let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Nat st IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) holds
( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )
set D = Int-Locations \/ FinSeq-Locations;
let I be MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Nat st IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) holds
( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )
let a be read-write Int-Location; :: thesis: for s being State of SCM+FSA
for k, n being Nat st IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) holds
( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )
let s be State of SCM+FSA; :: thesis: for k, n being Nat st IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) holds
( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )
let k, n be Nat; :: thesis: ( IC ((StepWhile=0 (a,I,P,s)) . k) = 0 & (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) implies ( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) ) )
set s1 = Initialize s;
set P1 = P +* (while=0 (a,I));
set sk = (StepWhile=0 (a,I,P,s)) . k;
set s2 = Initialize ((StepWhile=0 (a,I,P,s)) . k);
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))),(Initialize s),n) or ( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) ) )
assume A2: (StepWhile=0 (a,I,P,s)) . k = Comput ((P +* (while=0 (a,I))),(Initialize s),n) ; :: thesis: ( (StepWhile=0 (a,I,P,s)) . k = Initialize ((StepWhile=0 (a,I,P,s)) . k) & (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) )
(StepWhile=0 (a,I,P,s)) . k is 0 -started by A1;
then Start-At (0,SCM+FSA) c= (StepWhile=0 (a,I,P,s)) . k by MEMSTR_0:29;
hence Initialize ((StepWhile=0 (a,I,P,s)) . k) = (StepWhile=0 (a,I,P,s)) . k by FUNCT_4:98; :: thesis: (StepWhile=0 (a,I,P,s)) . (k + 1) = Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2)))
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),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2)) by Def1
.= Comput ((P +* (while=0 (a,I))),(Initialize s),(n + ((LifeSpan (((P +* (while=0 (a,I))) +* I),(Initialize ((StepWhile=0 (a,I,P,s)) . k)))) + 2))) by A2, EXTPRO_1:4 ;
:: thesis: verum