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fixes #1437, supersedes #1836 #1916

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fixes #1437, supersedes #1836 #1916

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dc0251d
fixes #1437
jamesmckinna Jan 25, 2023
a4dff80
`CHANGELOG`
jamesmckinna Jan 26, 2023
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18 changes: 17 additions & 1 deletion CHANGELOG.md
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Original file line number Diff line number Diff line change
Expand Up @@ -344,7 +344,7 @@ Non-backwards compatible changes
### Change in reduction behaviour of rationals

* Currently arithmetic expressions involving rationals (both normalised and
unnormalised) undergo disastorous exponential normalisation. For example,
unnormalised) undergo disastrous exponential normalisation. For example,
`p + q` would often be normalised by Agda to
`(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`. While the normalised form
of `p + q + r + s + t + u + v` would be ~700 lines long. This behaviour
Expand Down Expand Up @@ -677,6 +677,22 @@ Non-backwards compatible changes
exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
directly to use them.

* The names of the (in)equational reasoning combinators defined in the internal
modules `Data.Rational(.Unnormalised).Properties.≤-Reasoning` have been renamed
(issue #1437) to conform with the defined setoid equality `_≃_` on `Rational`s:
```agda
step-≈ ↦ step-≃
step-≃˘ ↦ step-≃˘
```
with corresponding associated syntax:
```agda
_≈⟨_⟩_ ↦ _≃⟨_⟩_
_≈˘⟨_⟩_ ↦ _≃˘⟨_⟩_
```
NB. It is not possible to rename or deprecate `syntax` declarations, so Agda will
only issue a "Could not parse the application `begin ...` when scope checking"
warning if the old combinators are used.

* The types of the proofs `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` in
`Data.Rational(.Unnormalised).Properties` have been switched, as the previous
naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`. For example
Expand Down
87 changes: 48 additions & 39 deletions src/Data/Rational/Properties.agda
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Expand Up @@ -429,9 +429,9 @@ toℚᵘ-cong refl = *≡* refl

fromℚᵘ-cong : fromℚᵘ Preserves _≃ᵘ_ ⟶ _≡_
fromℚᵘ-cong {p} {q} p≃q = toℚᵘ-injective (begin-equality
toℚᵘ (fromℚᵘ p) ≈⟨ toℚᵘ-fromℚᵘ p ⟩
p ≈⟨ p≃q ⟩
q ≈˘⟨ toℚᵘ-fromℚᵘ q ⟩
toℚᵘ (fromℚᵘ p) ≃⟨ toℚᵘ-fromℚᵘ p ⟩
p ≃⟨ p≃q ⟩
q ≃˘⟨ toℚᵘ-fromℚᵘ q ⟩
toℚᵘ (fromℚᵘ q) ∎)
where open ℚᵘ.≤-Reasoning

Expand Down Expand Up @@ -705,15 +705,24 @@ _>?_ = flip _<?_
------------------------------------------------------------------------

module ≤-Reasoning where
open import Relation.Binary.Reasoning.Base.Triple
import Relation.Binary.Reasoning.Base.Triple
≤-isPreorder
<-trans
(resp₂ _<_)
<⇒≤
<-≤-trans
≤-<-trans
public
hiding (step-≈; step-≈˘)
as Triple
open Triple public hiding (step-≈; step-≈˘)

infixr 2 step-≃ step-≃˘

step-≃ = Triple.step-≈
step-≃˘ = Triple.step-≈˘

syntax step-≃ x y∼z x≃y = x ≃⟨ x≃y ⟩ y∼z
syntax step-≃˘ x y∼z y≃x = x ≃˘⟨ y≃x ⟩ y∼z


------------------------------------------------------------------------
-- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_
Expand Down Expand Up @@ -950,9 +959,9 @@ neg-distrib-+ = +-Monomorphism.⁻¹-distrib-∙ ℚᵘ.+-0-isAbelianGroup (ℚ

