[Add] Reasoning combinator for semigroup #2688

jmougeot · 2025-04-01T19:56:36Z

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

JacquesCarette · 2025-04-02T00:01:49Z

src/Algebra/Reasoning/Semigroup.agda

+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra using (Semigroup)


tighter import?

JacquesCarette · 2025-04-02T00:04:47Z

src/Algebra/Reasoning/Semigroup.agda

+  Only reassociations that need two assoc steps are defined here.
+  -}
+  assoc²αδ : ((a ∙ b) ∙ c) ∙ d ≈ a ∙ ((b ∙ c) ∙ d)
+  assoc²αδ = trans (∙-cong (assoc a b c) refl) (assoc a (b ∙ c) d)


For the purposes of the standard library, I'm thinking that all of these should be equational proofs?

Also, I think that these will be simpler to define if done after the various push/pull combinators are defined and available.

As in my comments elsewhere, I think w x y z ought to be the preferred variable names, rather than a b c d, which (tend to be) reserved for Level variables... not sure what style-guide recommends here?

Now, you do seem to need at least 6 names, but still I would use u v w x y z rather than a b c d x y z...

As to @JacquesCarette 's

For the purposes of the standard library, I'm thinking that all of these should be equational proofs?

Actually, with one eye on #2629 , I think the opposite, as most of these are 'simple' instances of trans, esp. when written with ∙-congˡ and ∙-congʳ...

JacquesCarette · 2025-04-02T00:06:11Z

src/Algebra/Reasoning/Semigroup.agda

+    pullʳ : ∀ {x} → (x ∙ a) ∙ b ≈ x ∙ c
+    pullʳ {x = x} = begin
+        (x ∙ a) ∙ b ≈⟨ assoc x a b ⟩
+        x ∙ (a ∙ b) ≈⟨ ∙-cong refl ab≡c ⟩


isn't there a left/right version of cong already available? This explicit 'refl' everywhere is bothersome.

Algebra.Structures.IsMagma L58-L62:

∙-congˡ : LeftCongruent ∙ ∙-congˡ y≈z = ∙-cong refl y≈z ∙-congʳ : RightCongruent ∙ ∙-congʳ y≈z = ∙-cong y≈z refl

JacquesCarette · 2025-04-02T00:06:45Z

src/Algebra/Reasoning/Semigroup.agda

+
+    pullˡ : ∀ {x} → a ∙ (b ∙ x) ≈ c ∙ x
+    pullˡ {x = x} = begin
+        a ∙ (b ∙ x) ≈⟨ sym (assoc a b x) ⟩


You should use the combinator that builds-in symmetry rather than invoke it explicitly.

Here:

Suggested change

a ∙ (b ∙ x) ≈⟨ sym (assoc a b x) ⟩

a ∙ (b ∙ x) ≈⟨ assoc a b x ⟨

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

jamesmckinna · 2025-04-02T09:19:58Z

src/Algebra/Reasoning/Semigroup.agda

+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra using (Semigroup)


Suggested change

open import Algebra using (Semigroup)

open import Algebra.Bundles using (Semigroup)

jamesmckinna · 2025-04-02T09:37:23Z

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)


As in other comments, use also ∙-congˡ and ∙-congʳ... by which point you might as well not have a using qualification at all, as you've imported everything (I think)

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...

jamesmckinna · 2025-04-02T09:43:38Z

src/Algebra/Reasoning/Semigroup.agda

+    pull-center : ∀ {x y} → x ∙ (a ∙ (b ∙ y)) ≈ x ∙ (c ∙ y)
+    pull-center {x = x} {y = y} = ∙-cong refl (pullˡ)
+
+    -- could be called pull₃ʳ


But MUST not, IMNSVHO, because how on earth will a reader/user remember such names, without an additional mode d'emploi by their side?

Similarly for all the other names... I don't find the pull-X naming at all memorable/mnemonic.

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...

jamesmckinna · 2025-04-02T10:04:11Z

src/Algebra/Reasoning/Semigroup.agda

+-- operate in the 'center' instead (like pull/push)
+center : a ∙ b ≈ c → (d ∙ a) ∙ (b ∙ e) ≈ d ∙ (c ∙ e)
+center eq = pullʳ (pullˡ eq)
+
+-- operate on the left part, then the right part
+center⁻¹ : a ∙ b ≈ c → x ∙ y ≈ z →  a ∙ ((b ∙ x) ∙ y) ≈ c ∙ z
+center⁻¹ {a = a} {b = b} {c = c} {x = x} {y = y} {z = z} eq eq′ = begin
+  a ∙ ((b ∙ x) ∙ y) ≈⟨ ∙-cong refl (pullʳ eq′) ⟩
+  a ∙ (b ∙ z)       ≈⟨ pullˡ eq ⟩
+  c ∙ z             ∎
+
+push-center : a ∙ b ≈ c → x ∙ (c ∙ y) ≈ x ∙ (a ∙ (b ∙ y))
+push-center eq = sym (pull-center eq)


So... each of these is parametrised on an assumption of the form a ∙ b ≈ c, so why is this not abstracted out as a module parameter?

-- operate in the 'center' instead (like pull/push) module _ (a∙b≈c : a ∙ b ≈ c) where center : (d ∙ a) ∙ (b ∙ e) ≈ d ∙ (c ∙ e) center = pullʳ (pullˡ a∙b≈c) ... etc.

But see elsewhere about 'better' names!

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

jmougeot added 4 commits April 1, 2025 15:51

Adds reasonig combinator for semigroup

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Adds reasonig combinator for semigroup

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Adds reasonig combinator for semigroup

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Adds reasonig combinator for semigroup

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jmougeot changed the title ~~Add reasoning combinator for semigroup~~ [Add] Reasoning combinator for semigroup Apr 1, 2025

jmougeot added 4 commits April 1, 2025 17:35

Add some more missing reasoning combinators

63e88cc

add module Extends

bb7ce15

rename SemiGroup to Semigroup

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fix-whitespace

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

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[Add] Reasoning combinator for semigroup #2688

[Add] Reasoning combinator for semigroup #2688

jmougeot commented Apr 1, 2025

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		{-# OPTIONS --cubical-compatible --safe #-}

		open import Algebra using (Semigroup)

	a ∙ (b ∙ x) ≈⟨ sym (assoc a b x) ⟩
	a ∙ (b ∙ x) ≈⟨ assoc a b x ⟨

	open import Algebra using (Semigroup)
	open import Algebra.Bundles using (Semigroup)

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

[Add] Reasoning combinator for semigroup #2688

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