[Add] Consequences of associativity for `Semigroup`s #2688

jmougeot · 2025-04-01T19:56:36Z

Introduce a new module for equational reasoning in semigroups, providing utilities for associativity reasoning and operations.

src/Algebra/Reasoning/Semigroup.agda

JacquesCarette · 2025-04-02T00:10:13Z

For others: the names are taken from the combinators in agda-categories. Those were inherited from older versions of that library. I do not like them! So please do suggest better ones.

jamesmckinna · 2025-04-02T09:18:49Z

Introduce a new module for equational reasoning in semigroups, providing utilities for associativity reasoning and operations.

With an explicit callback/reference to #2288 ? This PR alone doesn't 'fix' that issue, but contributes a piece of the jigsaw to such a 'fix'.

CHANGELOG.md

src/Algebra/Reasoning/Semigroup.agda

jamesmckinna

Lots of comments, I'm requesting changes based on a single one, but can be summarised as:

please give lemmas names that can be remembered without needing a lookup table or other instruction manual to explain what they are/how they behave!
no need for a new module, we already have the Algebra.Properties.* sub-hierarchy...

src/Algebra/Reasoning/Semigroup.agda

jamesmckinna · 2025-04-02T09:50:31Z

src/Algebra/Reasoning/Semigroup.agda

+module Pulls (ab≡c : a ∙ b ≈ c) where
+    pullʳ : ∀ {x} → (x ∙ a) ∙ b ≈ x ∙ c


Elsewhere, the library would use lemma names of the form p∧q⇒r, with square brackets for grouping sub-terms, so here:

module Pulls (x∙y≈z : x ∙ y ≈ z) where x∙y≈z⇒[w∙x]∙y≈w∙z : (w ∙ x) ∙ y ≈ w ∙ z

NB. also: you have declared x as a variable, so there's no need to have the quantifier! Although here again, we have the problem that in any deployment in a concrete Semigroup the typechecker might not be able to infer w... so there are questions about how the quantifications should be handled.

If you can find examples in eg Data.Rational.Properties which might be simplified by these lemmas, then you might discover whether Agda can (or not) infer the various implicits?

Also, the use of ⇒ in the lemma names would avoid the need for the submodules here to be named; they could (more) simply be anonymous module _ (hyps) where...

While x∙y≈z⇒[w∙x]∙y≈w∙z is a logical name, it is also quite long. As these combinators that re-associate are book-keeping things, I'd like to find something shorter / less noisy.

I do hate push/pull. I'm partial to on-right.

Well, with my suggested notational optimisations, this would/could become...xy≈z⇒wx∙y≈wz, which is about as short as I can make it?!

As for on-right I'm really not sure what that is supposed to signify, given that the lemma is not cong like, nor is the action obviously happening on the right? (Never mind our ongoing debates about left/right distinctions giving rise to confusion...)

on-right as in apply this equality on the right, after a re-focus. It's very much cong-like to me, it just has a re-association done first, to put "the right" in focus.

I'd tried to reply, but my comment (seems to have) got lost: if you're wedded to an ASCII/prefix-Lisp name for these things, then I'd much prefer refocus-on-right, or even refocus-right (the -on isn't then doing much work), or even refocusʳ... in the style of other lemmas in the library.

I'm going to make some proposal on naming as a full comments instead of inline.

src/Algebra/Reasoning/Semigroup.agda

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>

jamesmckinna · 2025-04-03T07:14:26Z

src/Algebra/Reasoning/Semigroup.agda

+module Algebra.Reasoning.Semigroup {o ℓ} (S : Semigroup o ℓ) where
+
+open Semigroup S
+    using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong)


Indentation is by 2 spaces, not 4 as you have here... see style-guide.

Suggested change

using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong)

using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong; ∙-congˡ; ∙-congʳ)

jamesmckinna · 2025-04-03T07:23:04Z

src/Algebra/Reasoning/Semigroup.agda

+    pushʳ {x = x} = begin
+        x ∙ c       ≈⟨ sym (∙-cong refl ab≡c) ⟩
+        x ∙ (a ∙ b) ≈⟨ sym (assoc x a b) ⟩
+        (x ∙ a) ∙ b ∎


Successive appeals to sym suggest that it should be lifted out (permutation of sym through proofs), but in fact doing so quickly reveals the following:

