let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for a being read-write Int-Location
for I being really-closed MacroInstruction of SCM+FSA st ( ProperBodyWhile>0 a,I, Initialized s,p or I is parahalting ) & WithVariantWhile>0 a,I, Initialized s,p holds
DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s))))
let s be State of SCM+FSA; :: thesis: for a being read-write Int-Location
for I being really-closed MacroInstruction of SCM+FSA st ( ProperBodyWhile>0 a,I, Initialized s,p or I is parahalting ) & WithVariantWhile>0 a,I, Initialized s,p holds
DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s))))
let a be read-write Int-Location; :: thesis: for I being really-closed MacroInstruction of SCM+FSA st ( ProperBodyWhile>0 a,I, Initialized s,p or I is parahalting ) & WithVariantWhile>0 a,I, Initialized s,p holds
DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s))))
let I be really-closed MacroInstruction of SCM+FSA ; :: thesis: ( ( ProperBodyWhile>0 a,I, Initialized s,p or I is parahalting ) & WithVariantWhile>0 a,I, Initialized s,p implies DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s)))) )
set Ins = NAT ;
set WH = while>0 (a,I);
set Is = Initialized s;
set pds = p +* (while>0 (a,I));
A1: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;
assume A2: ( ( ProperBodyWhile>0 a,I, Initialized s,p or I is parahalting ) & WithVariantWhile>0 a,I, Initialized s,p ) ; :: thesis: DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s))))
then A3: ex k being Nat st
( ExitsAtWhile>0 (a,I,p,(Initialized s)) = k & ((StepWhile>0 (a,I,p,(Initialized s))) . k) . a <= 0 & ( for i being Nat st ((StepWhile>0 (a,I,p,(Initialized s))) . i) . a <= 0 holds
k <= i ) & DataPart (Comput ((p +* (while>0 (a,I))),(Initialize (Initialized s)),(LifeSpan ((p +* (while>0 (a,I))),(Initialize (Initialized s)))))) = DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . k) ) by Def6;
while>0 (a,I) is_halting_on Initialized s,p by A2, Th27, Th28;
then A4: p +* (while>0 (a,I)) halts_on Initialized s by A1, SCMFSA7B:def 7;
thus DataPart (IExec ((while>0 (a,I)),p,s)) = DataPart (Result ((p +* (while>0 (a,I))),(Initialized s))) by SCMFSA6B:def 1
.= DataPart ((StepWhile>0 (a,I,p,(Initialized s))) . (ExitsAtWhile>0 (a,I,p,(Initialized s)))) by A1, A4, A3, EXTPRO_1:23 ; :: thesis: verum