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11 changes: 11 additions & 0 deletions CHANGELOG.md
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Original file line number Diff line number Diff line change
Expand Up @@ -450,6 +450,17 @@ Additions to existing modules
p*q≡0⇒p≡0∨q≡0 : p * q ≡ 0ℚ → p ≡ 0ℚ ⊎ q ≡ 0ℚ
p*q≢0⇒p≢0 : p * q ≢ 0ℚ → p ≢ 0ℚ
p*q≢0⇒q≢0 : p * q ≢ 0ℚ → q ≢ 0ℚ
↥[i/1]≡i : (i : ℤ) → ↥ (i / 1) ≡ i
↧ₙ[i/1]≡1 : (i : ℤ) → ↧ₙ (i / 1) ≡ 1
n/n≡1 : ∀ (n : ℕ) .{{_ : ℕ.NonZero n}} → + n / n ≡ 1ℚ
-i/n≡-[i/n] : ∀ (i : ℤ) (n : ℕ) .{{_ : ℕ.NonZero n}} →
ℤ.- i / n ≡ - (i / n)
*-cancelˡ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (p ℕ.* r)}} →
(+ p ℤ.* q) / (p ℕ.* r) ≡ q / r
*-cancelʳ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (r ℕ.* p)}} →
(q ℤ.* + p) / (r ℕ.* p) ≡ q / r
i/n+j/n≡[i+j]/n : ∀ (i j : ℤ) (n : ℕ) .{{_ : ℕ.NonZero n }} →
i / n + j / n ≡ (i ℤ.+ j) / n
```

* In `Data.Rational.Show`:
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139 changes: 138 additions & 1 deletion src/Data/Rational/Properties.agda
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Expand Up @@ -31,7 +31,7 @@ open import Data.Maybe.Base using (Maybe; just; nothing)
open import Data.Integer.Base as ℤ using (ℤ; +_; -[1+_]; +[1+_]; +0; 0ℤ; 1ℤ; _◃_)
open import Data.Integer.Coprimality using (coprime-divisor)
import Data.Integer.Properties as ℤ
open import Data.Integer.GCD using (gcd; gcd[i,j]≡0⇒i≡0; gcd[i,j]≡0⇒j≡0)
open import Data.Integer.GCD using (gcd; gcd[i,j]≡0⇒i≡0; gcd[i,j]≡0⇒j≡0; gcd-zeroʳ)
open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
import Data.Nat.Properties as ℕ
Expand Down Expand Up @@ -394,12 +394,51 @@ normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡
↥p/↧p≡p (mkℚ (+ n) d-1 prf) = normalize-coprime prf
↥p/↧p≡p (mkℚ -[1+ n ] d-1 prf) = cong (-_) (normalize-coprime prf)

↥[i/1]≡i : (i : ℤ) → ↥ (i / 1) ≡ i
↥[i/1]≡i i = begin
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This feels like an awfully complicated proof for something so simple. I wonder if a "first principles" proof, by matching on i, would be simpler?

(↥ (i / 1)) ≡⟨ sym $ ℤ.*-identityʳ (↥ (i / 1)) ⟩
(↥ (i / 1)) ℤ.* 1ℤ ≡⟨ cong (↥ (i / 1) ℤ.*_) $ sym $ gcd-zeroʳ i ⟩
(↥ (i / 1)) ℤ.* gcd i 1ℤ ≡⟨ ↥-/ i 1 ⟩
i ∎
where open ≡-Reasoning

↧ₙ[i/1]≡1 : (i : ℤ) → ↧ₙ (i / 1) ≡ 1
↧ₙ[i/1]≡1 i = ℤ.+-injective $ begin
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Possibly the same here.

