[Agda] join 2 `with'|

[Christopher Jenkins]

On Mon, Nov 3, 2014 at 2:38 PM, Arseniy Alekseyev <
arseniy.alekseyev at gmail.com> wrote:

> You are right, there is not much point. I was trying to point out that
> it's the "proper expression" part that makes `with` complicated.
>
> Unfortunately, that's not even true. In the following example, I don't
> think you can fill the holes.
>
> f : Bool → Bool
> f x = true where
>
>   z : Bool → Bool
>   z true = x
>   z false = x
>
>   y : ∀ q → x ≡ z q
>   y with x
>   y | true = {!!}
>   y | false = {!!}
>
>
>
I'm not entirely sure why you would expect that approach to be fruitful.
Essentially, what you're trying to do is unwind the computational nature of
`z', which is defined by case analysis on its argument (in this case q). To
trigger the rules, you would have to bring q into scope and inspect q,
which would then lead to `z true' and `z false' in the equations,
respectively, and Agda will then go on to normalize the goals as `x \== x'

f : Bool → Bool
f x = true where
  z : Bool → Bool
  z true = x
  z false = x

  y : ∀ q → x ≡ z q
  y true = {!!} -- Goal: x ≡ x
  y false = {!!} -- Goal: x ≡ x

I have actually been thinking about writing a blog post targeted at newbies
(such as myself) on the sometimes unintuitive behaviour of `with' (and in
particular, `rewrite'). Should anyone be interested I can send a link on
this email chain once I have written it.



> On 3 November 2014 10:59, Andreas Abel <andreas.abel at ifi.lmu.de> wrote:
> > On 03.11.2014 09:47, Arseniy Alekseyev wrote:
> >> I think the problem goes away if you
> >> only use `with x` where `x` is a variable.
> >
> > Mmh, why use `with` at all then?  You could directly split on the
> variable
> > `x`.  I thought the only reason to use `with` was when you wanted to
> case on
> > a proper expression and refine your type in the with-branches...
> >
> >
> >
> > On 03.11.2014 09:47, Arseniy Alekseyev wrote:
> >>
> >> Indeed that's the usual Agda behaviour.
> >> Here the reason seems to be an additional occurrence of `lookup k ps`
> >> in the context produced by pattern-match on `no`.
> >> You see, `with expr` only rewrites the occurrences of `expr` present
> >> in the enclosing context, so the new occurrences inside the body of
> >> `with` don't get rewritten (and you do want it rewritten in this
> >> case).
> >>
> >> I do agree this is confusing. I think the problem goes away if you
> >> only use `with x` where `x` is a variable. That makes `with` much less
> >> useful though. You can also avoid `with` altogether and write the
> >> helper functions by hand (that's what `with` is doing after all!).
> >>
> >> On 2 November 2014 17:14,  <mechvel at scico.botik.ru> wrote:
> >>>
> >>> People,
> >>>
> >>> Having the proof
> >>>
> >>>    prove : (k∈ks : k ∈ ks) → proj₂ (lookupIf∈ k (p ∷ ps) k∈k':ks) ≡
> >>>                              proj₂ (lookupIf∈ k ps k∈ks)
> >>>    prove k∈ks  with  k ≟ k'
>  --
> >>> (I)
> >>>
> >>>    ... | yes k≡k' =  ⊥-elim $ k≉k' k≡k'
> >>>    ... | no _     with  lookup k ps
> >>>    ...                | inj₁ _    =  PE.refl
> >>>    ...                | inj₂ k∉ks =  ⊥-elim $ k∉ks k∈ks
> >>>
> >>>
> >>> (I do not give the complete code),
> >>>
> >>> I try to rewrite it in a bit simpler way by joining it into one `with'
> >>> clause:
> >>>
> >>>
> >>>    prove k∈ks  with  k ≟ k' | lookup k ps
> --
> >>> (II)
> >>>
> >>>    ... | yes k≡k' | _         =  ⊥-elim $ k≉k' k≡k'
> >>>    ... | no _     | inj₁ _    =  PE.refl
> >>>    ... | no _     | inj₂ k∉ks =  ⊥-elim $ k∉ks k∈ks
> >>>
> >>>
> >>> (I) is type-checked, and (II) is not.
> >>>
> >>> Is this natural for Agda to decide so?
> >>>
> >>> (I have spent 4 hours trying to prove a similar real and evident
> example.
> >>> Then tried to split `with' into two, and this has succeeded).
> >>>
> >>> Thanks,
> >>>
> >>> ------
> >>> Sergei
> >>> _______________________________________________
> >>> Agda mailing list
> >>> Agda at lists.chalmers.se
> >>> https://lists.chalmers.se/mailman/listinfo/agda
> >>
> >> _______________________________________________
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> >>
> >
> >
> > --
> > Andreas Abel  <><      Du bist der geliebte Mensch.
> >
> > Department of Computer Science and Engineering
> > Chalmers and Gothenburg University, Sweden
> >
> > andreas.abel at gu.se
> > http://www2.tcs.ifi.lmu.de/~abel/
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>



-- 
Christopher Jenkins
Computer Science 2013
Trinity University
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