# Re: [isabelle] Simplifying addition and subtraction of multisets

Yes, I've done simprocs to do this. They're based largely on those for natural numbers, but for a type class that has +, =
```obeying A + B - B = A but not (necessarily) A - B + B = A

It's all in http://users.cecs.anu.edu.au/~jeremy/isabelle/2005/pmgms/
specifically in pmg_cancel_sums.ML

```
It all works with Isabelle 2005, which is where I gave up changing everything I'd ever done to keep up with the changes in Isabelle
```
Jeremy

On 11/04/16 23:28, Andreas Lochbihler wrote:
```
```Hi Manuel,

I also think that you need a simproc, because the nesting can be
arbitrarily deep. Since multisets are similar to natural numbers in
terms of their algebraic properties, I'd suggest that you look at the
simprocs for natural numbers. I guess that you can generalise
cancel_diff_conv in src/HOL/Tools/Nat_Arith.ML appropriately to multisets.

Hope this helps,
Andreas

On 11/04/16 13:55, Manuel Eberl wrote:
```
```Well, that works for some cases, but not for all, e.g:

lemma "(A + B + C + D ) - (C :: 'a multiset) = A + B + D"

I don't think this is going to work without a simproc. The arithmetic
procedures do things
like that; maybe they can be adapted for this, or perhaps it's just a
matter of the right
setup?

I don't know who is the expert on these procedures.

Manuel

On 08/04/16 14:24, Julian Nagele wrote:
```
```Hello Manuel,

simplifying with subset_mset.diff_add_assoc works for me:

lemma "{#a, b, c, d#} - {#b#} = {#a, c, d#}"

or, more generally

lemma
fixes A B C :: "'a multiset"
shows "(A + C + B) - C = A + B"

Hope that helps,
Julian

Manuel Eberl writes:

```
```Hallo,

I have terms like "{#a,b,c,d#} - {#b#}", which desugars to "(single a +
single b + single c + single d) - single b". This is obviously equal to
"{#a,c,d#}".

However, the simplifier fails to prove this and I was not able to
find a
setup of existing simplification rules to solve it.

I ended up proving the following rule, which works in my particular
case:

lemma multiset_Diff_single_normalize:
fixes a c assumes "a â c"
shows   "({#a#} + B) - {#c#} = {#a#} + (B - {#c#})"

This, combined with add_ac, does the trick. (but only because I can, in
this particular instance, decide whether two elements are equal, i.e. I
know that b != c and b != d, even though the simplification of
"{#a,b,c,d#} - {#b#}" would be sound even if that were not the case)

Is there some existing simproc that can be set up to do this
automatically?

Cheers,

Manuel
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