[Agda] join 2 `with'|

[Andreas Abel]

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
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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/