agda / agda-stdlib

The Agda standard library

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Add certain proofs for ``_∤_``

mechvel opened this issue · comments

I suggest to add to Algebra.Definitions.RawMagma the proofs for _∤_ which are similar to the proofs
∣-respʳ, ∣-respˡ, ∣-resp .
For example, like this:

∤-respˡ  :  {x : C} →  (_∤ x) Respects _≈_
∤-respˡ y≈y' y∤x y'∣x =  y∤x y∣x      where
                                      y∣x = ∣-respˡ (sym y≈y') y'∣x
∤-respʳ :  {x : C} → (x ∤_) Respects _≈_
∤-respʳ = ...

∤-resp  :   _∤_  Respects₂  _≈_
∤-resp  =  ...

Otherwise almost everyone who programs algebra would have to implement them oneself.

They can't live in Algebra.Definitions.RawMagma as you don't have a proof that the relation is an equality.

Instead these already exist in Algebra.Properties.Magma.Divisibility:

https://github.com/agda/agda-stdlib/blob/master/src/Algebra/Properties/Magma/Divisibility.agda

I do not see these lemmas in Algebra.Properties.Magma.Divisibility in lib-2.0.
Do you mean that they have to appear in the next official version?

I see there that _∤_ is imported,
but I do not see there the three suggested proofs

∤-resp  :   _∤_  Respects₂  _≈_
...

For any occasion: the last nonempty line in this file is of number 54.

Lines 26-33

These lines concern _∣_ - divisibility.
And I talk about _∤_ - negation of divisibility.
Are you saying that ∣-respʳ is also a proof for _∤_ Respectsʳ _≈_ ?
(I think, it is not).
Or may be your GitHub editor does not show the symbol ("does not divide") correctly?

Ah sorry, yes my mistake, these are indeed missing and should be added just below.