let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being Program of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Element of NAT st IC ((StepWhile>0 (a,P,s,I)) . k) = 0 & (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) & ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 holds
( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) )
let I be Program of SCM+FSA; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Element of NAT st IC ((StepWhile>0 (a,P,s,I)) . k) = 0 & (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) & ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 holds
( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) )
let a be read-write Int-Location ; :: thesis: for s being State of SCM+FSA
for k, n being Element of NAT st IC ((StepWhile>0 (a,P,s,I)) . k) = 0 & (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) & ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 holds
( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) )
let s be State of SCM+FSA; :: thesis: for k, n being Element of NAT st IC ((StepWhile>0 (a,P,s,I)) . k) = 0 & (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) & ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 holds
( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) )
let k, n be Element of NAT ; :: thesis: ( IC ((StepWhile>0 (a,P,s,I)) . k) = 0 & (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) & ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 implies ( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) ) )
set D = Data-Locations SCM+FSA;
set s1 = s +* (Initialized (while>0 (a,I)));
set P1 = P +* (while>0 (a,I));
set sk = (StepWhile>0 (a,P,s,I)) . k;
set s0k = Initialized ((StepWhile>0 (a,P,s,I)) . k);
set At0 = Initialize (while>0 (a,I));
set s2 = (Initialized ((StepWhile>0 (a,P,s,I)) . k)) +* (Initialize (while>0 (a,I)));
set s3 = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I)));
assume A1: IC ((StepWhile>0 (a,P,s,I)) . k) = 0 ; :: thesis: ( not (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) or not ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 or ( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) ) )
A2: IC in dom (Initialize (while>0 (a,I))) by COMPOS_1:141;
A3: IC (((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I)))) = ((Initialized ((StepWhile>0 (a,P,s,I)) . k)) +* (Initialize (while>0 (a,I)))) . (IC ) by SCMFSA8A:13
.= IC (Initialize (while>0 (a,I))) by A2, FUNCT_4:14
.= IC ((StepWhile>0 (a,P,s,I)) . k) by A1, COMPOS_1:142 ;
assume A4: (StepWhile>0 (a,P,s,I)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),n) ; :: thesis: ( not ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 or ( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) ) )
then A5: ProgramPart ((StepWhile>0 (a,P,s,I)) . k) = ProgramPart (s +* (Initialized (while>0 (a,I)))) by AMI_1:123;
A6: ProgramPart (((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I)))) = (ProgramPart (s +* (Initialized (while>0 (a,I))))) +* (ProgramPart (Initialized (while>0 (a,I)))) by FUNCT_4:75, A5
.= ((ProgramPart s) +* (ProgramPart (Initialized (while>0 (a,I))))) +* (ProgramPart (Initialized (while>0 (a,I)))) by FUNCT_4:75
.= (ProgramPart s) +* (ProgramPart (Initialized (while>0 (a,I)))) by FUNCT_4:99
.= ProgramPart (s +* (Initialized (while>0 (a,I)))) by FUNCT_4:75 ;
assume A7: ((StepWhile>0 (a,P,s,I)) . k) . (intloc 0) = 1 ; :: thesis: ( (StepWhile>0 (a,P,s,I)) . k = ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) & (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) )
DataPart (((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I)))) = DataPart ((Initialized ((StepWhile>0 (a,P,s,I)) . k)) +* (Initialize (while>0 (a,I)))) by SCMFSA8A:13
.= DataPart (Initialized ((StepWhile>0 (a,P,s,I)) . k)) by SCMFSA8A:11
.= DataPart ((StepWhile>0 (a,P,s,I)) . k) by A1, A7, SCMFSA8C:14 ;
hence ((StepWhile>0 (a,P,s,I)) . k) +* (Initialized (while>0 (a,I))) = (StepWhile>0 (a,P,s,I)) . k by A3, SCMFSA_9:29, A6, A5; :: thesis: (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3)))
hence (StepWhile>0 (a,P,s,I)) . (k + 1) = Comput ((P +* (while>0 (a,I))),((StepWhile>0 (a,P,s,I)) . k),((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3)) by Def1
.= Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,P,s,I)) . k) +* (Initialized I)))) + 3))) by A4, EXTPRO_1:5 ;
:: thesis: verum