+-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
+-mono-≤ {p} {q} {r} {s} p≤q r≤s = toℚᵘ-cancel-≤ (begin
toℚᵘ(p + r) ≈⟨ toℚᵘ-homo-+ p r ⟩
toℚᵘ(p + r) ≃⟨ toℚᵘ-homo-+ p r ⟩
toℚᵘ(p) ℚᵘ.+ toℚᵘ(r) ≤⟨ ℚᵘ.+-mono-≤ (toℚᵘ-mono-≤ p≤q) (toℚᵘ-mono-≤ r≤s) ⟩
toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≈⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩
toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩
toℚᵘ(q + s) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -967,9 +976,9 @@ neg-distrib-+ = +-Monomorphism.⁻¹-distrib-∙ ℚᵘ.+-0-isAbelianGroup (ℚ

+-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_
+-mono-<-≤ {p} {q} {r} {s} p<q r≤s = toℚᵘ-cancel-< (begin-strict
toℚᵘ(p + r) ≈⟨ toℚᵘ-homo-+ p r ⟩
toℚᵘ(p + r) ≃⟨ toℚᵘ-homo-+ p r ⟩
toℚᵘ(p) ℚᵘ.+ toℚᵘ(r) <⟨ ℚᵘ.+-mono-<-≤ (toℚᵘ-mono-< p<q) (toℚᵘ-mono-≤ r≤s) ⟩
toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≈⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩
toℚᵘ(q) ℚᵘ.+ toℚᵘ(s) ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-+ q s) ⟩
toℚᵘ(q + s) ∎)
where open ℚᵘ.≤-Reasoning

Expand Down Expand Up @@ -1127,9 +1136,9 @@ private

*-inverseˡ : ∀ p .{{_ : NonZero p}} → (1/ p) * p ≡ 1ℚ
*-inverseˡ p = toℚᵘ-injective (begin-equality
toℚᵘ (1/ p * p) ≈⟨ toℚᵘ-homo-* (1/ p) p ⟩
toℚᵘ (1/ p) ℚᵘ.* toℚᵘ p ≈⟨ ℚᵘ.*-congʳ (toℚᵘ-homo-1/ p) ⟩
ℚᵘ.1/ (toℚᵘ p) ℚᵘ.* toℚᵘ p ≈⟨ ℚᵘ.*-inverseˡ (toℚᵘ p) ⟩
toℚᵘ (1/ p * p) ≃⟨ toℚᵘ-homo-* (1/ p) p ⟩
toℚᵘ (1/ p) ℚᵘ.* toℚᵘ p ≃⟨ ℚᵘ.*-congʳ (toℚᵘ-homo-1/ p) ⟩
ℚᵘ.1/ (toℚᵘ p) ℚᵘ.* toℚᵘ p ≃⟨ ℚᵘ.*-inverseˡ (toℚᵘ p) ⟩
ℚᵘ.1ℚᵘ ∎)
where open ℚᵘ.≤-Reasoning

Expand Down Expand Up @@ -1201,9 +1210,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-cancelʳ-≤-pos : ∀ r .{{_ : Positive r}} → p * r ≤ q * r → p ≤ q
*-cancelʳ-≤-pos {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-pos (toℚᵘ r) (begin
toℚᵘ p ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ≤⟨ toℚᵘ-mono-≤ pr≤qr ⟩
toℚᵘ (q * r) ≈⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ≃⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ∎))
where open ℚᵘ.≤-Reasoning

Expand All @@ -1212,9 +1221,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-monoʳ-≤-nonNeg : ∀ r .{{_ : NonNegative r}} → (_* r) Preserves _≤_ ⟶ _≤_
*-monoʳ-≤-nonNeg r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin
toℚᵘ (p * r) ≈⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ≃⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≤⟨ ℚᵘ.*-monoˡ-≤-nonNeg (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -1223,9 +1232,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-monoʳ-≤-nonPos : ∀ r .{{_ : NonPositive r}} → (_* r) Preserves _≤_ ⟶ _≥_
*-monoʳ-≤-nonPos r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin
toℚᵘ (q * r) ≈⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ≃⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ≤⟨ ℚᵘ.*-monoˡ-≤-nonPos (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -1234,9 +1243,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-cancelʳ-≤-neg : ∀ r .{{_ : Negative r}} → p * r ≤ q * r → p ≥ q
*-cancelʳ-≤-neg {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-neg _ (begin
toℚᵘ p ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ≤⟨ toℚᵘ-mono-≤ pr≤qr ⟩
toℚᵘ (q * r) ≈⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ≃⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ∎))
where open ℚᵘ.≤-Reasoning

