
let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being paraclosed Program of st P +* I halts_on s & Directed I c= P holds
IC (Comput (P,s,((LifeSpan ((P +* I),s)) + 1))) = card I
let I be paraclosed Program of ; :: thesis: ( P +* I halts_on s & Directed I c= P implies IC (Comput (P,s,((LifeSpan ((P +* I),s)) + 1))) = card I )
assume that
A1: P +* I halts_on s and
A3: Directed I c= P ; :: thesis: IC (Comput (P,s,((LifeSpan ((P +* I),s)) + 1))) = card I
A5: I c= P +* I by FUNCT_4:25;
set s2 = s;
set m = LifeSpan ((P +* I),s);
set l1 = IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s))));
A9: I c= P +* I by FUNCT_4:25;
A10: IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s)))) in dom I by A9, AMISTD_1:def 10;
set s1 = s;
A11: P +* (I ';' I) = P +* (I +* (I ';' I)) by SCMFSA6A:18
.= (P +* I) +* (I ';' I) by FUNCT_4:14 ;

now

let k be Element of NAT ; :: thesis: ( k <= LifeSpan ((P +* I),s) implies Comput ((P +* I),s,k) = Comput ((P +* (Directed I)),s,k) )
defpred S₁[ Nat] means ( $1 <= k implies Comput ((P +* (I ';' I)),s,$1) = Comput ((P +* (Directed I)),s,$1) );
assume A13: k <= LifeSpan ((P +* I),s) ; :: thesis: Comput ((P +* I),s,k) = Comput ((P +* (Directed I)),s,k)
A14: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A15: ( n <= k implies Comput ((P +* (I ';' I)),s,n) = Comput ((P +* (Directed I)),s,n) ) ; :: thesis: S₁[n + 1]
A16: Comput ((P +* (Directed I)),s,(n + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n)))),(Comput ((P +* (Directed I)),s,n))) ;
A17: Comput ((P +* (I ';' I)),s,(n + 1)) = Following ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n))) by EXTPRO_1:3
.= Exec ((CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n)))),(Comput ((P +* (I ';' I)),s,n))) ;
A18: n <= n + 1 by NAT_1:12;
assume A19: n + 1 <= k ; :: thesis: Comput ((P +* (I ';' I)),s,(n + 1)) = Comput ((P +* (Directed I)),s,(n + 1))
n <= k by A19, A18, XXREAL_0:2;
then Comput ((P +* I),s,n) = Comput ((P +* (I ';' I)),s,n) by A5, Th36, A11, A1, A13, XXREAL_0:2;
then IC (Comput ((P +* (I ';' I)),s,n)) in dom I by A5, AMISTD_1:def 10;
then A21: IC (Comput ((P +* (Directed I)),s,n)) in dom (Directed I) by A19, A15, A18, FUNCT_4:99, XXREAL_0:2;
dom (P +* (Directed I)) = NAT by PARTFUN1:def 2;
then A22: (P +* (Directed I)) /. (IC (Comput ((P +* (Directed I)),s,n))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),s,n))) by PARTFUN1:def 6;
A23: dom (P +* (I ';' I)) = NAT by PARTFUN1:def 2;
Directed I c= P +* (Directed I) by FUNCT_4:25;
then A24: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n))) = (Directed I) . (IC (Comput ((P +* (Directed I)),s,n))) by A21, A22, GRFUNC_1:2;
A25: ( dom I c= dom (I ';' I) & CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n))) = (P +* (I ';' I)) . (IC (Comput ((P +* (I ';' I)),s,n))) ) by A23, PARTFUN1:def 6, SCMFSA6A:17;
A26: Directed I c= I ';' I by SCMFSA6A:16;
I ';' I c= P +* (I ';' I) by FUNCT_4:25;
then Directed I c= P +* (I ';' I) by A26, XBOOLE_1:1;
hence Comput ((P +* (I ';' I)),s,(n + 1)) = Comput ((P +* (Directed I)),s,(n + 1)) by A15, A19, A18, A24, A17, A16, A21, A25, GRFUNC_1:2, XXREAL_0:2; :: thesis: verum

end;

( Comput ((P +* (I ';' I)),s,0) = s & Comput ((P +* (Directed I)),s,0) = s ) by EXTPRO_1:2;
then A28: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A28, A14);
then Comput ((P +* (I ';' I)),s,k) = Comput ((P +* (Directed I)),s,k) ;
hence Comput ((P +* I),s,k) = Comput ((P +* (Directed I)),s,k) by A13, Th36, A11, A1, FUNCT_4:25; :: thesis: verum

end;

then B30: Comput ((P +* I),s,(LifeSpan ((P +* I),s))) = Comput ((P +* (Directed I)),s,(LifeSpan ((P +* I),s))) ;
A31: dom (P +* I) = NAT by PARTFUN1:def 2;
I c= P +* I by FUNCT_4:25;
then A32: I . (IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s))))) = (P +* I) . (IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s))))) by A10, GRFUNC_1:2
.= CurInstr ((P +* I),(Comput ((P +* I),s,(LifeSpan ((P +* I),s))))) by A31, PARTFUN1:def 6
.= halt SCM+FSA by A1, EXTPRO_1:def 15 ;
IC (Comput ((P +* (Directed I)),s,(LifeSpan ((P +* I),s)))) in dom (Directed I) by A10, B30, FUNCT_4:99;
then A33: (P +* (Directed I)) . (IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s))))) = (Directed I) . (IC (Comput ((P +* I),s,(LifeSpan ((P +* I),s))))) by B30, FUNCT_4:13
.= goto (card I) by A10, A32, FUNCT_4:106 ;
A35: P +* (Directed I) = P by A3, FUNCT_4:98;
B36: dom (P +* (Directed I)) = NAT by PARTFUN1:def 2;
Comput ((P +* (Directed I)),s,((LifeSpan ((P +* I),s)) + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,(LifeSpan ((P +* I),s))))) by EXTPRO_1:3
.= Exec ((goto (card I)),(Comput ((P +* (Directed I)),s,(LifeSpan ((P +* I),s))))) by B36, B30, A33, PARTFUN1:def 6 ;
hence IC (Comput (P,s,((LifeSpan ((P +* I),s)) + 1))) = card I by A35, SCMFSA_2:69; :: thesis: verum