let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for cc, bb being Int-Location
for I being Program of {INT,(INT *)} st a <> cc holds
((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc
let a be read-write Int-Location ; :: thesis: for cc, bb being Int-Location
for I being Program of {INT,(INT *)} st a <> cc holds
((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc
let cc, bb be Int-Location ; :: thesis: for I being Program of {INT,(INT *)} st a <> cc holds
((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc
let I be Program of {INT,(INT *)}; :: thesis: ( a <> cc implies ((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc )
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
cc in {a,bb,cc} by ENUMSET1:def 1;
then cc in {a,bb,cc} \/ (UsedIntLoc I) by XBOOLE_0:def 3;
then A1: cc <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SFMASTR1:21;
assume a <> cc ; :: thesis: ((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc
then ((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . cc = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . cc by FUNCT_7:34
.= s . cc by A1, FUNCT_7:34 ;
hence ((StepForUp (a,bb,cc,I,s)) . 0) . cc = s . cc by SCMFSA_9:def 5; :: thesis: verum