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 +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))
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 +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))
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 +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)) )
set s1 = s +* (Initialize ((intloc 0) .--> 1));
set p1 = p +* I;
A1: I c= p +* I by FUNCT_4:26;
set s2 = s +* (Initialize ((intloc 0) .--> 1));
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 +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)) )
defpred S1[ Nat] means ( $1 <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) implies NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),$1)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),$1)) );
assume I is_halting_onInit s,p ; :: thesis: for m being Element of NAT st m <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) holds
NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))
then A4: p +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) 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 +* (Initialize ((intloc 0) .--> 1)))) implies NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)) ) ; :: thesis: S1[m + 1]
A7: IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom I by A3, Def4;
then A8: IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom (loop I) by FUNCT_4:105;
A9: (p +* I) /. (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) = (p +* I) . (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) by PBOOLE:158;
A10: CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) = I . (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) by A7, A9, A1, GRFUNC_1:8;
A11: Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1)) = Following ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)))),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) ;
A12: Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1)) = Following ((p +* I),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)))),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) ;
assume A13: m + 1 <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) ; :: thesis: NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1))) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1)))
then m < LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) by NAT_1:13;
then I . (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) <> halt SCM+FSA by A4, A10, EXTPRO_1:def 14;
then A14: I . (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) = (loop I) . (IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) by FUNCT_4:111;
A15: (p +* (loop I)) /. (IC (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) = (p +* (loop I)) . (IC (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) by PBOOLE:158;
IC (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) = IC (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m)) by A6, A13, COMPOS_1:230, NAT_1:13;
then CurInstr ((p +* I),(Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),m))) = CurInstr ((p +* (loop I)),(Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),m))) by A8, A10, A15, A14, A2, GRFUNC_1:8;
hence NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1))) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),(m + 1))) by A6, A13, A12, A11, AMISTD_2:def 20, NAT_1:13; :: thesis: verum
end;
A16: S1[ 0 ]
proof
assume 0 <= LifeSpan ((p +* I),(s +* (Initialize ((intloc 0) .--> 1)))) ; :: thesis: NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),0)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),0))
NPP (s +* (Initialize ((intloc 0) .--> 1))) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),0)) by EXTPRO_1:3;
hence NPP (Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),0)) = NPP (Comput ((p +* (loop I)),(s +* (Initialize ((intloc 0) .--> 1))),0)) 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