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,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) holds
( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize 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,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) holds
( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize 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,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) holds
( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) )
let s be State of SCM+FSA; :: thesis: for k, n being Element of NAT st IC ((StepWhile>0 (a,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) holds
( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) )
let k, n be Element of NAT ; :: thesis: ( IC ((StepWhile>0 (a,I,P,s)) . k) = 0 & (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) implies ( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) ) )
set D = Int-Locations \/ FinSeq-Locations;
set s1 = s +* (Initialize (while>0 (a,I)));
set P1 = P +* (while>0 (a,I));
set sk = (StepWhile>0 (a,I,P,s)) . k;
set s2 = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I)));
assume A1: IC ((StepWhile>0 (a,I,P,s)) . k) = 0 ; :: thesis: ( not (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) or ( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) ) )
A2: IC in dom (Initialize (while>0 (a,I))) by COMPOS_1:141;
assume A3: (StepWhile>0 (a,I,P,s)) . k = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n) ; :: thesis: ( (StepWhile>0 (a,I,P,s)) . k = ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) & (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) )
A4: ProgramPart (s +* (Initialize (while>0 (a,I)))) = (ProgramPart s) +* (ProgramPart (Initialize (while>0 (a,I)))) by FUNCT_4:75
.= (ProgramPart s) +* ((ProgramPart (while>0 (a,I))) +* (ProgramPart (Start-At (0,SCM+FSA)))) by FUNCT_4:75
.= (ProgramPart s) +* ((ProgramPart (while>0 (a,I))) +* {}) by COMPOS_1:27
.= (ProgramPart s) +* (ProgramPart (while>0 (a,I))) by FUNCT_4:22 ;
A5: ProgramPart (((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I)))) = (ProgramPart ((StepWhile>0 (a,I,P,s)) . k)) +* (ProgramPart (Initialize (while>0 (a,I)))) by FUNCT_4:75
.= (ProgramPart (s +* (Initialize (while>0 (a,I))))) +* (ProgramPart (Initialize (while>0 (a,I)))) by AMI_1:123, A3
.= (ProgramPart (s +* (Initialize (while>0 (a,I))))) +* ((ProgramPart (while>0 (a,I))) +* (ProgramPart (Start-At (0,SCM+FSA)))) by FUNCT_4:75
.= (ProgramPart (s +* (Initialize (while>0 (a,I))))) +* ((ProgramPart (while>0 (a,I))) +* {}) by COMPOS_1:27
.= (ProgramPart (s +* (Initialize (while>0 (a,I))))) +* (ProgramPart (while>0 (a,I))) by FUNCT_4:22
.= (ProgramPart s) +* (ProgramPart (while>0 (a,I))) by FUNCT_4:99, A4
.= (ProgramPart s) +* ((ProgramPart (while>0 (a,I))) +* {}) by FUNCT_4:22
.= (ProgramPart s) +* ((ProgramPart (while>0 (a,I))) +* (ProgramPart (Start-At (0,SCM+FSA)))) by COMPOS_1:27
.= (ProgramPart s) +* (ProgramPart (Initialize (while>0 (a,I)))) by FUNCT_4:75
.= ProgramPart (s +* (Initialize (while>0 (a,I)))) by FUNCT_4:75
.= ProgramPart (Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),n)) by AMI_1:123 ;
A6: ( DataPart (((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I)))) = DataPart ((StepWhile>0 (a,I,P,s)) . k) & P +* (while>0 (a,I)) = P +* (while>0 (a,I)) ) by SCMFSA8A:11;
IC (((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I)))) = IC (Initialize (while>0 (a,I))) by A2, FUNCT_4:14
.= IC ((StepWhile>0 (a,I,P,s)) . k) by A1, COMPOS_1:142 ;
hence ((StepWhile>0 (a,I,P,s)) . k) +* (Initialize (while>0 (a,I))) = (StepWhile>0 (a,I,P,s)) . k by A6, Th29, A5, A3; :: thesis: (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3)))
hence (StepWhile>0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while>0 (a,I))),((StepWhile>0 (a,I,P,s)) . k),((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3)) by Def5
.= Comput ((P +* (while>0 (a,I))),(s +* (Initialize (while>0 (a,I)))),(n + ((LifeSpan (((P +* (while>0 (a,I))) +* I),(((StepWhile>0 (a,I,P,s)) . k) +* (Initialize I)))) + 3))) by A3, EXTPRO_1:5 ;
:: thesis: verum