Modular arithmetic in terms of ideals #2729
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Would you want some help to get this further along? |
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Yes, actually. I've been working on a module for the special case of ideals of the ring of integers, and I've been struggling to prove that (for a non-zero modulus) it's finite, which I think it important for the "yes this is modular arithmetic as you know it" feel. I'll post a WIP commit shortly |
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Ok, once my students make further progress on the ones they are currently working on, I'll get them to look at this. |
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Some errors thrown up by checking with
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@jamesmckinna I've merged in master, are those two errors fixed now? |
| ι : RawGroup.Carrier N → Carrier | ||
| ι-monomorphism : IsGroupMonomorphism N rawGroup ι | ||
| -- every element of N commutes in G | ||
| normal : ∀ n g → ∃[ n′ ] g ∙ ι n ∙ g ⁻¹ ≈ ι n′ |
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So... I was a bit surprised that this was an 'easier' formulation than
| normal : ∀ n g → ∃[ n′ ] g ∙ ι n ∙ g ⁻¹ ≈ ι n′ | |
| normal : ∀ n g → ∃[ n′ ] g ∙ ι n ≈ ι n′ ∙ g |
which I wondered as to whether
- it would be easier/smoother to show gives rise to an equivalence relation on the quotient
- it would generalise (better) to
Loop,Quasigroupor evenMagma/Semigroup?
Cf. comments elsewhere from @JacquesCarette about defining 'ideal' via 'sink'...
| x * r * a ≈⟨ *-assoc x r a ⟩ | ||
| x * (r * a) ≈⟨ *-congˡ (*-comm r a) ⟩ | ||
| x * (a * r) ≈⟨ *-assoc x a r ⟨ | ||
| x * a * r ∎ |
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This kind of argument occurs in Algebra.Properties.CommutativeSemigroup, and might usefully be re-used here?
| x * a * r ∎ | ||
| } | ||
| } | ||
| ; injective = λ p → p |
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| ; injective = λ p → p | |
| ; injective = id |
???
| infix 0 _by_ | ||
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| data _≋_ (x y : Carrier) : Set (c ⊔ ℓ ⊔ c′) where | ||
| _by_ : ∀ g → x // y ≈ ι g → x ≋ y |
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Similarly to the type of NormalSubgroup.normal, is it 'easier' to write
| _by_ : ∀ g → x // y ≈ ι g → x ≋ y | |
| _by_ : ∀ g → x ≈ ι g ∙ y → x ≋ y |
???
| x // z ≈⟨ ∙-congʳ (identityʳ x) ⟨ | ||
| x ∙ ε // z ≈⟨ ∙-congʳ (∙-congˡ (inverseˡ y)) ⟨ | ||
| x ∙ (y \\ y) // z ≈⟨ ∙-congʳ (assoc x (y ⁻¹) y) ⟨ | ||
| (x // y) ∙ y // z ≈⟨ assoc (x // y) y (z ⁻¹) ⟩ | ||
| (x // y) ∙ (y // z) ≈⟨ ∙-cong x//y≈ιg y//z≈ιh ⟩ |
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Already covered by Algebra.Properties.Loop...? And if not, could/should be added?
| module _ .{{_ : NonZero m}} where | ||
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| from-% : ∀ {x y} → x % m ≡ y % m → x ≋ y | ||
| from-% {x} {y} x%m≡y%m = x / m - y / m by begin |
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How much of this argument recapitulates concretely reasoning steps already used abstractly in CRT?
Opening this PR to share my WIP. I've got a messy proof of the Chinese remainder theorem for arbitrary rings, but in porting it from my standalone library to this I've somehow made some parameters not infer properly