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[refactor]: +-∸-assoc as proposed by @jamesmckinna
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src/Data/Nat/Properties.agda

Lines changed: 5 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -1614,12 +1614,11 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
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∸-+-assoc (suc m) (suc n) o = ∸-+-assoc m n o
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+-∸-assoc : m {n o} o ≤ n (m + n) ∸ o ≡ m + (n ∸ o)
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+-∸-assoc m (z≤n {n = n}) = begin-equality m + n ∎
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+-∸-assoc m (s≤s {m = o} {n = n} o≤n) = begin-equality
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(m + suc n) ∸ suc o ≡⟨ cong (_∸ suc o) (+-suc m n) ⟩
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suc (m + n) ∸ suc o ≡⟨⟩
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(m + n) ∸ o ≡⟨ +-∸-assoc m o≤n ⟩
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m + (n ∸ o) ∎
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+-∸-assoc zero {n = _} {o = _} _ = refl
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+-∸-assoc (suc m) {n = n} {o = o} o≤n = begin-equality
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suc (m + n) ∸ o ≡⟨ ∸-suc (m≤n⇒m≤o+n m o≤n) ⟩
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suc ((m + n) ∸ o) ≡⟨ cong suc (+-∸-assoc m o≤n) ⟩
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suc (m + (n ∸ o)) ∎
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m≤n+o⇒m∸n≤o : m n {o} m ≤ n + o m ∸ n ≤ o
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m≤n+o⇒m∸n≤o m zero le = le

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