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finer-grained analysis of NonZero arguments #2230

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`breaking` changes to the analysis of `NonZero` arguments left over f…
jamesmckinna Dec 14, 2023
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knock-on viscosity
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2 changes: 1 addition & 1 deletion src/Data/Digit.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -56,7 +56,7 @@ toNatDigits base@(suc (suc _)) n = aux (<-wellFounded-fast n) []
aux {zero} _ xs = (0 ∷ xs)
aux {n@(suc _)} (acc wf) xs with does (0 <? n / base)
... | false = (n % base) ∷ xs -- Could this more simply be n ∷ xs here?
... | true = aux (wf (m/n<m n base sz<ss)) ((n % base) ∷ xs)
... | true = aux (wf (m/n<m n base)) ((n % base) ∷ xs)

------------------------------------------------------------------------
-- Converting between `ℕ` and expansions of `Digit base`
Expand Down
61 changes: 35 additions & 26 deletions src/Data/Nat/DivMod.agda
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Expand Up @@ -203,12 +203,13 @@ m/n≤m m n = *-cancelʳ-≤ (m / n) m n (begin
m ≤⟨ m≤m*n m n ⟩
m * n ∎)

m/n<m : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} →
1 < n → m / n < m
m/n<m m n 1<n = *-cancelʳ-< _ (m / n) m $ begin-strict
m/n<m : ∀ m n .{{_ : NonZero m}} .{{_ : NonTrivial n}} →
let instance _ = nonTrivial⇒nonZero n in m / n < m
m/n<m m n = *-cancelʳ-< _ (m / n) m $ begin-strict
m / n * n ≤⟨ m/n*n≤m m n ⟩
m <⟨ m<m*n m n 1<n ⟩
m <⟨ m<m*n m n (nonTrivial⇒n>1 n) ⟩
m * n ∎
where instance _ = nonTrivial⇒nonZero n

/-mono-≤ : .{{_ : NonZero o}} .{{_ : NonZero p}} →
m ≤ n → o ≥ p → m / o ≤ n / p
Expand Down Expand Up @@ -239,13 +240,13 @@ m≥n⇒m/n>0 {m@(suc _)} {n@(suc _)} m≥n = begin
m / n ∎

m/n≡0⇒m<n : ∀ {m n} .{{_ : NonZero n}} → m / n ≡ 0 → m < n
m/n≡0⇒m<n {m} {n@(suc _)} m/n≡0 with <-≤-connex m n
m/n≡0⇒m<n {m} {n} m/n≡0 with <-≤-connex m n
... | inj₁ m<n = m<n
... | inj₂ n≤m = contradiction m/n≡0 (≢-nonZero⁻¹ _)
where instance _ = >-nonZero (m≥n⇒m/n>0 n≤m)

m/n≢0⇒n≤m : ∀ {m n} .{{_ : NonZero n}} → m / n ≢ 0 → n ≤ m
m/n≢0⇒n≤m {m} {n@(suc _)} m/n≢0 with <-≤-connex m n
m/n≢0⇒n≤m {m} {n} m/n≢0 with <-≤-connex m n
... | inj₁ m<n = contradiction (m<n⇒m/n≡0 m<n) m/n≢0
... | inj₂ n≤m = n≤m

Expand Down Expand Up @@ -307,15 +308,15 @@ m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
o / pm * pm ≤⟨ m/n*n≤m o pm ⟩
o ∎

m*n/m*o≡n/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (m * o)}} →
m*n/m*o≡n/o : ∀ m n o .{{_ : NonZero m}} .{{_ : NonZero o}} →
let instance _ = m*n≢0 m o in
(m * n) / (m * o) ≡ n / o
m*n/m*o≡n/o m n o = helper (<-wellFounded n)
where
instance _ = m*n≢0 m o
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
... | 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 @@ -324,17 +325,17 @@ m*n/m*o≡n/o m n o = helper (<-wellFounded n)
n / o ∎
where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) n≥o

m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (o * n)}} →
m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} →
let instance _ = m*n≢0 o n in
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*n≢0}} (*-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 ∎
where instance
_ : NonZero n
_ = m*n≢0⇒n≢0 o
_ : NonZero (n * o)
o*n≢0 : NonZero (o * n)
o*n≢0 = m*n≢0 o n
_ = m*n≢0 n o

m<n*o⇒m/o<n : ∀ {m n o} .{{_ : NonZero o}} → m < n * o → m / o < n
Expand All @@ -357,8 +358,9 @@ m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict
m / o ∸ 1 ∸ n ≡⟨ ∸-+-assoc (m / o) 1 n ⟩
m / o ∸ suc n ∎

