```------------------------------------------------------------------------
-- The Agda standard library
--
-- Rational numbers in non-reduced form.
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Rational.Unnormalised.Base where

open import Algebra.Bundles.Raw
open import Data.Bool.Base using (Bool; true; false; if_then_else_)
open import Data.Integer.Base as ℤ
using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
hiding (module ℤ)
open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
open import Level using (0ℓ)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)
open import Relation.Unary using (Pred)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; _≢_; refl)

------------------------------------------------------------------------
-- Definition

-- Here we define rationals that are not necessarily in reduced form.
-- Consequently there are multiple ways of representing a given rational
-- number, and the performance of the arithmetic operations may suffer
-- due to blowup of the numerator and denominator.

-- Nonetheless they are much easier to reason about. In general proofs
-- are first proved for these unnormalised rationals and then translated
-- into the normalised rationals.

record ℚᵘ : Set where
-- We add "no-eta-equality; pattern" to the record to stop Agda
-- automatically unfolding rationals when arithmetic operations are
-- applied to them (see definition of operators below and Issue #1753
-- for details).
no-eta-equality; pattern

constructor mkℚᵘ
field
numerator     : ℤ
denominator-1 : ℕ

denominatorℕ : ℕ
denominatorℕ = suc denominator-1

denominator : ℤ
denominator = + denominatorℕ

open ℚᵘ public using ()
renaming
( numerator    to ↥_
; denominator  to ↧_
; denominatorℕ to ↧ₙ_
)

------------------------------------------------------------------------
-- Equality of rational numbers (does not coincide with _≡_)

infix 4 _≃_ _≠_

data _≃_ : Rel ℚᵘ 0ℓ where
*≡* : ∀ {p q} → (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p) → p ≃ q

_≄_ : Rel ℚᵘ 0ℓ
p ≄ q = ¬ (p ≃ q)

------------------------------------------------------------------------
-- Ordering of rationals

infix 4 _≤_ _<_ _≥_ _>_ _≰_ _≱_ _≮_ _≯_

data _≤_ : Rel ℚᵘ 0ℓ where
*≤* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.≤ (↥ q ℤ.* ↧ p) → p ≤ q

data _<_ : Rel ℚᵘ 0ℓ where
*<* : ∀ {p q} → (↥ p ℤ.* ↧ q) ℤ.< (↥ q ℤ.* ↧ p) → p < q

_≥_ : Rel ℚᵘ 0ℓ
x ≥ y = y ≤ x

_>_ : Rel ℚᵘ 0ℓ
x > y = y < x

_≰_ : Rel ℚᵘ 0ℓ
x ≰ y = ¬ (x ≤ y)

_≱_ : Rel ℚᵘ 0ℓ
x ≱ y = ¬ (x ≥ y)

_≮_ : Rel ℚᵘ 0ℓ
x ≮ y = ¬ (x < y)

_≯_ : Rel ℚᵘ 0ℓ
x ≯ y = ¬ (x > y)

------------------------------------------------------------------------
-- Boolean ordering

infix 4 _≤ᵇ_

_≤ᵇ_ : ℚᵘ → ℚᵘ → Bool
p ≤ᵇ q = (↥ p ℤ.* ↧ q) ℤ.≤ᵇ (↥ q ℤ.* ↧ p)

------------------------------------------------------------------------
-- Constructing rationals

-- An alternative constructor for ℚᵘ. See the constants section below
-- for examples of how to use this operator.

infixl 7 _/_

_/_ : (n : ℤ) (d : ℕ) .{{_ : ℕ.NonZero d}} → ℚᵘ
n / suc d = mkℚᵘ n d

------------------------------------------------------------------------
-- Some constants

0ℚᵘ : ℚᵘ
0ℚᵘ = + 0 / 1

1ℚᵘ : ℚᵘ
1ℚᵘ = + 1 / 1

½ : ℚᵘ
½ = + 1 / 2

-½ : ℚᵘ
-½ = ℤ.- (+ 1) / 2

------------------------------------------------------------------------
-- Simple predicates

NonZero : Pred ℚᵘ 0ℓ
NonZero p = ℤ.NonZero (↥ p)

Positive : Pred ℚᵘ 0ℓ
Positive p = ℤ.Positive (↥ p)

Negative : Pred ℚᵘ 0ℓ
Negative p = ℤ.Negative (↥ p)

NonPositive : Pred ℚᵘ 0ℓ
NonPositive p = ℤ.NonPositive (↥ p)

NonNegative : Pred ℚᵘ 0ℓ
NonNegative p = ℤ.NonNegative (↥ p)

-- Instances

open ℤ public
using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)

-- Constructors and destructors

-- Note: these could be proved more elegantly using the constructors
-- from ℤ but it requires importing `Data.Integer.Properties` which
-- we would like to avoid doing.

