# Re: [isabelle] conversions for symmetric/commutative operators (Peter Gammie)

• To: cl-isabelle-users at lists.cam.ac.uk, peteg42 at gmail.com
• Subject: Re: [isabelle] conversions for symmetric/commutative operators (Peter Gammie)
• From: Emin Karayel <me at eminkarayel.de>
• Date: Tue, 9 Nov 2021 16:06:35 +0100
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• References: <mailman.4977.1636465456.48434.cl-isabelle-users@lists.cam.ac.uk>

Hi Peter,

You can solve this using the OF attribute.

First proof (once):

lemma switch_eq:
assumes "(x=y) = z"
shows "(y=x) = z"
using assms by auto

Once you have that you can refer to the switched version of any lemma L of the form [(x=y)=z] like this:
switch_eq[OF L]

For example:
switch_eq[OF filter_empty_conv] is the same as your empty_filter_conv

I hope this helps,

Emin

Date: Tue, 9 Nov 2021 22:04:13 +0930
From: Peter Gammie <peteg42 at gmail.com>
Subject: [isabelle] conversions for symmetric/commutative operators
To: cl-isabelle-users <cl-isabelle-users at lists.cam.ac.uk>
Message-ID: <7711F679-C145-4B57-9DB7-16D7FEBDDF0D at gmail.com>
Content-Type: text/plain;       charset=utf-8

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

------------------------------

Message: 9
Date: Tue, 09 Nov 2021 14:28:23 +0100
From: Peter Lammich <lammich at in.tum.de>
Subject: Re: [isabelle] conversions for symmetric/commutative
operators
To: Peter Gammie <peteg42 at gmail.com>
Cc: cl-isabelle-users <cl-isabelle-users at lists.cam.ac.uk>
Message-ID: <207d7151-b15a-456e-9538-1aa3d9158635 at email.android.com>
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