let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for dd, bb, cc being Int-Location
for I being Program of {INT,(INT *)} st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd
let a be read-write Int-Location ; :: thesis: for dd, bb, cc being Int-Location
for I being Program of {INT,(INT *)} st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd
let dd, bb, cc be Int-Location ; :: thesis: for I being Program of {INT,(INT *)} st a <> dd & dd in UsedIntLoc I holds
((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd
let I be Program of {INT,(INT *)}; :: thesis: ( a <> dd & dd in UsedIntLoc I implies ((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd )
assume that
A1: a <> dd and
A2: dd in UsedIntLoc I ; :: thesis: ((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd
set aux = 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I));
dd in {a,bb,cc} \/ (UsedIntLoc I) by A2, XBOOLE_0:def 3;
then A3: dd <> 1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I)) by SFMASTR1:21;
set S = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb));
((s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) +* (a,(s . bb))) . dd = (s +* ((1 -stRWNotIn ({a,bb,cc} \/ (UsedIntLoc I))),(((s . cc) - (s . bb)) + 1))) . dd by A1, FUNCT_7:34
.= s . dd by A3, FUNCT_7:34 ;
hence ((StepForUp (a,bb,cc,I,s)) . 0) . dd = s . dd by SCMFSA_9:def 5; :: thesis: verum