let p be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_onInit s,p & I is_halting_onInit s,p holds
for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_onInit s,p & I is_halting_onInit s,p holds
for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT
let I be Program of SCM+FSA; :: thesis: ( I is_closed_onInit s,p & I is_halting_onInit s,p implies for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT )
set s1 = s +* (Initialized I);
set p1 = p +* I;
A1: I c= p +* I by FUNCT_4:26;
set s2 = s +* (Initialized (loop I));
set p2 = p +* (loop I);
A2: loop I c= p +* (loop I) by FUNCT_4:26;
assume A3: I is_closed_onInit s,p ; :: thesis: ( not I is_halting_onInit s,p or for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT )
defpred S₁[ Nat] means ( $1 <= LifeSpan ((p +* I),(s +* (Initialized I))) implies Comput ((p +* I),(s +* (Initialized I)),$1), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),$1) equal_outside NAT );
assume I is_halting_onInit s,p ; :: thesis: for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialized I))) holds
Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT
then A4: p +* I halts_on s +* (Initialized I) by Def5;
A5: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan ((p +* I),(s +* (Initialized I))) implies Comput ((p +* I),(s +* (Initialized I)),m), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
A7: IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom I by A3, Def4;
then A8: IC (Comput ((p +* I),(s +* (Initialized I)),m)) in dom (loop I) by FUNCT_4:105;
A9: (p +* I) /. (IC (Comput ((p +* I),(s +* (Initialized I)),m))) = (p +* I) . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by PBOOLE:158;
A10: CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) = I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by A7, A9, GRFUNC_1:8, A1;
A11: Comput ((p +* (loop I)),(s +* (Initialized (loop I))),(m + 1)) = Following ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m)))),(Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m))) ;
A12: Comput ((p +* I),(s +* (Initialized I)),(m + 1)) = Following ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m)))),(Comput ((p +* I),(s +* (Initialized I)),m))) ;
assume A13: m + 1 <= LifeSpan ((p +* I),(s +* (Initialized I))) ; :: thesis: Comput ((p +* I),(s +* (Initialized I)),(m + 1)), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),(m + 1)) equal_outside NAT
then m < LifeSpan ((p +* I),(s +* (Initialized I))) by NAT_1:13;
then I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) <> halt SCM+FSA by A4, A10, EXTPRO_1:def 14;
then A14: I . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) = (loop I) . (IC (Comput ((p +* I),(s +* (Initialized I)),m))) by FUNCT_4:111;
A15: (p +* (loop I)) /. (IC (Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m))) = (p +* (loop I)) . (IC (Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m))) by PBOOLE:158;
IC (Comput ((p +* I),(s +* (Initialized I)),m)) = IC (Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m)) by A6, A13, COMPOS_1:24, NAT_1:13;
then CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialized I)),m))) = CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialized (loop I))),m))) by A8, A10, A15, A14, GRFUNC_1:8, A2;
hence Comput ((p +* I),(s +* (Initialized I)),(m + 1)), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),(m + 1)) equal_outside NAT by A6, A13, A12, A11, NAT_1:13, AMISTD_2:def 20; :: thesis: verum

end;

A16: S₁[ 0 ]

proof

assume 0 <= LifeSpan ((p +* I),(s +* (Initialized I))) ; :: thesis: Comput ((p +* I),(s +* (Initialized I)),0), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),0) equal_outside NAT
( s +* I,s equal_outside NAT & s,s +* (loop I) equal_outside NAT ) by FUNCT_7:28, FUNCT_7:132;
then s +* I,s +* (loop I) equal_outside NAT by FUNCT_7:29;
then (s +* I) +* (Initialize ((intloc 0) .--> 1)),(s +* (loop I)) +* (Initialize ((intloc 0) .--> 1)) equal_outside NAT by FUNCT_7:106;
then s +* (I +* (Initialize ((intloc 0) .--> 1))),(s +* (loop I)) +* (Initialize ((intloc 0) .--> 1)) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Initialize ((intloc 0) .--> 1))),s +* ((loop I) +* (Initialize ((intloc 0) .--> 1))) equal_outside NAT by FUNCT_4:15;
then s +* (I +* (Initialize ((intloc 0) .--> 1))),s +* (Initialized (loop I)) equal_outside NAT by FUNCT_4:15;
then s +* (Initialized I),s +* (Initialized (loop I)) equal_outside NAT by FUNCT_4:15;
then s +* (Initialized I), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),0) equal_outside NAT by EXTPRO_1:3;
hence Comput ((p +* I),(s +* (Initialized I)),0), Comput ((p +* (loop I)),(s +* (Initialized (loop I))),0) equal_outside NAT by EXTPRO_1:3; :: thesis: verum

end;

thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A16, A5); :: thesis: verum