let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for I being Program of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations)
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being Int-Location
for I being Program of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations)
let a be Int-Location ; :: thesis: for I being Program of SCM+FSA st s . a <= 0 & s . (intloc 0) = 1 holds
(IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations)
let I be Program of SCM+FSA; :: thesis: ( s . a <= 0 & s . (intloc 0) = 1 implies (IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations) )
set FSL = FinSeq-Locations ;
assume that
A1: s . a <= 0 and
A2: s . (intloc 0) = 1 ; :: thesis: (IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations)
set UILI = UsedIntLoc I;
set au = 1 -stRWNotIn ({a} \/ (UsedIntLoc I));
set WH = while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0)))));
set s1 = Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s));
A3: ( a = intloc 0 or not a is read-only ) by SF_MASTR:def 5;
A4: Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s)) = IExec ((Macro ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a)),p,s) by SCMFSA6C:5;
A5: (Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))) . (1 -stRWNotIn ({a} \/ (UsedIntLoc I))) = (Initialized s) . a by SCMFSA_2:63
.= s . a by A2, A3, SCMFSA6A:38, SCMFSA6C:3 ;
then A6: ( while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0))))) is_closed_on IExec ((Macro ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a)),p,s),p & while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0))))) is_halting_on IExec ((Macro ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a)),p,s),p ) by A1, A4, SCMFSA_9:38;
(Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA6A:38 ;
then A7: DataPart (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))))) = DataPart (Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))) by A1, A5, SCMFSA9A:35;

A8: now

let x be FinSeq-Location ; :: thesis: ( x in FinSeq-Locations implies (IExec ((times (a,I)),p,s)) . x = s . x )
assume x in FinSeq-Locations ; :: thesis: (IExec ((times (a,I)),p,s)) . x = s . x
thus (IExec ((times (a,I)),p,s)) . x = (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))))) . x by A4, A6, SFMASTR1:15
.= (Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))) . x by A7, SCMFSA6A:7
.= (Initialized s) . x by SCMFSA_2:63
.= s . x by SCMFSA6C:3 ; :: thesis: verum

end;

A9: DataPart s = DataPart (Initialized s) by A2, SCMFSA8C:7;

A10: now

let x be Int-Location ; :: thesis: ( x in UsedIntLoc I implies (IExec ((times (a,I)),p,s)) . x = s . x )
A11: not 1 -stRWNotIn ({a} \/ (UsedIntLoc I)) in {a} \/ (UsedIntLoc I) by SFMASTR1:20;
assume x in UsedIntLoc I ; :: thesis: (IExec ((times (a,I)),p,s)) . x = s . x
then A12: 1 -stRWNotIn ({a} \/ (UsedIntLoc I)) <> x by A11, XBOOLE_0:def 3;
thus (IExec ((times (a,I)),p,s)) . x = (IExec ((while>0 ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(I ';' (SubFrom ((1 -stRWNotIn ({a} \/ (UsedIntLoc I))),(intloc 0)))))),p,(Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))))) . x by A4, A6, SFMASTR1:14
.= (Exec (((1 -stRWNotIn ({a} \/ (UsedIntLoc I))) := a),(Initialized s))) . x by A7, SCMFSA6A:7
.= (Initialized s) . x by A12, SCMFSA_2:63
.= s . x by A9, SCMFSA6A:7 ; :: thesis: verum

end;

[#] FinSeq-Locations = FinSeq-Locations ;
hence (IExec ((times (a,I)),p,s)) | ((UsedIntLoc I) \/ FinSeq-Locations) = s | ((UsedIntLoc I) \/ FinSeq-Locations) by A10, A8, Th6; :: thesis: verum