Suggested change

pushʳ {x = x} = begin

x ∙ c ≈⟨ sym (∙-cong refl ab≡c) ⟩

x ∙ (a ∙ b) ≈⟨ sym (assoc x a b) ⟩

(x ∙ a) ∙ b ∎

pushʳ = sym (pullʳ ab≡c)

src/Algebra/Reasoning/Semigroup.agda

jamesmckinna · 2025-04-04T08:23:34Z

OK... thanks for the recent commits, but I'm going to carry on nitpicking:

the Assoc4 proofs miss lots of symmetry, with 5 'independent' lemmas each with their symmetric counterpart, and since Reasoning.Setoid does have the syntax for applying the symmetric equivalent of an equation, we maybe don't even need the 5 equivalents...
- u∙[[v∙w]∙x]≈[[u∙v]∙w]∙x = sym [[u∙v]∙w]∙x≈u∙[[v∙w]∙x]
- u∙[v∙[w∙x]]≈[[u∙v]∙w]∙x = sym [[u∙v]∙w]∙x≈u∙[v∙[w∙x]]
- [u∙v]∙[w∙x]≈[u∙[v∙w]]∙x = sym [u∙[v∙w]]∙x≈[u∙v]∙[w∙x]
- u∙[v∙[w∙x]]≈[u∙[v∙w]]∙x = sym [u∙[v∙w]]∙x≈u∙[v∙[w∙x]]
- u∙[v∙w]]∙x]≈[u∙v]∙[w∙x] = sym [u∙v]∙[w∙x]≈u∙[[v∙w]]∙x]
  where this last one, [u∙v]∙[w∙x]≈u∙[[v∙w]]∙x] is defined as a step-by-step proof, rather than the others, just defined in terms of trans, so could instead consider
- [u∙v]∙[w∙x]≈u∙[[v∙w]]∙x] = trans (assoc u v (w ∙ x)) (sym (∙-congˡ (assoc v w x)))
similarly, while the Pushes are symmetric equivalents of something in Pulls, not all the Pulls have their counterparts... but as above, maybe (probably) we don't even need the Pushes at all...
the square brackets do improve things, I think, but you've overdone them, as there's no need to have each innermost group neither bracketed, nor mention the operation explicitly, cf. x∙yz≈xy∙z, so I should apologise for not expressing my suggestion clearly enough (or else falling into the same trap myself!), so eg. [[u∙v]∙w]∙x≈u∙[[v∙w]∙x] could be simplified to [uv∙w]∙x≈u∙[vw∙x] and that would be, I think, much more readable (with less ink!). Or even [[uv]w]x≈u[[vw]x] but that exchanges one character for two, so is perhaps suboptimal. Etc.

jamesmckinna · 2025-04-04T08:27:20Z

Also:

Introduce a new module for equational reasoning in semigroups, providing utilities for associativity reasoning and operations.

As with previous comments, we don't need a new module (even if it's moved location), these could all go in Algebra.Properties.Semigroup, and should, I think (another instance of Occam's Eraser: don't multiply modules unnecessarily)

jamesmckinna

Placeholder review: this will have to wait until at least the weekend before I can return to it.
But I think in the meantime there still seem to be spacing issues, the existing question of where these things should live, as well as those regarding whether explicitly-named proofs of 'symmetric' versions are really necessary.

MatthewDaggitt

I agree with a lot of what @jamesmckinna is saying.

This doesn't need it's own module, i.e. Algebra.Properties.Semigroup is fine.
I much prefer the current set of names.

MatthewDaggitt · 2025-04-10T05:23:52Z

CHANGELOG.md

@@ -123,6 +123,8 @@ New modules

 * `Data.Sign.Show` to show a sign

+* `AlgebraPropreties.Semigroup.Reasoning` adding reasoning combinators for semigroups


Lots of typos in the name?

And indeed, if we do as I have suggested several times, and not make this a new module, but under additions to existing modules, then it should be under Algebra.Properties.Semigroup...

src/Algebra/Properties/Semigroup/Reasoning.agda

MatthewDaggitt · 2025-04-10T05:25:06Z

CHANGELOG.md

@@ -123,6 +123,8 @@ New modules

 * `Data.Sign.Show` to show a sign

+* `AlgebraPropreties.Semigroup.Reasoning` adding reasoning combinators for semigroups
+


I'm also not sure whether we should refer to these as reasoning combinators. The Reasoning terminology has quite a specific use in the library for syntax that chains together nicely. I'm not sure these lemmas qualify?

