[Agda] overloading operations

[Sergei Meshveliani]

On Fri, 2018-11-09 at 19:49 +0000, Martín Hötzel Escardó wrote:
> Hi Sergei, I have a situation very similar to yours.
> 
> An ordinal is a type equipped with a relation _≺_ and a justification 
> that it is a well-order. Moreover, this order has an automatically 
> derived partial order _≼_.
> 
> If α is an ordinal, then I write ⟨ α ⟩ for the underlying type of α and
> x ≺⟨ α ⟩ y and x ≼⟨ α ⟩ y for its underlying orders (given and derived).
> 
> The notation ⟨_⟩ for the underlying type is defined directly, just as a 
> projection.
> 
> The notations for the given and derived orders are given by "syntax" 
> declarations,
> 
>    syntax underlying-order  α x y = x ≺⟨ α ⟩ y
>    syntax underlying-porder α x y = x ≼⟨ α ⟩ y
> 
> where the underlying things are again projections.
> 
> I found that this worked quite well (although it would have been much 
> better if Agda could infer the "subscript" α for the orders, given that 
> x and y are known to live in the type ⟨ α ⟩).
> 
> Here are some examples from
> http://www.cs.bham.ac.uk/~mhe/agda-new/OrdinalOfOrdinals.html
> 
> is-order-preserving
>   is-monotone
>   is-order-reflecting
>   is-order-embedding
>   is-order-equiv
>   is-initial-segment
>   is-simulation
>    : (α : Ordinal U) (β : Ordinal V) → (⟨ α ⟩ → ⟨ β ⟩) → U ⊔ V ̇
> 
> is-order-preserving α β f = (x y : ⟨ α ⟩) → x ≺⟨ α ⟩ y → f x ≺⟨ β ⟩ f y
> 
> is-monotone α β f         = (x y : ⟨ α ⟩) → x ≼⟨ α ⟩ y → f x ≼⟨ β ⟩ f y
> 
> is-order-reflecting α β f = (x y : ⟨ α ⟩) → f x ≺⟨ β ⟩ f y → x ≺⟨ α ⟩ y
> 
> ...
> 
> (Warning: I use U,V,W for type universes rather than "Set i" notation, 
> but this is another story. Additionally, U,V,W are "variables", a newly 
> introduced feature which is available in the development version only.)
> 
> I presume you could define a type "Setoid U" of setoids in the universe 
> U with very much the same underlying-type and underlying-equivalence 
> notation.
> 


Thank you.
1) I need to study using `syntax'.
   (is this thing essential for the subject?)

2) What is "type universe" ?
   Is it the Universe programming technique introduced in the book 
   by Norell & Chapman ?

"A universe is a set of types (or type formers) and a universe
construction consists of a type of codes and a decoding function mapping
codes to types in the universe. 
...
"
I never tried this thing. Is it essential for the subject?

3) Really, will it also support overloading for the operations
   _∙_, _⁻¹, _+_  in Group, Ring and such structures of Standard 
   library ?  

Regards,

------
Sergei



> [..]
> >> On 08/11/18 18:16, Sergei Meshveliani wrote:
> >>> Can anybody, please, explain how to arrange operation overloading
> >>> (something like classes) ?
> >>>
> >>> The first example is
> >>>
> >>> ----------------------------------------------------------------
> >>> open import Level           using (_⊔_)
> >>> open import Relation.Binary using (Setoid)
> >>>
> >>> module _ {α α= β β=} (A : Setoid α α=) (B : Setoid β β=)
> >>>    where
> >>>    open Setoid   -- {{...}}
> >>>
> >>>    C  = Setoid.Carrier A
> >>>    C' = Setoid.Carrier B
> >>>
> >>>    IsCongruent :  (C → C') → Set _
> >>>    IsCongruent f =
> >>>                  {x y : C} → _≈_ A x y → _≈_ B (f x) (f y)    -- (I)
> >>>
> >>>               -- {x y : C} → x ≈ y → (f x) ≈ (f y)            -- (II)
> >>>
> >>> ----------------------------------------------------------------
> >>>
> >>>
> >>> The line (I) does work.
> >>> Then I try the line (II), with un-commenting {{...}}.
> >>> And it is not type-checked.
> >>> I hoped for that it would find that the first ≈ is on C, while the
> >>> second ≈ is on C'. But it does not.
> >>>
> >>> And real examples are like this:
> >>>
> >>> -----------------------------------------------------------------------
> >>> module _ {α α= β β=} (G : Group α α=) (G' : Group β β=)
> >>>   where
> >>>   ...
> >>>   homomorphismPreservesInversion :
> >>>     (mHomo : MonoidHomomorphism)
> >>>     (let f = MonoidHomomorphism.carryMap mHomo) (x : C) →
> >>>                                                 f (x ⁻¹) ≈' (f x) ⁻¹'
> >>>   homomorphismPreservesInversion
> >>>                  (monoidHomo ((f , f-cong) , f∙homo) f-preserves-ε) x =
> >>>     begin≈'
> >>>       f x'                     ≈'[ ≈'sym (∙ε' fx') ]
> >>>       fx' ∙' ε'                ≈'[ ∙'cong2 (≈'sym (x∙'x⁻¹ fx)) ]
> >>>       fx' ∙' (fx ∙' fx ⁻¹')    ≈'[ ≈'sym (≈'assoc fx' fx (fx ⁻¹')) ]
> >>>       (fx' ∙' fx) ∙' fx ⁻¹'    ≈'[ ∙'cong1 (≈'sym (f∙homo x' x)) ]
> >>>       f (x' ∙ x) ∙' fx ⁻¹'     ≈'[ ∙'cong1 (f-cong (x⁻¹∙x x)) ]
> >>>       f ε ∙' fx ⁻¹'            ≈'[ ∙'cong1 f-preserves-ε ]
> >>>       ε' ∙' fx ⁻¹'             ≈'[ ε'∙ (fx ⁻¹') ]
> >>>       (f x) ⁻¹'
> >>>     end≈'
> >>>     where
> >>>     x' = x ⁻¹;   fx = f x;   fx' = f x'
> >>> ----------------------------------------------------------------------
> >>>
> >>> Here the carriers of G and G' are C and C',
> >>> ≈ is on C,  ≈' is on C' (by renaming),
> >>> _∙_ on C, _∙'_ on C',
> >>> ε is of G, ε' of G',
> >>> _⁻¹ is of G,  _⁻¹' of G',
> >>> and so on.
> >>>
> >>> It is desirable to make the code more readable by canceling some of the
> >>> above renaming. For example, to replace ε' with ε and _∙'_ with _∙_.
> >>> Is it possible to do this by using something like
> >>> open Group {{...}}
> >>> ?
> >>>
> >>> Thanks,
> >>>
> >>> ------
> >>> Sergei
> >>>
> >>>
> >>>
> >>>
> >>> _______________________________________________
> >>> Agda mailing list
> >>> Agda at lists.chalmers.se
> >>> https://lists.chalmers.se/mailman/listinfo/agda
> >>>
> >>
> > 
> > 
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