[Agda] Trying to use irrelevant arguments to get a nicer equality for algebraic structures

[Andreas Abel]

Hi Andrea,

since records do not support irrelevant fields yet, I suggest to use  
data instead:

   data SemiG : Set1 where
     _,_,_ : (M     : Set) ->
             (_+_   : M -> M -> M) ->
             .(∀ {x y z} ->  x + (y + z) ≡ (x + y) + z) ->
             SemiG

   dual : SemiG -> SemiG
   dual (M , _+_ , assoc) = M , (λ x y -> y + x) , sym assoc

   dual∘dual≡id : ∀ {M} -> dual (dual M) ≡ M
   dual∘dual≡id {_ , _ , _} = refl

Cheers,
Andreas

On Sep 23, 2010, at 5:58 PM, Andrea Vezzosi wrote:

>
> {-# OPTIONS --universe-polymorphism #-}
> module Monoid where
>
> data Squash {a} (A : Set a) : Set a where
>  squash : .A -> Squash A
>
> open import Relation.Binary.PropositionalEquality
>
> record SemiG : Set1 where
>  constructor _,_,_
>  field
>    M : Set
>    _+_ : M -> M -> M
>    assoc : Squash (∀ {x y z} ->  x + (y + z) ≡ (x + y) + z)
>
>
>
> dual : SemiG -> SemiG
> dual (M , _+_ , assoc) = M , (λ x y -> y + x) , lemma where
>     lemma : Squash ({x y z : M} → (z + y) + x ≡ z + (y + x))
>     lemma with assoc
>     lemma | squash y = squash (\ {_} {_} {_} -> sym y)
>
>
> dual∘dual≡id : ∀ {M} -> dual (dual M) ≡ M
> dual∘dual≡id {_ , _ , squash _} = refl

Andreas Abel  <><      Du bist der geliebte Mensch.

Theoretical Computer Science, University of Munich
Oettingenstr. 67, D-80538 Munich, GERMANY

andreas.abel at ifi.lmu.de
http://www2.tcs.ifi.lmu.de/~abel/