↧ (i / 1) ≡⟨ sym $ ℤ.*-identityʳ (↧ (i / 1)) ⟩
↧ (i / 1) ℤ.* 1ℤ ≡⟨ cong (↧ (i / 1) ℤ.*_) $ sym $ gcd-zeroʳ i ⟩
↧ (i / 1) ℤ.* gcd i 1ℤ ≡⟨ ↧-/ i 1 ⟩
1ℤ ∎
where open ≡-Reasoning

0/n≡0 : ∀ n .{{_ : ℕ.NonZero n}} → 0ℤ / n ≡ 0ℚ
0/n≡0 n@(suc n-1) {{n≢0}} = mkℚ+-cong {{n/n≢0}} {c₂ = 0-cop-1} (ℕ.0/n≡0 (ℕ.gcd 0 n)) (ℕ.n/n≡1 n)
where
0-cop-1 = C.sym (C.1-coprimeTo 0)
n/n≢0 = ℕ.>-nonZero (subst (ℕ._> 0) (sym (ℕ.n/n≡1 n)) (ℕ.z<s))

n/n≡1 : ∀ (n : ℕ) .{{_ : ℕ.NonZero n}} → + n / n ≡ 1ℚ
n/n≡1 n {{nz}} = mkℚ+-cong n/gcd[n,n]≡1 n/gcd[n,n]≡1
where
instance g≢0 = ℕ.≢-nonZero (ℕ.gcd[m,n]≢0 n n (inj₂ (ℕ.≢-nonZero⁻¹ n)))
n/g≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 n n {{gcd≢0 = g≢0}})

gcd[n,n]≡n : ∀ n → ℕ.gcd n n ≡ n
gcd[n,n]≡n n rewrite sym (ℕ.*-identityʳ n)
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We tend not to use rewrite in stdlib. Also this lemma should go in to Data.Nat.GCD.

= sym (ℕ.c*gcd[m,n]≡gcd[cm,cn] n 1 1)

n/gcd[n,n]≡1
= trans (ℕ./-congʳ {ℕ.gcd n n} (gcd[n,n]≡n n)) (ℕ.n/n≡1 n {{nz}})

-i/n≡-[i/n] : ∀ (i : ℤ) (n : ℕ) .{{_ : ℕ.NonZero n}} →
ℤ.- i / n ≡ - (i / n)
-i/n≡-[i/n] +0 n = trans (0/n≡0 n) (cong -_ (sym (0/n≡0 n)))
-i/n≡-[i/n] +[1+ m ] n = refl
-i/n≡-[i/n] -[1+ m ] n
with +[1+ m ] / n
... | mkℚ -[1+ a ] d prf = refl
... | mkℚ +0 d prf = refl
... | mkℚ +[1+ a ] d prf = refl

/-cong : ∀ {p₁ q₁ p₂ q₂} .{{_ : ℕ.NonZero q₁}} .{{_ : ℕ.NonZero q₂}} →
p₁ ≡ p₂ → q₁ ≡ q₂ → p₁ / q₁ ≡ p₂ / q₂
/-cong {+ n} refl = normalize-cong {n} refl
Expand Down Expand Up @@ -1361,6 +1400,40 @@ module _ where
heytingField : HeytingField 0ℓ 0ℓ 0ℓ
heytingField = record { isHeytingField = isHeytingField }

------------------------------------------------------------------------
-- Properties of _*_ and _/_

*-cancelˡ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (p ℕ.* r)}} →
(+ p ℤ.* q) / (p ℕ.* r) ≡ q / r
*-cancelˡ-/ p {q} {r} = proof q
where
*-cancelˡ-/-helper : ∀ qₙ → normalize (p ℕ.* qₙ) (p ℕ.* r) ≡ + qₙ / r
*-cancelˡ-/-helper qₙ = mkℚ+-cong (lemma qₙ) (lemma r)
where
instance
p≢0 = ℕ.m*n≢0⇒m≢0 p
g≢0 = ℕ.≢-nonZero $ ℕ.gcd[m,n]≢0 (p ℕ.* qₙ) (p ℕ.* r) $ inj₂
$ ℕ.≢-nonZero⁻¹ $ p ℕ.* r
n/g≢0 = ℕ.≢-nonZero $ ℕ.n/gcd[m,n]≢0 (p ℕ.* qₙ) (p ℕ.* r) {{gcd≢0 = g≢0}}
g≢0' = ℕ.≢-nonZero $ ℕ.gcd[m,n]≢0 qₙ r $ inj₂ $ ℕ.≢-nonZero⁻¹ r
