[Agda] rewrite on the right?

[Victor Miraldo]

Hello,

I did implement a library for doing exactly that as my Msc thesis. My code
is not up-to-date with the new API for reflection aswell,
and, in fact, this new API provides a lot of interesting oportunities to
improve it. If you're using a "not so recent" version of Agda
it might be worth to check it out.

https://github.com/VictorCMiraldo/agda-rw

Cheers,
Victor

On Wed, Jan 20, 2016 at 6:59 PM, Jesper Cockx <Jesper at sikanda.be> wrote:

> I implemented a simple rewrite tactic using the reflection in Agda back in
> 2014, see https://lists.chalmers.se/pipermail/agda/2014/007032.html. The
> code is not up-to-date with the new reflection API (and you could now
> probably make something that has a much nicer syntax anyway, using the new
> macro system) but it shows that there is nothing preventing you to do this
> using reflection. So I would say there is no need to implement something as
> a language extension if it is already possible as a library.
>
> cheers,
> Jesper
>
> On Wed, Jan 20, 2016 at 6:33 PM, Andrea Vezzosi <sanzhiyan at gmail.com>
> wrote:
>
>> Yes, having rewrite on the right would be very nice.
>>
>> It'd have to generate an helper like you did manually, that's how
>> rewrite on the left is implemented too, so it might not make sense as
>> an expression because it mangles the context, maybe it can have the
>> same status as the macros of reflection.
>>
>> In your case you cannot really get away with just a subst, however you
>> can get somewhere if you have proven that get commutes with subst, so
>> you can fill the open goal here:
>>
>> (λ { gbx≡gax → trans gbx≡gax (trans (cong (subst V a≡b) gax≡gay) ?) })
>>
>> Best,
>> Andrea
>>
>>
>>
>>
>>
>>
>> On Wed, Jan 20, 2016 at 1:39 AM, Martin Stone Davis
>> <martin.stone.davis at gmail.com> wrote:
>> > I've found myself wishing that 'rewrite' or with-abstractions were
>> invokable
>> > from lambda expressions. A (contrived) case in point is as follows,
>> where a
>> > helper function is used to complete 'foo'. NB, I realize that, in this
>> case,
>> > I could have simply used a with-abstraction (as in 'bar'), but there are
>> > other cases (not shown here) where one of the variables being ≟-compared
>> > only exists on the rhs, thus requiring a call to 'case_of_'.
>> >
>> > Is there a way to 'subst' my way out of the need for foo-helper?
>> >
>> > module RewriteOnRight where
>> >   open import Data.Product using (∃ ; _,_)
>> >   open import Relation.Nullary using (Dec ; yes ; no)
>> >   open import Function using (case_of_)
>> >   open import Relation.Nullary.Negation using (contradiction)
>> >   open import Relation.Binary using (IsDecEquivalence)
>> >   open import Relation.Binary.PropositionalEquality using (_≡_ ; subst ;
>> > trans ; refl)
>> >
>> >   postulate
>> >     K : Set
>> >     V : K → Set
>> >     M : Set
>> >     _∈_ : K → M → Set
>> >     get : ∀ {k : K} {m : M} → k ∈ m → V k
>> >     isDecEquivalence/K : IsDecEquivalence {A = K} _≡_
>> >
>> >   open IsDecEquivalence isDecEquivalence/K using (_≟_ ; sym)
>> >
>> >   foo-helper :
>> >     ∀ {x : M}
>> >       {y : M}
>> >       {a : K}
>> >       {b : K}
>> >       {a∈x : a ∈ x}
>> >       {b∈x : b ∈ x}
>> >       {a∈y : a ∈ y}
>> >       (gax≡gay : get a∈x ≡ get a∈y)
>> >       (a≡b : a ≡ b)
>> >       (gbx≡gax : get b∈x ≡ subst V a≡b (get a∈x))
>> >       → get b∈x ≡ get (subst (λ k → k ∈ y) a≡b a∈y)
>> >   foo-helper gax≡gay a≡b gbx≡gax rewrite a≡b = trans gbx≡gax gax≡gay
>> >
>> >   foo : ∀ {x : M}
>> >           {y : M}
>> >           (a : K)
>> >           (b : K)
>> >           {a∈x : a ∈ x}
>> >           {b∈x : b ∈ x}
>> >           {a∈y : a ∈ y}
>> >           (gax≡gay : get a∈x ≡ get a∈y)
>> >         → Dec (∃ λ (a≡b : a ≡ b) → (gbx≡gax : get b∈x ≡ subst V a≡b (get
>> > a∈x)) → get b∈x ≡ get (subst (λ k → k ∈ y) a≡b a∈y))
>> >   foo a b gax≡gay = case a ≟ b of λ
>> >     { (yes a≡b) → yes (a≡b , (λ {gbx≡gax → foo-helper gax≡gay a≡b
>> gbx≡gax}))
>> >     ; (no a≢b) → no (λ { (a≡b , _) → contradiction a≡b a≢b })
>> >     }
>> >
>> >   bar : ∀ {x : M}
>> >           {y : M}
>> >           (a : K)
>> >           (b : K)
>> >           {a∈x : a ∈ x}
>> >           (b∈x : b ∈ x)
>> >           {a∈y : a ∈ y}
>> >           (gax≡gay : get a∈x ≡ get a∈y)
>> >         → Dec (∃ λ (a≡b : a ≡ b) → (gbx≡gax : get b∈x ≡ subst V a≡b (get
>> > a∈x)) → get b∈x ≡ get (subst (λ k → k ∈ y) a≡b a∈y))
>> >   bar a  b _   _       with a ≟ b
>> >   bar _ ._ b∈x gax≡gay | yes refl = yes (refl , (λ {gbx≡gax → trans
>> gbx≡gax
>> > gax≡gay}))
>> >   bar _  _ _   _       | no a≢b = no (λ { (a≡b , _) → contradiction a≡b
>> a≢b
>> > })
>> >
>> > --
>> > Martin Stone Davis
>> >
>> > Postal/Residential:
>> > 1223 Ferry St
>> > Apt 5
>> > Eugene, OR 97401
>> > Talk / Text / Voicemail: (310) 699-3578
>> > Electronic Mail: martin.stone.davis at gmail.com
>> > Website: martinstonedavis.com
>> >
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