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Fill in some implicit arguments in Data.Nat.DivMod #3122

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Fill in some implicit arguments in Data.Nat.DivMod #3122

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20 changes: 10 additions & 10 deletions src/Data/Nat/DivMod.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -306,7 +306,7 @@ m∣n⇒o%n%m≡o%m : ∀ m n o .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣
o % n % m ≡ o % m
m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
o % n % m ≡⟨⟩
o % pm % m ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩
o % pm % m ≡⟨ %-congˡ {o = m} (m%n≡m∸m/n*n o pm) ⟩
(o ∸ o / pm * pm) % m ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨
(o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩
o % m ∎
Expand All @@ -327,7 +327,7 @@ m*n/m*o≡n/o m n o = helper (<-wellFounded n)
helper : ∀ {n} → Acc _<_ n → (m * n) / (m * o) ≡ n / o
helper {n} (acc rec) with <-≤-connex n o
... | inj₁ n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))
where instance _ = m*n≢0⇒m≢0 m
where instance _ = m*n≢0⇒m≢0 m {o}
... | inj₂ n≥o = begin-equality
(m * n) / (m * o) ≡⟨ m/n≡1+[m∸n]/n (*-monoʳ-≤ m n≥o) ⟩
1 + (m * n ∸ m * o) / (m * o) ≡⟨ cong (suc ∘ (_/ (m * o))) (*-distribˡ-∸ m n o) ⟨
Expand All @@ -339,7 +339,7 @@ m*n/m*o≡n/o m n o = helper (<-wellFounded n)
m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (o * n)}} →
m * n / (o * n) ≡ m / o
m*n/o*n≡m/o m n o = begin-equality
m * n / (o * n) ≡⟨ /-congˡ (*-comm m n) ⟩
m * n / (o * n) ≡⟨ /-congˡ {o = o * n} (*-comm m n) ⟩
n * m / (o * n) ≡⟨ /-congʳ (*-comm o n) ⟩
n * m / (n * o) ≡⟨ m*n/m*o≡n/o n m o ⟩
m / o ∎
Expand All @@ -363,7 +363,7 @@ m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict
[m∸n*o]/o≡m/o∸n : ∀ m n o .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n
[m∸n*o]/o≡m/o∸n m zero o = refl
[m∸n*o]/o≡m/o∸n m (suc n) o = begin-equality
(m ∸ (o + n * o)) / o ≡⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟨
(m ∸ (o + n * o)) / o ≡⟨ /-congˡ {o = o} (∸-+-assoc m o (n * o)) ⟨
(m ∸ o ∸ n * o) / o ≡⟨ [m∸n*o]/o≡m/o∸n (m ∸ o) n o ⟩
(m ∸ o) / o ∸ n ≡⟨ cong (_∸ n) ([m∸n]/n≡m/n∸1 m o) ⟩
m / o ∸ 1 ∸ n ≡⟨ ∸-+-assoc (m / o) 1 n ⟩
Expand All @@ -372,10 +372,10 @@ m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict
m/n/o≡m/[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}}
.{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)
m/n/o≡m/[n*o] m n o = begin-equality
m / n / o ≡⟨ /-congˡ {o = o} (/-congˡ (m≡m%n+[m/n]*n m n*o)) ⟩
(m % n*o + m / n*o * n*o) / n / o ≡⟨ /-congˡ (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩
m / n / o ≡⟨ /-congˡ {o = o} (/-congˡ {o = n} (m≡m%n+[m/n]*n m n*o)) ⟩
(m % n*o + m / n*o * n*o) / n / o ≡⟨ /-congˡ {o = o} (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩
(m % n*o / n + m / n*o * n*o / n) / o ≡⟨ cong (λ # → (m % n*o / n + #) / o) lem₂ ⟩
(m % n*o / n + m / n*o * o) / o ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) (divides-refl (m / n*o)) ⟩
(m % n*o / n + m / n*o * o) / o ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) {d = o} (divides-refl (m / n*o)) ⟩
m % n*o / n / o + m / n*o * o / o ≡⟨ cong (m % n*o / n / o +_) (m*n/n≡m (m / n*o) o) ⟩
m % n*o / n / o + m / n*o ≡⟨ cong (_+ m / n*o) (m<n⇒m/n≡0 (m<n*o⇒m/o<n {n = o} lem₃)) ⟩
m / n*o ∎
Expand All @@ -392,7 +392,7 @@ m/n/o≡m/[n*o] m n o = begin-equality
lem₂ : m / n*o * n*o / n ≡ m / n*o * o
lem₂ = begin-equality
m / n*o * n*o / n ≡⟨ cong (λ # → m / n*o * # / n) (*-comm n o) ⟩
m / n*o * o*n / n ≡⟨ /-congˡ (*-assoc (m / n*o) o n) ⟨
m / n*o * o*n / n ≡⟨ /-congˡ {o = n} (*-assoc (m / n*o) o n) ⟨
m / n*o * o * n / n ≡⟨ m*n/n≡m (m / n*o * o) n ⟩
m / n*o * o ∎

Expand All @@ -414,7 +414,7 @@ m/n/o≡m/[n*o] m n o = begin-equality
/-*-interchange {o} {p} {m@.(q * o)} {n@.(r * p)} (divides-refl q) (divides-refl r)
= begin-equality
(m * n) / (o * p) ≡⟨⟩
q * o * (r * p) / (o * p) ≡⟨ /-congˡ ([m*n]*[o*p]≡[m*o]*[n*p] q o r p) ⟩
q * o * (r * p) / (o * p) ≡⟨ /-congˡ {o = o * p} ([m*n]*[o*p]≡[m*o]*[n*p] q o r p) ⟩
q * r * (o * p) / (o * p) ≡⟨ m*n/n≡m (q * r) (o * p) ⟩
q * r ≡⟨ cong₂ _*_ (m*n/n≡m q o) (m*n/n≡m r p) ⟨
(q * o / o) * (r * p / p) ≡⟨⟩
Expand All @@ -431,7 +431,7 @@ m*n/m!≡n/[m∸1]! m′@(suc m) n = m*n/m*o≡n/o m′ n (m !)
m%[n*o]/o≡m/o%n : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} →
{{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n
m%[n*o]/o≡m/o%n m n o = begin-equality
m % (n * o) / o ≡⟨ /-congˡ (m%n≡m∸m/n*n m (n * o)) ⟩
m % (n * o) / o ≡⟨ /-congˡ {o = o} (m%n≡m∸m/n*n m (n * o)) ⟩
(m ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨
(m ∸ (m / (n * o) * n * o)) / o ≡⟨ [m∸n*o]/o≡m/o∸n m (m / (n * o) * n) o ⟩
m / o ∸ m / (n * o) * n ≡⟨ cong (λ # → m / o ∸ # * n) (/-congʳ (*-comm n o)) ⟩
Expand Down