[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.

src/Algebra/Reasoning/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

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)

src/Algebra/Properties/Semigroup/Reasoning.agda

JacquesCarette · 2025-04-04T19:05:52Z

src/Algebra/Properties/Semigroup/Reasoning.agda

+
+module Assoc4  {u v w x : Carrier} where
+  [[u∙v]∙w]∙x≈u∙[[v∙w]∙x] : ((u ∙ v) ∙ w) ∙ x ≈ u ∙ ((v ∙ w) ∙ x)
+  [[u∙v]∙w]∙x≈u∙[[v∙w]∙x] = trans (∙-congʳ (assoc u v w)) (assoc u (v ∙ w) x)


The strong reason to have push/pull defined before these is that they can be used here. You should indeed move those up above Assoc4 and refactor the proofs here to use push/pull.

src/Algebra/Properties/Semigroup/Reasoning.agda

JacquesCarette · 2025-04-09T01:44:17Z

The 'new' names in Assoc4 are better. The rest are hideous and make things quite unreadable. It was a nice experiment... which I would call a clear failure. I most definitely prefer push/pull over these.

jamesmckinna · 2025-04-09T09:56:50Z

The 'new' names in Assoc4 are better. The rest are hideous and make things quite unreadable. It was a nice experiment... which I would call a clear failure. I most definitely prefer push/pull over these.

Again, agree to disagree. No 'clear' failure on my side, but happy to try to find a good outcome to the naming discussion.

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 <[email protected]>

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.

jamesmckinna · 2025-04-12T07:29:26Z

CHANGELOG.md

@@ -123,7 +123,7 @@ New modules

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

-Additions to existing modules
+* `Algebra.Properties.Semigroup` adding consequences for associtvity for semigroups


Typo

Suggested change

* `Algebra.Properties.Semigroup` adding consequences for associtvity for semigroups

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

And, in case it wasn't clear before: you do need to list, explicitly, every single lemma as an addition to that module (follow the model of the entries below), so this line should also move downwards to a separate block between Algebra.Properties.Monoid.Divisibility and Algebra.Properties.Semigroup.Divisibility...
... AND you've somehow deleted the line
Additions to existing modules at L126... which you need to restore, sigh.

jamesmckinna · 2025-04-12T07:30:39Z

src/Algebra/Properties/Semigroup.agda

+    uv≈w⇒[xu∙v]y≈x∙wy x y = trans (∙-congʳ (uv≈w⇒xu∙v≈xw x)) (assoc _ _ _)
+
+    uv≈w⇒[xy∙u]v≈x∙yw : ∀ x y → ((x ∙ y) ∙ u) ∙ v ≈ x ∙ (y ∙ w)
+    uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x  y )


whitespace

Suggested change

uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x y )

uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x y)

jamesmckinna · 2025-04-12T07:32:24Z

src/Algebra/Properties/Semigroup.agda

+    uv≈w⇒xu∙vy≈x∙wy : (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)
+    uv≈w⇒xu∙vy≈x∙wy = uv≈w⇒xu∙v≈xw (uv≈w⇒u∙vx≈wx uv≈w _) _


I think that this needs, for the usual reasons about the unifier/typechecker

Suggested change

uv≈w⇒xu∙vy≈x∙wy : (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)

uv≈w⇒xu∙vy≈x∙wy = uv≈w⇒xu∙v≈xw (uv≈w⇒u∙vx≈wx uv≈w _) _

uv≈w⇒xu∙vy≈x∙wy : ∀ x y → (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)

uv≈w⇒xu∙vy≈x∙wy x y = uv≈w⇒xu∙v≈xw (uv≈w⇒u∙vx≈wx uv≈w x) y

jamesmckinna · 2025-04-12T07:33:51Z

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 _ _)


And here

Suggested change

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 _ _)

uv≈w⇒x∙wy≈x∙[u∙vy] : ∀ x y → x ∙ (w ∙ y) ≈ x ∙ (u ∙ (v ∙ y))

uv≈w⇒x∙wy≈x∙[u∙vy] x y = sym (uv≈w⇒x[uv∙y]≈x∙wy uv≈w x y)

jamesmckinna · 2025-04-12T07:37:40Z

src/Algebra/Properties/Semigroup.agda

+    uv≈w⇒xy≈z⇒u[vx∙y]≈wz : ∀ z → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z
+    uv≈w⇒xy≈z⇒u[vx∙y]≈wz z xy≈z = trans (∙-congˡ (uv≈w⇒xu∙v≈xw xy≈z v)) (uv≈w⇒u∙vx≈wx uv≈w z)


And here:

Suggested change

uv≈w⇒xy≈z⇒u[vx∙y]≈wz : ∀ z → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z

uv≈w⇒xy≈z⇒u[vx∙y]≈wz z xy≈z = trans (∙-congˡ (uv≈w⇒xu∙v≈xw xy≈z v)) (uv≈w⇒u∙vx≈wx uv≈w z)

uv≈w⇒xy≈z⇒u[vx∙y]≈wz : ∀ x y → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z

uv≈w⇒xy≈z⇒u[vx∙y]≈wz x y xy≈z = trans (∙-congˡ (uv≈w⇒xu∙v≈xw xy≈z v)) (uv≈w⇒u∙vx≈wx uv≈w _)

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

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...

jmougeot added 4 commits April 1, 2025 15:51

Adds reasonig combinator for semigroup

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Add some more missing reasoning combinators

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[Add] Consequences of identity for monoids #2692

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jamesmckinna changed the title ~~[Add] Reasoning combinator for semigroup~~ [Add] Consequences of associativity for Semigroups Apr 10, 2025

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Implementation of consequences and lemmas for Groups #2695

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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

	* `Algebra.Properties.Semigroup` adding consequences for associtvity for semigroups
	* `Algebra.Properties.Semigroup` adding consequences for associativity for semigroups

	uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x y )
	uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x y)

		uv≈w⇒xu∙vy≈x∙wy : (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)
		uv≈w⇒xu∙vy≈x∙wy = uv≈w⇒xu∙v≈xw (uv≈w⇒u∙vx≈wx uv≈w _) _

		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 _ _)

		uv≈w⇒xy≈z⇒u[vx∙y]≈wz : ∀ z → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z
		uv≈w⇒xy≈z⇒u[vx∙y]≈wz z xy≈z = trans (∙-congˡ (uv≈w⇒xu∙v≈xw xy≈z v)) (uv≈w⇒u∙vx≈wx uv≈w z)

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

[Add] Consequences of associativity for Semigroups #2688

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[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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