We won't need CHANGELOG text if it moves to Additions to..., but the opening comment block should be rephrased...

jamesmckinna

Lots of comments and suggestion, but mostly a do-over of things I have already commented on before, and were not (yet) changed. TL;DR:

move all the lemmas to Algebra.Properties.Semigroup
anonymous modules
fix implicit/explicit quantification

It would be better for you to do (all) this, but if you are having problems with it, it's easy enough for me to take a copy of your branch and push the various changes to it...

jamesmckinna · 2025-04-10T09:18:54Z

CHANGELOG.md

@@ -123,6 +123,8 @@ New modules

 * `Data.Sign.Show` to show a sign

+* `AlgebraPropreties.Semigroup.Reasoning` adding reasoning combinators for semigroups


And indeed, if we do as I have suggested several times, and not make this a new module, but under additions to existing modules, then it should be under Algebra.Properties.Semigroup...

jamesmckinna · 2025-04-10T09:20:14Z

CHANGELOG.md

@@ -123,6 +123,8 @@ New modules

 * `Data.Sign.Show` to show a sign

+* `AlgebraPropreties.Semigroup.Reasoning` adding reasoning combinators for semigroups
+


We won't need CHANGELOG text if it moves to Additions to..., but the opening comment block should be rephrased...

src/Algebra/Properties/Semigroup/Reasoning.agda

jamesmckinna · 2025-04-10T14:02:13Z

Following @MatthewDaggitt 's comment about 'reasoning combinators', I've taken the liberty of renaming the PR to hopefully better reflect the intention of the contribution(s) here.

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>

jmougeot · 2025-04-11T15:32:04Z

I moved everything into Semigroups and made the variables explicit. I tried to address all the comments, but I probably missed some. Sorry for not taking these changes into account earlier

jamesmckinna

Last round of changes, I hope, because I've looked at this to many times now... law of diminishing returns...

NB if you adopt my suggestions, there will no doubt be knock-on consequences (wrt parametrisation) which you should fix before committing... for the sake of any other reviewers!

UPDATED: and you still have a 4-space indentation for all the definitions, instead of the mandated 2... PLEASE FIX, along with the CHANGELOG formatting, which has now broken.

CHANGELOG.md

src/Algebra/Properties/Semigroup.agda

jamesmckinna · 2025-04-12T07:39:00Z

src/Algebra/Properties/Semigroup.agda

+    uv≈w⇒x∙wy≈x∙[u∙vy] : x ∙ (w ∙ y) ≈ x ∙ (u ∙ (v ∙ y))
+    uv≈w⇒x∙wy≈x∙[u∙vy] = sym (uv≈w⇒x[uv∙y]≈x∙wy uv≈w _ _)
+
+module _ {u v w x : Carrier} where


These will need to be explicit... for the usual reasons, but there's no need to make Carrier explicit:

Suggested change

module _ {u v w x : Carrier} where

module _ u v w x where

I'm not sure why this was marked as resolved?

jamesmckinna · 2025-04-12T07:40:15Z

src/Algebra/Properties/Semigroup.agda

+    u[v∙wx]≈[u∙vw]x : u ∙ (v ∙ (w ∙ x)) ≈ (u ∙ (v ∙ w)) ∙ x
+    u[v∙wx]≈[u∙vw]x = sym [u∙vw]x≈u[v∙wx]
+
+module _ {u v w x : Carrier} (uv≈wx : u ∙ v ≈ w ∙ x) where


Finally, here:

Suggested change

module _ {u v w x : Carrier} (uv≈wx : u ∙ v ≈ w ∙ x) where

module _ (uv≈wx : u ∙ v ≈ w ∙ x) where

because the variable declarations take care of the initial implicit prefix...

Nor this, because it hasn't been fixed yet.

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>

CHANGELOG.md

jamesmckinna · 2025-04-19T11:05:03Z

CHANGELOG.md

@@ -126,6 +126,8 @@ New modules
 Additions to existing modules
 -----------------------------

+* In `Algebra.Properties.Semigroup` adding consequences for associativity for semigroups
+


Need to insert here all the new names, once we have (eventually) agree them...

