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# Re: [tlaplus] Re: How to understand the concept "step simulation"

 Let me try another attempt for explaining the issue. To make things simpler, let's leave aside the issue of refinement mapping, that is, consider just the identity refinement mapping. We want to prove(0) InitL /\ [][NextL]_varsL => InitH /\ [][NextH]_varsHObviously, this can be reduced to proving(1) InitL => InitH(2) NextL \/ varsL' = varsL => NextH \/ varsH' = varsHHowever, the obligation (2) is very likely going to be unprovable because really, we only have to prove that implication for all reachable states (i.e., reachable in runs of the low-level spec) rather than for completely arbitrary states, and that's where the invariant comes in. Let's look at a stupid example: our high-level specification has one variable initialized to zero and that keeps growing while remaining an even number:Even == {n \in Int : n % 2 = 0}InitH == x = 0NextH == x' > x /\ x' \in EvenOur low-level specification has two variables x and y that evolve as follows:InitL == x = 0 /\ y = 0NextL == x' = x+y /\ y' = y+2Condition (2) now requires us to show(x' = x+y /\ y'=y+2) \/ (x'=x /\ y'=y) => (x' > x /\ x' \in Even) \/ x' = xbut this implication is not true and therefore cannot be proved: we have no information about the "types" of x and y, so they could be strings for example. Even if they are integers, we could have x=42 and y=7, x' = 49 and y'=9, and the right-hand side will be false. And indeed, such a state cannot be reached because the low-level spec ensures that x and y are always even. Therefore we defineInv == x \in Even /\ y \in Evenand relax our proof obligations to be(1') InitL => InitH /\ Inv(2') Inv /\ (NextL \/ varsL' = varsL) => Inv' /\ (NextH \/ varsH' = varsH)which still implies our high-level goal (0). I leave proving (1') and (2') for our example as an exercise to you.Indeed, Inv can be an arbitrary state predicate and has to be "invented" by the system designer / verifier. But it must be an invariant of the low-level specification, and FALSE is unlikely to be one.You may also want to read section 6.8 of the Hyperbook.Hope this helps,StephanOn 5 Dec 2019, at 09:58, ss.nedunuri@xxxxxxxxx wrote:I have to admit I'm a bit confused by the explanation in the text too and don't quite see what Inv is representing. For example, can I set Inv to anything that allows me to prove the formula, what if I set it to False?ThanksOn Sunday, March 10, 2019 at 12:52:43 PM UTC-7, Leslie Lamport wrote:(1) If we remove the Inv from the formula Inv /\ Next => ..., it would assert    that a step starting in any state that satisfies Next satisfies "..." --    for example a state in which memQ is a sequence of imaginary numbers.    I have no idea if that assertion is true for such a starting state.    However, it suffices to prove the assertion for steps starting in a    reachable state.  Conjoining the invariant Inv allows you to prove    the assertion only for reachable states.  You have to choose Inv so    it asserts what is true about reachable states that makes the    implication true.  To do this, you have to understand why the theorem    you're trying to prove is true.(2) That mapping isn't derived; you have to invent it.  The sentence     beginning "Intuitively" that starts on line 9 of page 63 tells you    what condition that substitution must satisfy.  To be able to choose     the necessary mapping, you need to understand why the theorem you're     trying to prove is true .LeslieOn Wednesday, March 6, 2019 at 10:53:35 PM UTC-8, Oliver Yang wrote:Hi All,In Section 5.8 of book "Specifying Systems", the "Proving Impl" is introduced. I have a rough understanding of refinement mapping, whichessentially maps states of Spec A to the states of Spec B. However, I have a hard time understanding "step simulation". 1) What's the purpose of introducing the invariant Inv in Formula 5.3? What are we trying to achieve here?2) How do we derive the mapping: omem = vmem, octl = ..., obuf = buf? It looks like we jumped to the conclusion without showing any proof?Thanks,Oliver -- You received this message because you are subscribed to the Google Groups "tlaplus" group. To unsubscribe from this group and stop receiving emails from it, send an email to tlaplus+unsubscribe@xxxxxxxxxxxxxxxx. To view this discussion on the web visit https://groups.google.com/d/msgid/tlaplus/369feaf8-c1ad-49c5-9915-d13e8687105c%40googlegroups.com. -- You received this message because you are subscribed to the Google Groups "tlaplus" group. To unsubscribe from this group and stop receiving emails from it, send an email to tlaplus+unsubscribe@xxxxxxxxxxxxxxxx. To view this discussion on the web visit https://groups.google.com/d/msgid/tlaplus/D3586299-C7D2-44B5-8A7A-8B8EAC5AAAE0%40gmail.com.