n/g≢0' = ℕ.≢-nonZero $ ℕ.n/gcd[m,n]≢0 qₙ r {{gcd≢0 = g≢0'}}
p*g≢0 = ℕ.m*n≢0 p (ℕ.gcd qₙ r)

lemma : ∀ n → (p ℕ.* n) ℕ./ ℕ.gcd (p ℕ.* qₙ) (p ℕ.* r) ≡ n ℕ./ ℕ.gcd qₙ r
lemma n = trans (ℕ./-congʳ $ sym $ ℕ.c*gcd[m,n]≡gcd[cm,cn] p qₙ r)
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an equational proof would be more readable here

(ℕ.m*n/m*o≡n/o p n $ ℕ.gcd qₙ r)

proof : ∀ q → (+ p ℤ.* q) / (p ℕ.* r) ≡ q / r
proof (+ qₙ) rewrite sym (ℤ.pos-* p qₙ) = *-cancelˡ-/-helper qₙ
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Please make these explicit instead of using rewrite

proof -[1+ qₙ ] rewrite sym (ℤ.neg-distribʳ-* (+ p) +[1+ qₙ ])
| sym (ℤ.pos-* p (suc qₙ))
= trans (-i/n≡-[i/n] (+ (p ℕ.* suc qₙ)) (p ℕ.* r))
(cong (-_) $ *-cancelˡ-/-helper $ suc qₙ)

*-cancelʳ-/ : ∀ p {q r} .{{_ : ℕ.NonZero r}} .{{_ : ℕ.NonZero (r ℕ.* p)}} →
(q ℤ.* + p) / (r ℕ.* p) ≡ q / r
*-cancelʳ-/ p {q} {r}
rewrite ℕ.*-comm r p | ℤ.*-comm q (ℤ.+ p) = *-cancelˡ-/ p

------------------------------------------------------------------------
-- Properties of _*_ and _≤_
Expand Down Expand Up @@ -1844,6 +1917,70 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl
∣∣p∣∣≡∣p∣ : ∀ p → ∣ ∣ p ∣ ∣ ≡ ∣ p ∣
∣∣p∣∣≡∣p∣ p = 0≤p⇒∣p∣≡p (0≤∣p∣ p)

------------------------------------------------------------------------
-- Other properties of _+_
------------------------------------------------------------------------

i/n+j/n≡[i+j]/n : ∀ (i j : ℤ) (n : ℕ) .{{_ : ℕ.NonZero n }} →
i / n + j / n ≡ (i ℤ.+ j) / n
i/n+j/n≡[i+j]/n i j n = proof
where
pᵢ = i / n
qⱼ = j / n
gcd[i,n]ₙ = ℕ.gcd ℤ.∣ i ∣ n
gcd[i,n] = + gcd[i,n]ₙ
gcd[j,n]ₙ = ℕ.gcd ℤ.∣ j ∣ n
gcd[j,n] = + gcd[j,n]ₙ

instance
_ = ℕ.≢-nonZero $ ℕ.gcd[m,n]≢0 ℤ.∣ i ∣ n $ inj₂ $ ℕ.≢-nonZero⁻¹ n
_ = ℕ.≢-nonZero $ ℕ.gcd[m,n]≢0 ℤ.∣ j ∣ n $ inj₂ $ ℕ.≢-nonZero⁻¹ n
_ = ℕ.m*n≢0 (↧ₙ pᵢ ℕ.* ↧ₙ qⱼ) gcd[j,n]ₙ
_ = ℕ.m*n≢0 (↧ₙ pᵢ ℕ.* ↧ₙ qⱼ ℕ.* gcd[j,n]ₙ) gcd[i,n]ₙ
_ = ℕ.m*n≢0 n n

+-def : pᵢ + qⱼ ≡ (↥ pᵢ ℤ.* ↧ qⱼ ℤ.+ ↥ qⱼ ℤ.* ↧ pᵢ) / (↧ₙ pᵢ ℕ.* ↧ₙ qⱼ)
+-def with record{} ← pᵢ with record{} ← qⱼ = refl

↥≡ : (↥ pᵢ ℤ.* ↧ qⱼ ℤ.+ ↥ qⱼ ℤ.* ↧ pᵢ) ℤ.* gcd[j,n] ℤ.* gcd[i,n]
≡ (i ℤ.+ j) ℤ.* + n
↥≡ rewrite ℤ.*-distribʳ-+ gcd[j,n] (↥ pᵢ ℤ.* ↧ qⱼ) (↥ qⱼ ℤ.* ↧ pᵢ)
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most definitely needs to be done equationally!

