# Re: [isabelle] conversions for symmetric/commutative operators

```Peter,

```
This is really funny, I was in the middle of implementing the first part of what you suggested. There is now the lemma eq_iff_swap: (x = y) = P ==> (y = x) = P and I have simplified a number of proofs in List as a result. I have also added a handful of lemmas that were missing. Strangely enough, I missed your empty_filter_conv. Or maybe I tried and it did more harm than good - it took a couple of iterations to smooth things out. I'll double-check.
```
```
Note that one should not apply eq_iff_swap blindly: one may want to swap some equations in P as well.
```
Tobias

On 09/11/2021 13:34, Peter Gammie wrote:
```
```Hello,

I find myself with a sea of conversions of the form exhibited by `filter_empty_conv`. `HOL.List` is missing its obvious friend:

lemma empty_filter_conv:
shows "([] = filter P xs) = (\<forall>x\<in>set xs. \<not> P x)"
by (induct xs) simp_all

which could also be derived via a rule of the form:

(x = y) = z ==> (y = x) = z

(and so forth for other symmetric or commutative operators such as inf and sup).

One could imagine defining an attribute `symconv` like `symmetric` to handle this:

lemmas empty_filter_conv = filter_empty_conv[symconv]

But in my ideal world I wouldn’t need to type this out and make up another name, but would instead bind both theorems to the same name, e.g.

lemma empty_filter_conv[symconv]:
shows "([] = filter P xs) = (\<forall>x\<in>set xs. \<not> P x)”
…

thm empty_filter_conv
```
```([] = filter P xs) = (\<forall>x\<in>set xs. \<not> P x)
(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)
```
```
However AIUI attributes map a single theorem to a single theorem, so this isn’t going to work.

All I can think of is to create another keyword like `lemma` that does a bit of post-processing before the binding is made. This feels a bit heavyweight.

Has anyone got a better solution? It might help to make these sorts of lemma pairs more systematic in the distribution.

cheers,
peter

```

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