let P be Instruction-Sequence of SCM+FSA; for s being State of SCM+FSA
for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let s be State of SCM+FSA; for I being really-closed MacroInstruction of SCM+FSA
for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let I be really-closed MacroInstruction of SCM+FSA ; for a being read-write Int-Location st I is_halting_onInit s,P & s . a > 0 holds
for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
let a be read-write Int-Location; ( I is_halting_onInit s,P & s . a > 0 implies for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )
set s0 = Initialized s;
set IA = Start-At (0,SCM+FSA);
assume
I is_halting_onInit s,P
; ( not s . a > 0 or for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )
then A1:
P +* I halts_on Initialized s
;
Initialized s = Initialize (Initialized s)
by MEMSTR_0:44;
then A2:
I is_halting_on Initialized s,P
by A1, SCMFSA7B:def 7;
assume
s . a > 0
; for k being Nat st k <= (LifeSpan ((P +* I),(Initialized s))) + 2 holds
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
then A3:
(Initialized s) . a > 0
by SCMFSA_M:37;
hereby verum
let k be
Nat;
( k <= (LifeSpan ((P +* I),(Initialized s))) + 2 implies IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I)) )A4:
Initialized s = Initialize (Initialized s)
by MEMSTR_0:44;
assume
k <= (LifeSpan ((P +* I),(Initialized s))) + 2
;
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))hence
IC (Comput ((P +* (while>0 (a,I))),(Initialized s),k)) in dom (while>0 (a,I))
by A4, A2, A3, SCMFSA_9:42;
verum
end;