Attempt at the INT construction #2891
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Taneb
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| record _≃_ (i j : ℤ) : Set where | ||
| constructor mk≃ | ||
| field | ||
| eq : ℤ.minuend i ℕ.+ ℤ.subtrahend j ≡ ℤ.minuend j ℕ.+ ℤ.subtrahend i |
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First observation from Properties: use of sym to flip rewrites in the RHS of such equations... so suggest instead
| eq : ℤ.minuend i ℕ.+ ℤ.subtrahend j ≡ ℤ.minuend j ℕ.+ ℤ.subtrahend i | |
| eq : ℤ.minuend i ℕ.+ ℤ.subtrahend j ≡ ℤ.subtrahend i ℕ.+ ℤ.minuend j |
?
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The reason I did it this way was so that reflexivity is exactly refl. But you're right, that doesn't seem to be worth the tradeoff
| 1ℤ : ℤ | ||
| 1ℤ = 1 ⊖ 0 | ||
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Also:
| 1ℤ : ℤ | |
| 1ℤ = 1 ⊖ 0 | |
| 1ℤ : ℤ | |
| 1ℤ = 1 ⊖ 0 | |
| -1ℤ : ℤ | |
| -1ℤ = 0 ⊖ 1 |
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Easy enough, but does it pass the Fairbairn threshold? -1ℤ = - 1ℤ, after all
| infix 8 ⁻_ ⁺_ | ||
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| ⁺_ : ℕ → ℤ | ||
| ⁺ n = n ⊖ 0 | ||
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| ⁻_ : ℕ → ℤ | ||
| ⁻ n = 0 ⊖ n |
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The ⁻ symbol is in particular quite hard to discern as distinct form -. Would it be better to use Sign here (and reimplement the corresponding signAbs/SignAbs plumbing)?
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I mostly got annoyed at Agda not being able to tell if a section (x +_) was _+_ x or x (+_), so I wanted different symbols for them... the superscript-minus symbol is used in APL's number syntax for a negative, and that's what inspired the use of these here. I'm not attached to them, though.
| a ℕ.% n ℕ.+ 0 ≡⟨ ℕ.+-identityʳ (a ℕ.% n) ⟩ | ||
| a ℕ.% n ≡⟨ ℕ.m∸n+n≡m p ⟨ | ||
| a ℕ.% n ℕ.∸ b ℕ.% n ℕ.+ b ℕ.% n ∎ | ||
| ... | no b%n≰a%n = ≃-trans property′ $ mk≃ $ begin |
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Oh. Tempting to use a (Semi)Ring solver here?
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Do those work well when I'm using monus? Not that I ever really can manage to get them to work well for me...
| open import Data.Product.Base | ||
| open import Function.Base | ||
| open import Relation.Binary.PropositionalEquality | ||
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is it easier to derive each of these properties from the 'adjoint equivalence'/'graph relation' forms:
fromINT i ≡ j → i ≃ toINT j
i ≃ toINT j → fromINT i ≡ j(plus, perhaps, some additional lemmas to 'explain' these two graph relations)?
(I think even the congs follow from this, but I'd need to check that...)
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The second one is easy enough, it's hidden in my surjective proof. I'll have a go at the first one later.
As weak evidence that the congs don't follow from this: they're separate fields in the IsInverse structure in Function.Structures
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Closes #2890 (eventually)
The proofs, while purely algebraic, are often very tedious, especially
*-cong.I've got basic arithmetic, ordering with addition, and divmod working!