let s be State of SCM+FSA; for P being Instruction-Sequence of SCM+FSA
for I being really-closed Program of
for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let P be Instruction-Sequence of SCM+FSA; for I being really-closed Program of
for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let I be really-closed Program of ; for a being Int-Location st not I destroys a holds
for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
let a be Int-Location; ( not I destroys a implies for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a )
assume A1:
not I destroys a
; for k being Nat holds (Comput ((P +* I),(Initialize s),k)) . a = s . a
defpred S1[ Nat] means (Comput ((P +* I),(Initialize s),$1)) . a = s . a;
A2:
I c= P +* I
by FUNCT_4:25;
A3:
now for k being Nat st S1[k] holds
S1[k + 1]let k be
Nat;
( S1[k] implies S1[k + 1] )assume A4:
S1[
k]
;
S1[k + 1]set l =
IC (Comput ((P +* I),(Initialize s),k));
IC (Initialize s) = 0
by MEMSTR_0:47;
then
IC (Initialize s) in dom I
by AFINSQ_1:65;
then A5:
IC (Comput ((P +* I),(Initialize s),k)) in dom I
by A2, AMISTD_1:21;
then
(P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) = I . (IC (Comput ((P +* I),(Initialize s),k)))
by A2, GRFUNC_1:2;
then
(P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) in rng I
by A5, FUNCT_1:def 3;
then A6:
not
(P +* I) . (IC (Comput ((P +* I),(Initialize s),k))) destroys a
by A1;
A7:
dom (P +* I) = NAT
by PARTFUN1:def 2;
(Comput ((P +* I),(Initialize s),(k + 1))) . a =
(Following ((P +* I),(Comput ((P +* I),(Initialize s),k)))) . a
by EXTPRO_1:3
.=
(Exec (((P +* I) . (IC (Comput ((P +* I),(Initialize s),k)))),(Comput ((P +* I),(Initialize s),k)))) . a
by A7, PARTFUN1:def 6
.=
(Comput ((P +* I),(s +* (Start-At (0,SCM+FSA))),k)) . a
by A6, Th19
.=
s . a
by A4
;
hence
S1[
k + 1]
;
verum end;
A8:
not a in dom (Start-At (0,SCM+FSA))
by SCMFSA_2:102;
(Comput ((P +* I),(Initialize s),0)) . a =
(Initialize s) . a
.=
s . a
by A8, FUNCT_4:11
;
then A9:
S1[ 0 ]
;
thus
for k being Nat holds S1[k]
from NAT_1:sch 2(A9, A3); verum