Re: [tlaplus] Meta-theorem (induction lemma) in TLA+

Hello,

for a lemma involving temporal operators to be useful, its hypotheses should be "boxed", i.e. occur in the scope of an always operator. For example, you could state

LEMMA BoxImplies ==

ASSUME NEW TEMPORAL F, NEW TEMPORAL G,

[]F, [](F => G)

PROVE []G

and then use this lemma somewhere in a proof about temporal properties. For details, please see section 8.3 of [1]. In practice, we never state lemmas of (propositional) temporal logic because the PTL decision procedure proves them automatically.

Also note that your proof only goes through only because "NEW Invariant" implicitly means "NEW CONSTANT Invariant", hence the formula Invariant' in steps <1>2 and <1>3 gets rewritten to Invariant, and these steps are therefore tautologies. It is certainly not what you had in mind.

Regards,

Stephan

[1] http://lamport.azurewebsites.net/tla/tla2-guide.pdf

On 25 Jul 2019, at 18:55, shinsa82 <shinsa82@xxxxxxxxx> wrote:

I want to prove a meta-theorem (or induction lemma, tactic) for a specification and reuse it in other proofs, like Coq.
I wonder if its possible or not.

Consider the following spec:

----
vars == <state variables>

Act1 == ...
Act2 == ...

Next == Act1 \/ Act2
Spec == Init /\ [][Next]_vars
----

Then we expect the following lemma hold for any non-temporal formula "Invariant", and we want to use it to prove, say, its type invariance.
(I know its too simple. Actually I want to consider more complicated cases)
I was able to prove the lemma, but I could not "apply" the lemma to the type invariant theorem.
Is there anything wrong?

----
LEMMA SpecInduction ==
  ASSUME
  NEW Invariant,
  ASSUME Init PROVE Invariant,
  ASSUME Invariant, Act1 PROVE Invariant',
  ASSUME Invariant, Act2 PROVE Invariant',
  PROVE
  Spec => []Invariant
PROOF
  <1>1. Init => Invariant OBVIOUS
  <1>2. Invariant /\ Next => Invariant' OBVIOUS
  <1>3. Invariant /\ UNCHANGED vars => Invariant' OBVIOUS

  <1> QED BY PTL, <1>1, <1>2, <1>3 DEF Spec
----

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