Expand All @@ -1248,9 +1257,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-monoˡ-<-pos : ∀ r .{{_ : Positive r}} → (_* r) Preserves _<_ ⟶ _<_
*-monoˡ-<-pos r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict
toℚᵘ (p * r) ≈⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ≃⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r <⟨ ℚᵘ.*-monoˡ-<-pos (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -1259,9 +1268,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-cancelˡ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → ∀ {p q} → r * p < r * q → p < q
*-cancelˡ-<-nonNeg r {p} {q} rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonNeg (toℚᵘ r) (begin-strict
toℚᵘ r ℚᵘ.* toℚᵘ p ≈˘⟨ toℚᵘ-homo-* r p ⟩
toℚᵘ r ℚᵘ.* toℚᵘ p ≃˘⟨ toℚᵘ-homo-* r p ⟩
toℚᵘ (r * p) <⟨ toℚᵘ-mono-< rp<rq ⟩
toℚᵘ (r * q) ≈⟨ toℚᵘ-homo-* r q ⟩
toℚᵘ (r * q) ≃⟨ toℚᵘ-homo-* r q ⟩
toℚᵘ r ℚᵘ.* toℚᵘ q ∎))
where open ℚᵘ.≤-Reasoning

Expand All @@ -1270,9 +1279,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-monoˡ-<-neg : ∀ r .{{_ : Negative r}} → (_* r) Preserves _<_ ⟶ _>_
*-monoˡ-<-neg r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict
toℚᵘ (q * r) ≈⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ (q * r) ≃⟨ toℚᵘ-homo-* q r ⟩
toℚᵘ q ℚᵘ.* toℚᵘ r <⟨ ℚᵘ.*-monoˡ-<-neg (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≈˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ p ℚᵘ.* toℚᵘ r ≃˘⟨ toℚᵘ-homo-* p r ⟩
toℚᵘ (p * r) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -1281,9 +1290,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i

*-cancelˡ-<-nonPos : ∀ r .{{_ : NonPositive r}} → r * p < r * q → p > q
*-cancelˡ-<-nonPos {p} {q} r rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonPos (toℚᵘ r) (begin-strict
toℚᵘ r ℚᵘ.* toℚᵘ p ≈˘⟨ toℚᵘ-homo-* r p ⟩
toℚᵘ r ℚᵘ.* toℚᵘ p ≃˘⟨ toℚᵘ-homo-* r p ⟩
toℚᵘ (r * p) <⟨ toℚᵘ-mono-< rp<rq ⟩
toℚᵘ (r * q) ≈⟨ toℚᵘ-homo-* r q ⟩
toℚᵘ (r * q) ≃⟨ toℚᵘ-homo-* r q ⟩
toℚᵘ r ℚᵘ.* toℚᵘ q ∎))
where open ℚᵘ.≤-Reasoning

Expand Down Expand Up @@ -1578,7 +1587,7 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl

∣p∣≡p⇒0≤p : ∀ {p} → ∣ p ∣ ≡ p → 0ℚ ≤ p
∣p∣≡p⇒0≤p {p} ∣p∣≡p = toℚᵘ-cancel-≤ (ℚᵘ.∣p∣≃p⇒0≤p (begin-equality
ℚᵘ.∣ toℚᵘ p ∣ ≈⟨ ℚᵘ.≃-sym (toℚᵘ-homo-∣-∣ p) ⟩
ℚᵘ.∣ toℚᵘ p ∣ ≃⟨ ℚᵘ.≃-sym (toℚᵘ-homo-∣-∣ p) ⟩
toℚᵘ ∣ p ∣ ≡⟨ cong toℚᵘ ∣p∣≡p ⟩
toℚᵘ p ∎))
where open ℚᵘ.≤-Reasoning
Expand All @@ -1589,11 +1598,11 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl

