Standardise implicit arguments of cancellation properties.

[FR-vdash-bot]

In Algebra.Definitions

LeftCancellative _•_ = ∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z
RightCancellative _•_ = ∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z
AlmostLeftCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z
AlmostRightCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z

In Data.Rational.Unnormalised.Properties

+-cancelˡ : ∀ {r p q} → r + p ≃ r + q → p ≃ q
+-cancelʳ : ∀ {r p q} → p + r ≃ q + r → p ≃ q

Perhaps we should make them uniform.

[MatthewDaggitt]

Hey, thanks for the spot! Yup, so for starters the rational proofs should be using the LeftCancellative and RightCancellative types.

As for the AlmostX and X definitions, the reason for the differences in the implicits is Agda's ability to infer them. LeftCancellative and RightCancellative were introduced early on in the library's life and the implicit arguments are slightly broken, see the code below. To fix this I think none of the arguments to those definitions should be implicit. The Almost versions were introduced very recently and I think have the right implicits (x is inferrable from the equality).

open import Data.Rational.Base
open import Algebra.Definitions
open import Relation.Binary.PropositionalEquality.Core

postulate
  rc : RightCancellative _≡_ _+_
  arc : AlmostRightCancellative _≡_ 0ℚ _+_
  x y z : ℚ
  x≢0 : x ≢ 0ℚ
  eq : y + x ≡ z + x

-- Inference error
test : y ≡ z
test = rc y z eq

-- No inference error
test2 : y ≡ z
test2 = arc y z x≢0 eq

Therefore I think the correct definitions are:

LeftCancellative _•_ = ∀ x y z → (x • y) ≈ (x • z) → y ≈ z
RightCancellative _•_ = ∀ x y z → (y • x) ≈ (z • x) → y ≈ z
AlmostLeftCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z
AlmostRightCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z

but we will need to wait until v2.0 to make such a breaking change.

[FR-vdash-bot]

I have just noticed something similar.

In Data.Nat.Properties

≤-stepsˡ : ∀ {m n} o → m ≤ n → m ≤ o + n
≤-stepsʳ : ∀ {m n} o → m ≤ n → m ≤ n + o
m≤m+n : ∀ m n → m ≤ m + n
m≤n+m : ∀ m n → m ≤ n + m

In Data.Integer.Properties

≤-steps : ∀ {m n} p → m ≤ n → m ≤ + p + n
m≤m+n : ∀ {m} n → m ≤ m + + n
n≤m+n : ∀ m {n} → n ≤ + m + n

In Data.Rational.Unnormalised.Properties

≤-steps : ∀ {p q r} → NonNegative r → p ≤ q → p ≤ r + q
p≤p+q : ∀ {p q} → NonNegative q → p ≤ p + q
p≤q+p : ∀ {p} → NonNegative p → ∀ {q} → q ≤ p + q

[MatthewDaggitt]

Okay so

• we should be explicit everywhere in Algebra.Definitions
• we should change Data.Rational.Properties (and elsewhere) to be use those definitions

[MatthewDaggitt]

We should check that the subsequent inconsistencies in Data.Nat/Integer/Rational.Properties been resolved by the NonZero rewrite everywhere, as their types have changed.

[JacquesCarette]

So there's a problem with using the definitions in Data.Rational.Properties: the notion of NonZero that it uses is different than the notion that AlmostLeftCancellative uses. So I'm not sure we can make that change. To make things consistent, what we'd need to do is to parametrize AlmostLeftCancellative by a "non zero" predicate.

Also, by inconsistencies, do you mean like the ones pointed out above regarding implicit/explicit? For example, in ≤-stepsˡ it does make sense to make them implicit, as they'll always be inferable. So I'm not quite sure what to do for part 2 of this.

[jamesmckinna]

Is there a common resolution available of this issue with (the implied need for non-zeroness distinct from NonZero brought up by) #1562? And #1175?

Eg. see developing branch towards a possible PR...

... where I will shortly push some breaking (and speculative) changes.