≢-nonZero : ∀ {p} → p ≄ 0ℚᵘ → NonZero p
≢-nonZero {mkℚᵘ -[1+ _ ] _      } _   = _
≢-nonZero {mkℚᵘ +[1+ _ ] _      } _   = _
≢-nonZero {mkℚᵘ +0       zero   } p≢0 = contradiction (*≡* refl) p≢0
≢-nonZero {mkℚᵘ +0       (suc d)} p≢0 = contradiction (*≡* refl) p≢0

>-nonZero : ∀ {p} → p > 0ℚᵘ → NonZero p
>-nonZero {mkℚᵘ +0       _} (*<* (+<+ ()))
>-nonZero {mkℚᵘ +[1+ n ] _} (*<* _) = _

<-nonZero : ∀ {p} → p < 0ℚᵘ → NonZero p
<-nonZero {mkℚᵘ +[1+ n ] _} (*<* _) = _
<-nonZero {mkℚᵘ +0 _}       (*<* (+<+ ()))
<-nonZero {mkℚᵘ -[1+ n ] _} (*<* _) = _

positive : ∀ {p} → p > 0ℚᵘ → Positive p
positive {mkℚᵘ +[1+ n ]   _} (*<* _) = _
positive {mkℚᵘ +0         _} (*<* (+<+ ()))
positive {mkℚᵘ (-[1+_] n) _} (*<* ())

negative : ∀ {p} → p < 0ℚᵘ → Negative p
negative {mkℚᵘ +[1+ n ]   _} (*<* (+<+ ()))
negative {mkℚᵘ +0         _} (*<* (+<+ ()))
negative {mkℚᵘ (-[1+_] n) _} (*<* _       ) = _

nonPositive : ∀ {p} → p ≤ 0ℚᵘ → NonPositive p
nonPositive {mkℚᵘ +[1+ n ] _} (*≤* (+≤+ ()))
nonPositive {mkℚᵘ +0       _} (*≤* _) = _
nonPositive {mkℚᵘ -[1+ n ] _} (*≤* _) = _

nonNegative : ∀ {p} → p ≥ 0ℚᵘ → NonNegative p
nonNegative {mkℚᵘ +0       _} (*≤* _) = _
nonNegative {mkℚᵘ +[1+ n ] _} (*≤* _) = _

------------------------------------------------------------------------
-- Operations on rationals

-- Explanation for `@record{}` everywhere: combined with no-eta-equality
-- on the record definition of ℚᵘ above, these annotations prevent the
-- operations from automatically expanding unless their arguments are
-- explicitly pattern matched on.
--
-- For example prior to their addition, `p + q` would often be
-- normalised by Agda to `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`.
-- While in this small example this isn't a big problem, it leads to an
-- exponential blowup when you have large arithmetic expressions which
-- would often choke both type-checking and the display code. For
-- example, the normalised form of `p + q + r + s + t + u` would be
-- ~300 lines long.
--
-- This is fundementally a problem with Agda, so if over-eager
-- normalisation is ever fixed in Agda (e.g. with glued representation
-- of terms) these annotations can be removed.