| ℤ.*-distribʳ-+ gcd[i,n] (↥ pᵢ ℤ.* ↧ qⱼ ℤ.* gcd[j,n])
(↥ qⱼ ℤ.* ↧ pᵢ ℤ.* gcd[j,n])
| ℤ.*-assoc (↥ pᵢ) (↧ qⱼ) gcd[j,n]
| cong (↥ pᵢ ℤ.*_) (↧-/ j n)
| ℤ.*-assoc (↥ qⱼ) (↧ pᵢ) gcd[j,n]
| ℤ.*-comm (↧ pᵢ) gcd[j,n]
| sym (ℤ.*-assoc (↥ qⱼ) gcd[j,n] (↧ pᵢ))
| cong (ℤ._* ↧ pᵢ) (↥-/ j n)
| ℤ.*-assoc j (↧ pᵢ) gcd[i,n]
| cong (j ℤ.*_) (↧-/ i n)
| ℤ.*-comm (↥ pᵢ) (+ n)
| ℤ.*-assoc (+ n) (↥ pᵢ) gcd[i,n]
| cong (+ n ℤ.*_) (↥-/ i n)
| ℤ.*-comm (+ n) i
| sym (ℤ.*-distribʳ-+ (+ n) i j)
= refl

↧≡ : ↧ₙ pᵢ ℕ.* ↧ₙ qⱼ ℕ.* gcd[j,n]ₙ ℕ.* gcd[i,n]ₙ ≡ n ℕ.* n
↧≡ rewrite ℕ.*-assoc (↧ₙ pᵢ) (↧ₙ qⱼ) gcd[j,n]ₙ
| sym (ℤ.abs-* (↧ qⱼ) (gcd j (+ n)))
| cong ℤ.∣_∣ (↧-/ j n)
| ℕ.*-comm (↧ₙ pᵢ) n
| ℕ.*-assoc n (↧ₙ pᵢ) gcd[i,n]ₙ
| sym (ℤ.abs-* (↧ pᵢ) (gcd i (+ n)))
| cong ℤ.∣_∣ (↧-/ i n)
= refl

proof : i / n + j / n ≡ (i ℤ.+ j) / n
proof rewrite +-def
| sym (*-cancelʳ-/ gcd[j,n]ₙ
{↥ pᵢ ℤ.* ↧ qⱼ ℤ.+ ↥ qⱼ ℤ.* ↧ pᵢ} { ↧ₙ pᵢ ℕ.* ↧ₙ qⱼ })
| sym (*-cancelʳ-/ gcd[i,n]ₙ
{ (↥ pᵢ ℤ.* ↧ qⱼ ℤ.+ ↥ qⱼ ℤ.* ↧ pᵢ) ℤ.* gcd[j,n] }
{ ↧ₙ pᵢ ℕ.* ↧ₙ qⱼ ℕ.* gcd[j,n]ₙ })
| sym (*-cancelʳ-/ n {i ℤ.+ j} {n})
= /-cong ↥≡ ↧≡

------------------------------------------------------------------------
-- DEPRECATED NAMES
Expand Down
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