∣p+q∣≤∣p∣+∣q∣ : ∀ p q → ∣ p + q ∣ ≤ ∣ p ∣ + ∣ q ∣
∣p+q∣≤∣p∣+∣q∣ p q = toℚᵘ-cancel-≤ (begin
toℚᵘ ∣ p + q ∣ ≈⟨ toℚᵘ-homo-∣-∣ (p + q) ⟩
ℚᵘ.∣ toℚᵘ (p + q) ∣ ≈⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-+ p q) ⟩
toℚᵘ ∣ p + q ∣ ≃⟨ toℚᵘ-homo-∣-∣ (p + q) ⟩
ℚᵘ.∣ toℚᵘ (p + q) ∣ ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-+ p q) ⟩
ℚᵘ.∣ toℚᵘ p ℚᵘ.+ toℚᵘ q ∣ ≤⟨ ℚᵘ.∣p+q∣≤∣p∣+∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩
ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.+ ℚᵘ.∣ toℚᵘ q ∣ ≈˘⟨ ℚᵘ.+-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
toℚᵘ ∣ p ∣ ℚᵘ.+ toℚᵘ ∣ q ∣ ≈˘⟨ toℚᵘ-homo-+ ∣ p ∣ ∣ q ∣ ⟩
ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.+ ℚᵘ.∣ toℚᵘ q ∣ ≃˘⟨ ℚᵘ.+-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
toℚᵘ ∣ p ∣ ℚᵘ.+ toℚᵘ ∣ q ∣ ≃˘⟨ toℚᵘ-homo-+ ∣ p ∣ ∣ q ∣ ⟩
toℚᵘ (∣ p ∣ + ∣ q ∣) ∎)
where open ℚᵘ.≤-Reasoning

Expand All @@ -1606,11 +1615,11 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl

∣p*q∣≡∣p∣*∣q∣ : ∀ p q → ∣ p * q ∣ ≡ ∣ p ∣ * ∣ q ∣
∣p*q∣≡∣p∣*∣q∣ p q = toℚᵘ-injective (begin-equality
toℚᵘ ∣ p * q ∣ ≈⟨ toℚᵘ-homo-∣-∣ (p * q) ⟩
ℚᵘ.∣ toℚᵘ (p * q) ∣ ≈⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-* p q) ⟩
ℚᵘ.∣ toℚᵘ p ℚᵘ.* toℚᵘ q ∣ ≈⟨ ℚᵘ.∣p*q∣≃∣p∣*∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩
ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.* ℚᵘ.∣ toℚᵘ q ∣ ≈˘⟨ ℚᵘ.*-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
toℚᵘ ∣ p ∣ ℚᵘ.* toℚᵘ ∣ q ∣ ≈˘⟨ toℚᵘ-homo-* ∣ p ∣ ∣ q ∣ ⟩
toℚᵘ ∣ p * q ∣ ≃⟨ toℚᵘ-homo-∣-∣ (p * q) ⟩
ℚᵘ.∣ toℚᵘ (p * q) ∣ ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-* p q) ⟩
ℚᵘ.∣ toℚᵘ p ℚᵘ.* toℚᵘ q ∣ ≃⟨ ℚᵘ.∣p*q∣≃∣p∣*∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩
ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.* ℚᵘ.∣ toℚᵘ q ∣ ≃˘⟨ ℚᵘ.*-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
toℚᵘ ∣ p ∣ ℚᵘ.* toℚᵘ ∣ q ∣ ≃˘⟨ toℚᵘ-homo-* ∣ p ∣ ∣ q ∣ ⟩
toℚᵘ (∣ p ∣ * ∣ q ∣) ∎)
where open ℚᵘ.≤-Reasoning

Expand Down
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