let P be Instruction-Sequence of SCM+FSA; :: thesis: 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; :: thesis: 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 ; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( 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 ; :: thesis: 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;

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; :: thesis: 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 ; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( 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 ; :: thesis: 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 :: thesis: verum

let k be Nat; :: thesis: ( 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 ; :: thesis: 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; :: thesis: verum

end;A4: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;

assume k <= (LifeSpan ((P +* I),(Initialized s))) + 2 ; :: thesis: 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; :: thesis: verum