infix  8 -_ 1/_
infixl 7 _*_ _÷_ _⊓_
infixl 6 _-_ _+_ _⊔_

-- negation

-_ : ℚᵘ → ℚᵘ
- mkℚᵘ n d = mkℚᵘ (ℤ.- n) d

_+_ : ℚᵘ → ℚᵘ → ℚᵘ
p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)

-- multiplication

_*_ : ℚᵘ → ℚᵘ → ℚᵘ
p@record{} * q@record{} = (↥ p ℤ.* ↥ q) / (↧ₙ p ℕ.* ↧ₙ q)

-- subtraction

_-_ : ℚᵘ → ℚᵘ → ℚᵘ
p - q = p + (- q)

-- reciprocal: requires a proof that the numerator is not zero

1/_ : (p : ℚᵘ) → .{{_ : NonZero p}} → ℚᵘ
1/ mkℚᵘ +[1+ n ] d = mkℚᵘ +[1+ d ] n
1/ mkℚᵘ -[1+ n ] d = mkℚᵘ -[1+ d ] n

-- division: requires a proof that the denominator is not zero

_÷_ : (p q : ℚᵘ) → .{{_ : NonZero q}} → ℚᵘ
p@record{} ÷ q@record{} = p * (1/ q)

-- max
_⊔_ : (p q : ℚᵘ) → ℚᵘ
p@record{} ⊔ q@record{} = if p ≤ᵇ q then q else p

-- min
_⊓_ : (p q : ℚᵘ) → ℚᵘ
p@record{} ⊓ q@record{} = if p ≤ᵇ q then p else q

-- absolute value
∣_∣ : ℚᵘ → ℚᵘ
∣ mkℚᵘ p q ∣ = mkℚᵘ (+ ℤ.∣ p ∣) q

------------------------------------------------------------------------
-- Rounding functions

-- Floor (round towards -∞)
floor : ℚᵘ → ℤ
floor p@record{} = ↥ p ℤ./ ↧ p

-- Ceiling (round towards +∞)
ceiling : ℚᵘ → ℤ
ceiling p@record{} = ℤ.- floor (- p)

-- Truncate  (round towards 0)
truncate : ℚᵘ → ℤ
truncate p with p ≤ᵇ 0ℚᵘ
... | true  = ceiling p
... | false = floor p

-- Round (to nearest integer)
round : ℚᵘ → ℤ
round p with p ≤ᵇ 0ℚᵘ
... | true  = ceiling (p - ½)
... | false = floor (p + ½)

-- Fractional part (remainder after floor)
fracPart : ℚᵘ → ℚᵘ
fracPart p@record{} = ∣ p - truncate p / 1 ∣

-- Extra notations  ⌊ ⌋ floor,  ⌈ ⌉ ceiling,  [ ] truncate
syntax floor    p = ⌊ p ⌋
syntax ceiling  p = ⌈ p ⌉
syntax truncate p = [ p ]

------------------------------------------------------------------------
-- Raw bundles for _+_

+-rawMagma : RawMagma 0ℓ 0ℓ
+-rawMagma = record
{ _≈_ = _≃_
; _∙_ = _+_
}

+-0-rawMonoid : RawMonoid 0ℓ 0ℓ
+-0-rawMonoid = record
{ _≈_ = _≃_
; _∙_ = _+_
; ε   = 0ℚᵘ
}

+-0-rawGroup : RawGroup 0ℓ 0ℓ
+-0-rawGroup = record
{ Carrier = ℚᵘ
; _≈_ = _≃_
; _∙_ = _+_
; ε = 0ℚᵘ
; _⁻¹ = -_
}

+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
+-*-rawNearSemiring = record
{ Carrier = ℚᵘ
; _≈_ = _≃_
; _+_ = _+_
; _*_ = _*_
; 0# = 0ℚᵘ
}

+-*-rawSemiring : RawSemiring 0ℓ 0ℓ
+-*-rawSemiring = record
{ Carrier = ℚᵘ
; _≈_ = _≃_
; _+_ = _+_
; _*_ = _*_
; 0# = 0ℚᵘ
; 1# = 1ℚᵘ
}

+-*-rawRing : RawRing 0ℓ 0ℓ
+-*-rawRing = record
{ Carrier = ℚᵘ
; _≈_ = _≃_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0ℚᵘ
; 1# = 1ℚᵘ
}

------------------------------------------------------------------------
-- Raw bundles for _*_

*-rawMagma : RawMagma 0ℓ 0ℓ
*-rawMagma = record
{ _≈_ = _≃_
; _∙_ = _*_
}

*-1-rawMonoid : RawMonoid 0ℓ 0ℓ
*-1-rawMonoid = record
{ _≈_ = _≃_
; _∙_ = _*_
; ε   = 1ℚᵘ
}

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 2.0

+-rawMonoid = +-0-rawMonoid
{-# WARNING_ON_USAGE +-rawMonoid
"Warning: +-rawMonoid was deprecated in v2.0