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reverted export; removed anonymous module
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src/Data/Nat/Divisibility.agda

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@@ -39,43 +39,32 @@ private
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open import Data.Nat.Divisibility.Core public hiding (*-pres-∣)
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module _ (m∣n : m ∣ n) where
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open _∣_ m∣n
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quotient≢0 : .{{NonZero n}} NonZero quotient
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quotient≢0 rewrite equality = m*n≢0⇒m≢0 quotient
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equalityᵒ : n ≡ m * quotient
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equalityᵒ rewrite equality = *-comm quotient m
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quotient-∣ : quotient ∣ n
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quotient-∣ = divides m equalityᵒ
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quotient>1 : m < n 1 < quotient
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quotient>1 m<n = *-cancelˡ-< m 1 quotient $ begin-strict
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m * 1 ≡⟨ *-identityʳ m ⟩
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m <⟨ m<n ⟩
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n ≡⟨ equalityᵒ ⟩
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m * quotient ∎
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where open ≤-Reasoning
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quotient-< : .{{NonTrivial m}} .{{NonZero n}} quotient < n
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quotient-< = begin-strict
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quotient <⟨ m<m*n quotient m (nonTrivial⇒n>1 m) ⟩
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quotient * m ≡⟨ equality ⟨
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n ∎
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where
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open ≤-Reasoning
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instance _ = quotient≢0
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-- Exports
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open _∣_ using (quotient) public
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quotient≢0 : (m∣n : m ∣ n) .{{NonZero n}} NonZero (quotient m∣n)
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quotient≢0 m∣n rewrite _∣_.equality m∣n = m*n≢0⇒m≢0 (quotient m∣n)
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equalityᵒ : (m∣n : m ∣ n) n ≡ m * (quotient m∣n)
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equalityᵒ {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m
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quotient-∣ : (m∣n : m ∣ n) (quotient m∣n) ∣ n
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quotient-∣ {m = m} m∣n = divides m (equalityᵒ m∣n)
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quotient>1 : (m∣n : m ∣ n) m < n 1 < quotient m∣n
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quotient>1 {m} {n} m∣n m<n = *-cancelˡ-< m 1 (quotient m∣n) $ begin-strict
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m * 1 ≡⟨ *-identityʳ m ⟩
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m <⟨ m<n ⟩
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n ≡⟨ equalityᵒ m∣n ⟩
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m * quotient m∣n ∎
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where open ≤-Reasoning
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quotient-< : (m∣n : m ∣ n) .{{NonTrivial m}} .{{NonZero n}} quotient m∣n < n
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quotient-< {m} {n} m∣n = begin-strict
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quotient m∣n <⟨ m<m*n (quotient m∣n) m (nonTrivial⇒n>1 m) ⟩
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quotient m∣n * m ≡⟨ _∣_.equality m∣n ⟨
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n ∎
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where open ≤-Reasoning; instance _ = quotient≢0 m∣n
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------------------------------------------------------------------------
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-- Relating _/_ and _∣_.quotient
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-- Relating _/_ and quotient
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n/m≡quotient : (m∣n : m ∣ n) .{{_ : NonZero m}} n / m ≡ quotient m∣n
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n/m≡quotient {m = m} (divides-refl q) = m*n/n≡m q m

src/Data/Nat/Divisibility/Core.agda

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@@ -46,6 +46,8 @@ m ∤ n = ¬ (m ∣ n)
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pattern divides-refl q = divides q refl
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open _∣_ using (quotient) public
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------------------------------------------------------------------------
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-- Restricted divisor relation
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