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 .{{_ : NonZero n}} .{{_ : NonZero o}} →
let instance _ = m*n≢0 n o in
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) ⟩
Expand All @@ -370,6 +372,8 @@ m/n/o≡m/[n*o] m n o = begin-equality
where
n*o = n * o
o*n = o * n
instance
_ = m*n≢0 n o

lem₁ : n ∣ m / n*o * n*o
lem₁ = divides (m / n*o * o) $ begin-equality
Expand Down Expand Up @@ -397,10 +401,11 @@ m/n/o≡m/[n*o] m n o = begin-equality
n / d + (m * n) / d ≡⟨ cong (n / d +_) (*-/-assoc m d∣n) ⟩
n / d + m * (n / d) ∎

/-*-interchange : .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} →
o ∣ m → p ∣ n → (m * n) / (o * p) ≡ (m / o) * (n / p)
/-*-interchange : .{{_ : NonZero o}} .{{_ : NonZero p}} →
let instance _ = m*n≢0 o p in o ∣ m → p ∣ n →
(m * n) / (o * p) ≡ (m / o) * (n / p)
/-*-interchange {o} {p} {m@.(q * o)} {n@.(r * p)} (divides-refl q) (divides-refl r)
= begin-equality
= let instance _ = m*n≢0 o p in 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 * r * (o * p) / (o * p) ≡⟨ m*n/n≡m (q * r) (o * p) ⟩
Expand All @@ -411,13 +416,11 @@ m/n/o≡m/[n*o] m n o = begin-equality
m*n/m!≡n/[m∸1]! : ∀ m n .{{_ : NonZero m}} →
let instance _ = m !≢0 ; instance _ = (pred m) !≢0 in
(m * n / m !) ≡ (n / (pred m) !)
m*n/m!≡n/[m∸1]! m′@(suc m) n = m*n/m*o≡n/o m′ n (m !)
where instance
_ = m !≢0
_ = m′ !≢0
m*n/m!≡n/[m∸1]! m′@(suc m) n = let instance _ = m !≢0 in 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
let instance _ = m*n≢0 n o in
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 ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨
Expand All @@ -426,9 +429,12 @@ m%[n*o]/o≡m/o%n m n o = begin-equality
m / o ∸ m / (o * n) * n ≡⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟨
m / o ∸ m / o / n * n ≡⟨ m%n≡m∸m/n*n (m / o) n ⟨
m / o % n ∎
where instance _ = m*n≢0 o n
where instance
_ = m*n≢0 n o
_ = m*n≢0 o n

m%n*o≡m*o%[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero (n * o)}} →
m%n*o≡m*o%[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} →
let instance _ = m*n≢0 n o in
m % n * o ≡ m * o % (n * o)
m%n*o≡m*o%[n*o] m n o = begin-equality
m % n * o ≡⟨ cong (_* o) (m%n≡m∸m/n*n m n) ⟩
Expand All @@ -437,8 +443,10 @@ m%n*o≡m*o%[n*o] m n o = begin-equality
m * o ∸ m * o / (n * o) * n * o ≡⟨ cong (m * o ∸_) (*-assoc (m * o / (n * o)) n o) ⟩
m * o ∸ m * o / (n * o) * (n * o) ≡⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟨
m * o % (n * o) ∎
where instance _ = m*n≢0 n o

[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p .{{_ : NonZero (p * n)}} → o < n →
[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p .{{_ : NonZero n}} .{{_ : NonZero p}} →
let instance _ = m*n≢0 p n in o < n →
(m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
[m*n+o]%[p*n]≡[m*n]%[p*n]+o m {n} {o} p@(suc p-1) o<n = begin-equality
(mn + o) % pn ≡⟨ %-distribˡ-+ mn o pn ⟩
Expand All @@ -448,6 +456,7 @@ m%n*o≡m*o%[n*o] m n o = begin-equality
where
mn = m * n
pn = p * n
instance _ = m*n≢0 p n

lem₁ : mn % pn ≤ p-1 * n
lem₁ = begin
Expand Down