[Agda] Associativity and Heterogeneous Equality

[Alan Jeffrey]

Oooh, I can convert a propositional equality to a definitional equality. 
Cool.

Can I use this to build non-terminating programs?

   p : foo ≡ bar
   p = ...
   {-# REWRITE p #-}

   q : bar ≡ foo
   q = ?
   {-# REWRITE q #-}

Is the semantics of REWRITE "you'd better know what you're doing", or 
"programs whose typechecking terminates are sound"?

A.

On 06/10/2015 04:21 PM, Andreas Abel wrote:
> This is more a feature /preview/ than a mature feature, but in Agda
> 2.4.2.3 you can add rewrite rules to the *definitional* equality, making
> more types equal.
>
> The following type-checks:
>
> open import Relation.Binary.PropositionalEquality
>
> -- Rewrite rules for a definition currently must be in the same module
> -- as the definition, so we roll our own _++_.
>
> infixr 5 _∷_ _++_
>
> data List {a} (A : Set a) : Set a where
>    [] : List A
>    _∷_ : (x : A) (xs : List A) → List A
>
> _++_ : ∀{a} {A : Set a} (xs ys : List A) → List A
> [] ++ ys = ys
> (x ∷ xs) ++ ys = x ∷ (xs ++ ys)
>
> assoc-++ : ∀ (a b c : List Set) -> (a ++ b) ++ c ≡ a ++ b ++ c
> assoc-++ [] b c = refl
> assoc-++ (x ∷ a) b c = cong (_∷_ x) (assoc-++ a b c)
>
> {-# BUILTIN REWRITE _≡_ #-}
> {-# REWRITE assoc-++ #-}
>
> postulate View : List Set -> Set -> Set
>
> data HList : List Set -> Set₁ where
>    [] : HList []
>    _∷_ : ∀ {xs x} -> x -> (ts : HList xs) -> HList (x ∷ xs)
>
> _+++_ : ∀ {xs ys} -> HList xs -> HList ys -> HList (xs ++ ys)
> [] +++ bs = bs
> (a ∷ as) +++ bs = a ∷ (as +++ bs)
>
> infixr 5 _+++_
>
> assoc-+++ : ∀ {xs ys zs} (a : HList xs) (b : HList ys) (c : HList zs) ->
> a +++ b +++ c ≡ (a +++ b) +++ c
> assoc-+++ [] b c = refl
> assoc-+++ {_ ∷ xs} {ys} {zs} (x ∷ a) b c = cong (_∷_ x) (assoc-+++ {xs}
> {ys} {zs} a b c)
>
>
> On 07.06.2015 16:33, Marco Vassena wrote:
>> Given an heterogeneous list data-type:
>>
>> data HList : List Set -> Set₁ where
>>    [] : HList []
>>    _∷_ : ∀ {xs x} -> x -> (ts : HList xs) -> HList (x ∷ xs)
>>
>> I have defined the append function:
>> _+++_ : ∀ {xs ys} -> HList xs -> HList ys -> HList (xs ++ ys)
>> [] +++ bs = bs
>> (t ∷ as) +++ bs = t ∷ (as +++ bs)
>>
>> Now I want to prove that _+++_ is associative: a +++ b +++ c = (a +++
>> b) +++ c.
>>
>> I could not state this lemma using propositional equality, because _+++_
>> appends the list of types using _++_ :
>> a +++ b +++ c :: HList (xs ++ ys ++ zs)
>> (a +++ b) +++ c :: HList ((xs ++ ys) ++ zs)
>>
>> the two indexes are not considered equal, leading to different types,
>> preventing the use of propositional equality.
>> In order to solve this I should use associativity of _++_ in the
>> signature itself, but I could not find any
>> way to do that.
>>
>> I tried then to use heterogeneous equality.
>>
>> assoc-+++ : ∀ {xs ys zs} (a : HList xs) (b : HList ys) (c : HList zs)
>> -> a +++ b +++ c ≅ (a +++ b) +++ c
>> assoc-+++ [] b c = refl
>> assoc-+++ {_ ∷ xs} {ys} {zs} (x ∷ a) b c = H.cong (_∷_ x) (assoc-+++ a
>> b c)  -- (xs +++ ys) != xs (needs associativity)
>>
>> The term (assoc-+++ a b c) is ill-typed because of the associativity
>> of _++_ .
>> At this point I have tried to use firstly associativity of _++_, but
>> this failed with an error about the generated with function being
>> ill-typed.
>>
>> Is there a way to prove associativity of _+++_ ?
>>
>> I have attached a file with this problem.
>>
>> All the best,
>> Marco
>>
>>
>>
>>
>>
>>
>>
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