let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being Int-Location
for J being good Program of SCM+FSA
for k being Element of NAT st ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . (intloc 0) = 1 or not a is read-only ) holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations)
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being Int-Location
for J being good Program of SCM+FSA
for k being Element of NAT st ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . (intloc 0) = 1 or not a is read-only ) holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations)
let a be Int-Location ; :: thesis: for J being good Program of SCM+FSA
for k being Element of NAT st ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . (intloc 0) = 1 or not a is read-only ) holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations)
let J be good Program of SCM+FSA; :: thesis: for k being Element of NAT st ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . (intloc 0) = 1 or not a is read-only ) holds
((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations)
let k be Element of NAT ; :: thesis: ( ( ProperTimesBody a,J,s,p or J is parahalting ) & k < s . a & ( s . (intloc 0) = 1 or not a is read-only ) implies ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations) )
set UFLI = FinSeq-Locations ;
set I = J;
assume that
A1: ( ProperTimesBody a,J,s,p or J is parahalting ) and
A2: k < s . a and
A3: ( s . (intloc 0) = 1 or not a is read-only ) ; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations)
set ST = StepTimes (a,J,p,s);
A4: ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 by A1, A2, Th15, Th16;
set au = 1 -stRWNotIn ({a} \/ (UsedIntLoc J));
A5: ProperTimesBody a,J,s,p by A1, Th15;
then A6: (((StepTimes (a,J,p,s)) . k) . (1 -stRWNotIn ({a} \/ (UsedIntLoc J)))) + k = s . a by A2, A3, Th17;
A7: J is_closed_on (StepTimes (a,J,p,s)) . k,p +* (times* (a,J)) by A2, A5, Def4;
then A8: J is_closed_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) by A4, Th4;
A9: k - k < (s . a) - k by A2, XREAL_1:9;
J is_halting_on (StepTimes (a,J,p,s)) . k,p +* (times* (a,J)) by A2, A5, Def4;
then J is_halting_on Initialized ((StepTimes (a,J,p,s)) . k),p +* (times* (a,J)) by A4, A7, Th5;
hence ((StepTimes (a,J,p,s)) . (k + 1)) | ((UsedIntLoc J) \/ FinSeq-Locations) = (IExec (J,(p +* (times* (a,J))),((StepTimes (a,J,p,s)) . k))) | ((UsedIntLoc J) \/ FinSeq-Locations) by A4, A8, A6, A9, Th20; :: thesis: verum
set UILI = UsedIntLoc J;