Suggested change

```agda

uv≈w⇒xu∙v≈xw : u ∙ v ≈ w → ∀ x → (x ∙ u) ∙ v ≈ x ∙ w

...

etc.

JacquesCarette · 2025-04-21T16:22:40Z

As I've indicated, I'm not a big fan of push/pull because they are not very mnemomic; the superscripts don't help either. Indeed, if one looks at

module Pulls (x∙y≈z : x ∙ y ≈ z) where
  x∙y≈z⇒[w∙x]∙y≈w∙z : (w ∙ x) ∙ y ≈ w ∙ z

what it really does is

refocus on the right
apply an equality there.

So that's what I'd like the naming to reflect. Now, I quite like the 'synonym' _⟩∙⟨_ for ∙-cong (hoping to introduce that synonym too.) So then if we take ⭆ to be refocus (on the right), then the above could be ⭆∙⟨_. We would similarly have a refocus on the left then act on that side as ⭅⟨_∙. I would be fine with putting in a marker for where "nothing happens". So we could have:

⭆-I∙⟨_ : refocus right then apply equation there
⭅-_⟩∙I : refocus left then apply equation there
I∙_⟩-⭅ : apply equation on right then refocus left
_⟩∙I-⭆ : apply equation on left then refocus right

Note that these are meant to have the 3 variables be implicit, on purpose: if they cannot be figured out by Agda, then it's probably better to use the 'long form' of the reasoning (i.e. explicit calls to assoc).

jamesmckinna · 2025-04-22T11:58:58Z

As I've indicated, I'm not a big fan of push/pull because they are not very mnemomic; the superscripts don't help either. Indeed, if one looks at
module Pulls (x∙y≈z : x ∙ y ≈ z) where
  x∙y≈z⇒[w∙x]∙y≈w∙z : (w ∙ x) ∙ y ≈ w ∙ z
what it really does is
1. refocus on the right

2. apply an equality there.
So that's what I'd like the naming to reflect. Now, I quite like the 'synonym' _⟩∙⟨_ for ∙-cong (hoping to introduce that synonym too.) So then if we take ⭆ to be refocus (on the right), then the above could be ⭆∙⟨_. We would similarly have a refocus on the left then act on that side as ⭅⟨_∙. I would be fine with putting in a marker for where "nothing happens". So we could have:
* `⭆-I∙⟨_` : refocus right then apply equation there

* `⭅-_⟩∙I` : refocus left then apply equation there

* `I∙_⟩-⭅` : apply equation on right then refocus left

* `_⟩∙I-⭆` : apply equation on left then refocus right
Note that these are meant to have the 3 variables be implicit, on purpose: if they cannot be figured out by Agda, then it's probably better to use the 'long form' of the reasoning (i.e. explicit calls to assoc).

OK, I guess it is/was good to see how far apart the various contributors are on how to think about these naming issues, and their intended semantic underpinnings. But I have to say that I find the gap between the symbolic names you propose ((NB in my Firefox font for GitHub, your symbol ⭅ doesn't even display!) and those intended meanings so large that I simply cannot imagine ever using these principles except via uncomprehending cut-and-paste from the module, and with the above explanation by my side as user manual.

In the face of this, I am going to suggest

either that we roll back to a text-based name, with your clear desire to inject 'refocus' as an idiom as a basis for the name (and I even suggested names based around that above)...
or that we redraw the 'implicational' forms I suggested to make the x∙y≈z⇒prefix implicit as an infix argument position, with eg.
- x∙y≈z⇒[w∙x]∙y≈w∙z : (w ∙ x) ∙ y ≈ w ∙ z becoming [wx]y≈[_]wz or [wx]y≈⟨_⟩wz, but again, we have up until now avoided making operators for equational proofs into infixes/mixfixes... except in the 'equation over equation' presentation of equality-modulo-cast in Data.Vec.Relation.Binary.Equality.Cast
or simply that I withdraw from even expressing a view about this (much less merge the PR), other than to say that adopting such a change as you propose would mark such a meta-level breaking change from existing practice in stdlib that... I am completely lost for words!

Barn-burning.

JacquesCarette · 2025-04-22T15:33:24Z

My firefox displays them all just fine!

I would be fine with a name that involves refocus instead of symbology. We could use superscripts for indicating left/right refocusing. The one thing I don't know how to make nice: refocus-then-cong vs cong-then-refocus.

Your mixfix suggestion might have legs. In fact, by eliding the common prefix entirely, using it as a purely prefix operation might result in a name that's close to readable.

If my mnemonic symbology is not acceptable, I would think a text-based name based around 'refocus' is probably better than other unicode soup.

jmougeot added 4 commits April 1, 2025 15:51

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[Add] Consequences of identity for monoids #2692

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		module Pulls (ab≡c : a ∙ b ≈ c) where
		pullʳ : ∀ {x} → (x ∙ a) ∙ b ≈ x ∙ c

	using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong)
	using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong; ∙-congˡ; ∙-congʳ)

		@@ -123,6 +123,8 @@ New modules

		* `Data.Sign.Show` to show a sign

		* `AlgebraPropreties.Semigroup.Reasoning` adding reasoning combinators for semigroups

	module _ {u v w x : Carrier} (uv≈wx : u ∙ v ≈ w ∙ x) where
	module _ (uv≈wx : u ∙ v ≈ w ∙ x) where

+  ```agda
+  uv≈w⇒xu∙v≈xw : u ∙ v ≈ w → ∀ x → (x ∙ u) ∙ v ≈ x ∙ w
+  ...

[Add] Consequences of associativity for Semigroups #2688

Are you sure you want to change the base?

[Add] Consequences of associativity for Semigroups #2688

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[Add] Consequences of associativity for `Semigroup`s #2688

[Add] Consequences of associativity for `Semigroup`s #2688

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