[Agda] How can we define a type of symmetric binary operations in Agda ?

[Andreas Abel]

You can create a record with two fields, one is your binary operation, 
the other the proof of symmetry.

record SymBinOp : Set where
   field
     _⊙_ : BinaryOperation
     proof-of-symmetry : ∀ x y → x ⊙ y == y ⊙ x

That's it!

On 30.04.2015 17:35, Sergey Kirgizov wrote:
> Hello,
> 	
> I don't  understand how  we can  define in Agda  a type  of "Symmetric
> Binary Relation". Can you help me?
> Imagine I have something like:
>
> =======================================================
>
> {-
>    At first we reaname  Set
>
> {-
>    First, we define a polymorphic idenity
> -}
> data _==_ {A : 𝓤} (a : A) : A → 𝓤 where
>    definition-of-idenity : a == a
> infix 30 _==_
>
> {-
>   Next we define the finite set Ω = {A,B,C}
> -}
> data Ω : 𝓤 where
>    A B C : Ω
>
> {-
>   We add a type "Binary operation"
> -}
> BinaryOperation = Ω → Ω → Ω
>
> {-
>   I can actually define an example of Binary operation.
> -}
> _⊙_ : BinaryOperation
> x ⊙ x' = A
> infix 40 _⊙_
>
> {-
>   And then I can proove that ⊙ is Symmetric
> -}
> proof-of-symmetry : ∀ x y → x ⊙ y == y ⊙ x
> proof-of-symmetry = λ x y → definition-of-idenity
>
>
> ========================================================
>
> How we  can define a  type "Symmetric Binary Operation"?   Having this
> type we will be able to define ⊙ as
>
> _⊙_ : SymmetricBinaryOperation
> x ⊙ y = A
>
> and proof that ⊙ is symmetric will no be required.
>
>
> Best regards,
> Sergey
> _______________________________________________
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> Agda at lists.chalmers.se
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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/