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_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m))
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s,P & I is_halting_on s,P holds
for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m))
set A = NAT ;
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P implies for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m)) )
set s1 = Initialize s;
set P1 = P +* I;
A2: I c= P +* I by FUNCT_4:26;
set s2 = Initialize s;
set P2 = P +* (loop I);
A3: loop I c= P +* (loop I) by FUNCT_4:26;
assume A4: I is_closed_on s,P ; :: thesis: ( not I is_halting_on s,P or for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m)) )
defpred S₁[ Nat] means ( $1 <= LifeSpan ((P +* I),(Initialize s)) implies NPP (Comput ((P +* I),(Initialize s),$1)) = NPP (Comput ((P +* (loop I)),(Initialize s),$1)) );
assume I is_halting_on s,P ; :: thesis: for m being Element of NAT st m <= LifeSpan ((P +* I),(Initialize s)) holds
NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m))
then A5: P +* I halts_on Initialize s by SCMFSA7B:def 8;
A6: 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 A7: ( m <= LifeSpan ((P +* I),(Initialize s)) implies NPP (Comput ((P +* I),(Initialize s),m)) = NPP (Comput ((P +* (loop I)),(Initialize s),m)) ) ; :: thesis: S₁[m + 1]
A8: Comput ((P +* I),(Initialize s),(m + 1)) = Following ((P +* I),(Comput ((P +* I),(Initialize s),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m)))),(Comput ((P +* I),(Initialize s),m))) ;
A9: Comput ((P +* (loop I)),(Initialize s),(m + 1)) = Following ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m)))),(Comput ((P +* (loop I)),(Initialize s),m))) ;
A10: IC (Comput ((P +* I),(Initialize s),m)) in dom I by A4, SCMFSA7B:def 7;
A11: (P +* I) /. (IC (Comput ((P +* I),(Initialize s),m))) = (P +* I) . (IC (Comput ((P +* I),(Initialize s),m))) by PBOOLE:158;
A12: CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = I . (IC (Comput ((P +* I),(Initialize s),m))) by A10, A11, A2, GRFUNC_1:8;
assume A13: m + 1 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: NPP (Comput ((P +* I),(Initialize s),(m + 1))) = NPP (Comput ((P +* (loop I)),(Initialize s),(m + 1)))
then m < LifeSpan ((P +* I),(Initialize s)) by NAT_1:13;
then B14: I . (IC (Comput ((P +* I),(Initialize s),m))) <> halt SCM+FSA by A5, A12, EXTPRO_1:def 14;
A15: (P +* (loop I)) /. (IC (Comput ((P +* (loop I)),(Initialize s),m))) = (P +* (loop I)) . (IC (Comput ((P +* (loop I)),(Initialize s),m))) by PBOOLE:158;
A16: IC (Comput ((P +* I),(Initialize s),m)) in dom (loop I) by A10, FUNCT_4:105;
CurInstr ((P +* I),(Comput ((P +* I),(Initialize s),m))) = (P +* I) . (IC (Comput ((P +* I),(Initialize s),m))) by PBOOLE:158
.= I . (IC (Comput ((P +* I),(Initialize s),m))) by A2, A10, GRFUNC_1:8
.= (loop I) . (IC (Comput ((P +* I),(Initialize s),m))) by B14, FUNCT_4:111
.= (P +* (loop I)) . (IC (Comput ((P +* I),(Initialize s),m))) by A3, A16, GRFUNC_1:8
.= CurInstr ((P +* (loop I)),(Comput ((P +* (loop I)),(Initialize s),m))) by A7, A13, A15, COMPOS_1:230, NAT_1:13 ;
hence NPP (Comput ((P +* I),(Initialize s),(m + 1))) = NPP (Comput ((P +* (loop I)),(Initialize s),(m + 1))) by A7, A13, A8, A9, AMISTD_2:def 20, NAT_1:13; :: thesis: verum

end;

A17: S₁[ 0 ]

proof

assume 0 <= LifeSpan ((P +* I),(Initialize s)) ; :: thesis: NPP (Comput ((P +* I),(Initialize s),0)) = NPP (Comput ((P +* (loop I)),(Initialize s),0))
NPP (Initialize s) = NPP (Comput ((P +* (loop I)),(Initialize s),0)) by EXTPRO_1:3;
hence NPP (Comput ((P +* I),(Initialize s),0)) = NPP (Comput ((P +* (loop I)),(Initialize s),0)) by EXTPRO_1:3; :: thesis: verum

end;

thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A17, A6); :: thesis: verum