[Add] Consequences of identity for monoids
#2692
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Co-authored-by: jamesmckinna <[email protected]>
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Probably the 'best' thing to do is to mark this as blocked (on #2688 ), otherwise you'll end up with merge conflicts when that PR gets merged, and indeed to start afresh with this PR being on a branch based on |
jamesmckinna
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status: blocked-by-issue
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| module Algebra.Properties.Monoid.Reasoning {o ℓ} (M : Monoid o ℓ) where | ||
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| open import Algebra.Structures using (IsMagma) |
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As with #2688 there's no need to make this explicit, it's even an abstraction failure/violation to do so
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Suggest corresponding title change for the PR in the spirit of that for #2688 ... |
jmougeot
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jmougeot
changed the title
monoids
Co-authored-by: jamesmckinna <[email protected]>
| uv≈wx⇒u∙vy≈w∙xy y = trans (uv≈w⇒u∙vx≈wx uv≈wx y) (assoc w x y) | ||
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| uv≈wx⇒yu∙vz≈yw∙xz : ∀ y z → (y ∙ u) ∙ (v ∙ z) ≈ (y ∙ w) ∙ (x ∙ z) | ||
| uv≈wx⇒yu∙vz≈yw∙xz y z = trans (uv≈w⇒xu∙v≈xw (uv≈wx⇒u∙vy≈w∙xy z) y)(sym (assoc y w (x ∙ z))) |
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Space
| uv≈wx⇒yu∙vz≈yw∙xz y z = trans (uv≈w⇒xu∙v≈xw (uv≈wx⇒u∙vy≈w∙xy z) y)(sym (assoc y w (x ∙ z))) | |
| uv≈wx⇒yu∙vz≈yw∙xz y z = trans (uv≈w⇒xu∙v≈xw (uv≈wx⇒u∙vy≈w∙xy z) y) (sym (assoc y w (x ∙ z))) |
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
| id-comm-sym : ∀ a → ε ∙ a ≈ a ∙ ε | ||
| id-comm-sym a = sym (id-comm a) | ||
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| module _ {a b : Carrier} (a≈ε : a ≈ ε) where |
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the variable block above means you don't need {a b : Carrier} anymore
| elimˡ b = trans (∙-congʳ a≈ε) (identityˡ b) | ||
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| introʳ : ∀ b → b ≈ b ∙ a | ||
| introʳ b = sym (elimʳ b) |
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eta-reduce, here and next
| introˡ b = sym (elimˡ b) | ||
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| introcenter : ∀ c → b ∙ c ≈ b ∙ (a ∙ c) | ||
| introcenter c = trans (∙-congˡ (sym (identityˡ c))) (∙-congˡ (∙-congʳ (sym a≈ε))) |
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I wonder if we should use a superscript m (for 'middle') or c (for, indeed, 'center') instead of spelling it out?
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Meta: a good general point that, besides 'left' and 'right' (modulo collective mutual misunderstanding/inconsistency about what such conventions might even mean #2654 ), we don't have good heuristics about super-/sub-scripts and their intended semantics. I think I'd be happy with ᶜ or ᵐ... but nervous about establishing a precedent without further thought/discussion?
CHANGELOG.md
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| @@ -123,7 +123,7 @@ New modules | |||
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| * `Data.Sign.Show` to show a sign | |||
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| Additions to existing modules | |||
| * `Algebra.Properties.Semigroup` adding consequences for associatvity for semigroups | |||
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No, you need to revert this change.
The lemmas added to Algebra.Properties.Semigroup should each be added in an Agda code block under this line moved to be a sub-heading underneath this section heading...
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| open import Algebra.Bundles using (Monoid) | ||
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| module Algebra.Properties.Monoid.Reasoning {o ℓ} (M : Monoid o ℓ) where |
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As usual: these lemmas belong in Algebra.Properties.Monoid, not in a separate module as you have it.
By all means do initial development in that style before raising a PR, or else mark it as DRAFT (and hence not ready-for-review) but once you do raise the PR/ask for review, we do need it to be compatible both with the existing stdlib hierarchy/structure, but also with the style-guide, or else we will expend a lot of extra reviewing energy on things which should have been fixed ahead of time.
Co-authored-by: jamesmckinna <[email protected]>
Co-authored-by: jamesmckinna <[email protected]>
| module _ (inv : a ∙ c ≈ ε) where | ||
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| cancelʳ : ∀ b → (b ∙ a) ∙ c ≈ b | ||
| cancelʳ b = trans (assoc b a c) (trans (∙-congˡ inv) (identityʳ b)) |
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I'm quite sure this can be done in 2 steps via one of the Semigroup combinators. Same for on the left.
Introduce reasoning combinators for monoids.
This PR is intended to focus on the
Monoid.Reasoningfilr as the discussion regarding theSemigroup.reasonigfile is in the PR : #2688I went with the names from agda-categories since they seemed to make sense.