[ProofPower] Theorem QA

[Rob Arthan]

On 25 Mar 2012, at 12:31, Phil Clayton wrote:

> On 24/03/12 09:32, Rob Arthan wrote:
>> The scripts for the ProofPower mathematical case studies have a little tool called "check_thms" which does a quality assurance check on the the theorems in a theory. It checks against:
>> 
>> a) Theorems with free variables. Typically this means you forgot an outer universal quantifier. Later on you will be puzzled when tools like the rewriting tools think you don't want the free variables to be instantiated.
>> 
>> b) Theorems with variables bound by logical quantifiers (universal, existential and unique existential) that are not used in the body of the abstraction. This happens for various reasons (often hand in hand with (a)). It is misleading for the reader and can be confusing when you try to use the theorem.
>> 
>> It outputs a little report on any problems it finds.
>> 
>> I am considering putting a bug-fixed and documented version of check_thms in the next working release of ProofPower. Any comments or suggestions for other things to check for would be welcome.
> 
> I am wondering about a stronger version of check b for individual conjuncts of a theorem.  In the past, I have found that e.g. rewriting can be awkward when an equational conjunct of a theorem does not mention a universally quantified variable (that is mentioned by other conjuncts).  I think the issue was when the unmentioned variable was quantified over a non-maximal set, so this is probably most relevant to Z.  For example, given
> 
> │ _ ^ _ : ℤ × ℕ → ℤ
> ├──────
> │ ∀ i : ℤ; j : ℕ ⦁ i ^ 0 = 1 ∧ i ^ (j + 1) = i * i ^ j
> 
> I think rewriting with the base case requires manual intervention to provide a value for j,

Indeed. ∀ i : ℤ; j : X ⦁ i ^ 0 = 1  amounts to a convoluted way of saying "either X is empty or ∀ i : ℤ ^ 0 = 1". 

> so the following would be preferable:
> 
> ├──────
> │ ∀ i : ℤ ⦁ i ^ 0 = 1 ∧ (∀ j : ℕ ⦁ i ^ (j + 1) = i * i ^ j)
> 

> I expect that this sort of check would be dependent on the current proof context (perhaps making use of canonicalization support) so may not be desirable as part of the same utility.

I think a simple heuristic would work. I think is reasonable to say that in a predicate of the form:

	∀ ...  x : X  | P ⦁ Q1 ∧ Q2 ∧ ...

If there is an i such that x doesn't appear free in P or Qi, then report a possible problem. There are many cases (e.g., law of transivity) where a theorem has an implication with an antecedent that contains variables that are not in the succedent, but it is a reasonable style rule in Z not to disguise such an implication by burying the antecedent in an implication.

> 
> 
>> This is currently just for HOL, but I could do something similar for Z too.
> 
> Certainly this would be a useful facility for HOL users but I could only make use of a Z version.
> 
> Regards,
> 
> Phil
> 
> 
> P.S.  The above formal text is UTF8 encoded!  Hopefully that is not a problem these days.  It would be useful to know if any mail systems aren't displaying it properly.
> 

It looks good to me.  I must get round to some more work on Unicode and UTF-8 for ProofPower.

Thanks for the input.

Regards,

Rob.