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| ∣m+n∣n⇒∣m {d} {m} {n} (divides p m*n≡p*d) (divides-refl q) = | ||
| divides (p ∸ q) $ begin-equality | ||
| m ≡⟨ m+n∸n≡m m n ⟨ | ||
| m + n ∸ n ≡⟨ cong (_∸ q * d) m*n≡p*d ⟩ | ||
| p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨ | ||
| (p ∸ q) * d ∎ |
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Based on the other proof, I presume that this is a typo on the variable name.
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| ∣m+n∣n⇒∣m {d} {m} {n} (divides p m*n≡p*d) (divides-refl q) = | |
| divides (p ∸ q) $ begin-equality | |
| m ≡⟨ m+n∸n≡m m n ⟨ | |
| m + n ∸ n ≡⟨ cong (_∸ q * d) m*n≡p*d ⟩ | |
| p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨ | |
| (p ∸ q) * d ∎ | |
| ∣m+n∣n⇒∣m {d} {m} {n} (divides p m+n≡p*d) (divides-refl q) = | |
| divides (p ∸ q) $ begin-equality | |
| m ≡⟨ m+n∸n≡m m n ⟨ | |
| m + n ∸ n ≡⟨ cong (_∸ q * d) m+n≡p*d ⟩ | |
| p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨ | |
| (p ∸ q) * d ∎ |
Also, why prove it by going back to the core argument instead of invoking ∣m+n∣m⇒∣n and
the fact _+_ is commutative?
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Also, why prove it by going back to the core argument instead of invoking
∣m+n∣m⇒∣nand
the fact_+_is commutative?
I felt like this new proof is the more natural variant (the other already uses +-comm in its proof) and I don't like using subst if I can help it
Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
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Also slightly simplify the proof of
∣m+n∣m⇒∣n, which is closely related