------------------------------------------------------------------------
-- The Agda standard library
--
-- Rational numbers
------------------------------------------------------------------------

module Data.Rational where

import Algebra
import Data.Bool.Properties as Bool
open import Function
open import Data.Integer as  using (; ∣_∣; +_; -_)
open import Data.Integer.Divisibility as ℤDiv using (Coprime)
import Data.Integer.Properties as
open import Data.Nat.Divisibility as ℕDiv using (_∣_)
import Data.Nat.Coprimality as C
open import Data.Nat as  using (; zero; suc)
open import Data.Sum
import Level
open import Relation.Nullary.Decidable
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P
using (_≡_; refl; cong; cong₂)
open P.≡-Reasoning

------------------------------------------------------------------------
-- The definition

-- Rational numbers in reduced form. Note that there is exactly one
-- representative for every rational number. (This is the reason for
-- using "True" below. If Agda had proof irrelevance, then it would
-- suffice to use "isCoprime : Coprime numerator denominator".)

record  : Set where
field
numerator     :
denominator-1 :
isCoprime     : True (C.coprime?  numerator  (suc denominator-1))

denominator :
denominator = + suc denominator-1

coprime : Coprime numerator denominator
coprime = toWitness isCoprime

-- Constructs rational numbers. The arguments have to be in reduced
-- form.

infixl 7 _÷_

_÷_ : (numerator : ) (denominator : )
{coprime : True (C.coprime?  numerator  denominator)}
{≢0 : False (ℕ._≟_ denominator 0)}

(n ÷ zero) {≢0 = ()}
(n ÷ suc d) {c} =
record { numerator = n; denominator-1 = d; isCoprime = c }

private

-- Note that the implicit arguments do not need to be given for
-- concrete inputs:

0/1 :
0/1 = + 0 ÷ 1

:
= - + 1 ÷ 2

------------------------------------------------------------------------
-- Equality

-- Equality of rational numbers.

infix 4 _≃_

_≃_ : Rel  Level.zero
p  q = numerator p ℤ.* denominator q
numerator q ℤ.* denominator p
where open

-- _≃_ coincides with propositional equality.

≡⇒≃ : _≡_  _≃_
≡⇒≃ refl = refl

≃⇒≡ : _≃_  _≡_
≃⇒≡ {i = p} {j = q} =
helper (numerator p) (denominator-1 p) (isCoprime p)
(numerator q) (denominator-1 q) (isCoprime q)
where
open

helper :  n₁ d₁ c₁ n₂ d₂ c₂
n₁ ℤ.* + suc d₂  n₂ ℤ.* + suc d₁
(n₁ ÷ suc d₁) {c₁}  (n₂ ÷ suc d₂) {c₂}
helper n₁ d₁ c₁ n₂ d₂ c₂ eq
with Poset.antisym ℕDiv.poset 1+d₁∣1+d₂ 1+d₂∣1+d₁
where
1+d₁∣1+d₂ : suc d₁  suc d₂
1+d₁∣1+d₂ = ℤDiv.coprime-divisor (+ suc d₁) n₁ (+ suc d₂)
(C.sym \$ toWitness c₁) \$
ℕDiv.divides  n₂  (begin
n₁ ℤ.* + suc d₂   ≡⟨ cong ∣_∣ eq
n₂ ℤ.* + suc d₁   ≡⟨ ℤ.abs-*-commute n₂ (+ suc d₁)
n₂  ℕ.* suc d₁    )

1+d₂∣1+d₁ : suc d₂  suc d₁
1+d₂∣1+d₁ = ℤDiv.coprime-divisor (+ suc d₂) n₂ (+ suc d₁)
(C.sym \$ toWitness c₂) \$
ℕDiv.divides  n₁  (begin
n₂ ℤ.* + suc d₁   ≡⟨ cong ∣_∣ (P.sym eq)
n₁ ℤ.* + suc d₂   ≡⟨ ℤ.abs-*-commute n₁ (+ suc d₂)
n₁  ℕ.* suc d₂    )

helper n₁ d c₁ n₂ .d c₂ eq | refl with ℤ.cancel-*-right
n₁ n₂ (+ suc d)  ()) eq
helper n  d c₁ .n .d c₂ eq | refl | refl with Bool.proof-irrelevance c₁ c₂
helper n  d c  .n .d .c eq | refl | refl | refl = refl

------------------------------------------------------------------------
-- Equality is decidable

infix 4 _≟_

_≟_ : Decidable {A = } _≡_
p  q with ℚ.numerator p ℤ.* ℚ.denominator q ℤ.≟
ℚ.numerator q ℤ.* ℚ.denominator p
p  q | yes pq≃qp = yes (≃⇒≡ pq≃qp)
p  q | no ¬pq≃qp = no (¬pq≃qp  ≡⇒≃)

------------------------------------------------------------------------
-- Ordering

infix 4 _≤_ _≤?_

data _≤_ :     Set where
*≤* :  {p q}
ℚ.numerator p ℤ.* ℚ.denominator q ℤ.≤
ℚ.numerator q ℤ.* ℚ.denominator p
p  q

drop-*≤* :  {p q}  p  q
ℚ.numerator p ℤ.* ℚ.denominator q ℤ.≤
ℚ.numerator q ℤ.* ℚ.denominator p
drop-*≤* (*≤* pq≤qp) = pq≤qp

_≤?_ : Decidable _≤_
p ≤? q with ℚ.numerator p ℤ.* ℚ.denominator q ℤ.≤?
ℚ.numerator q ℤ.* ℚ.denominator p
p ≤? q | yes pq≤qp = yes (*≤* pq≤qp)
p ≤? q | no ¬pq≤qp = no  { (*≤* pq≤qp)  ¬pq≤qp pq≤qp })

decTotalOrder : DecTotalOrder _ _ _
decTotalOrder = record
{ Carrier         =
; _≈_             = _≡_
; _≤_             = _≤_
; isDecTotalOrder = record
{ isTotalOrder = record
{ isPartialOrder = record
{ isPreorder = record
{ isEquivalence = P.isEquivalence
; reflexive     = refl′
; trans         = trans
}
; antisym = antisym
}
; total = total
}
; _≟_  = _≟_
; _≤?_ = _≤?_
}
}
where
module ℤO = DecTotalOrder ℤ.decTotalOrder

refl′ : _≡_  _≤_
refl′ refl = *≤* ℤO.refl

trans : Transitive _≤_
trans {i = p} {j = q} {k = r} (*≤* le₁) (*≤* le₂)
= *≤* (ℤ.cancel-*-+-right-≤ _ _ _
(lemma
(ℚ.numerator p) (ℚ.denominator p)
(ℚ.numerator q) (ℚ.denominator q)
(ℚ.numerator r) (ℚ.denominator r)
(ℤ.*-+-right-mono (ℚ.denominator-1 r) le₁)
(ℤ.*-+-right-mono (ℚ.denominator-1 p) le₂)))
where
open Algebra.CommutativeRing ℤ.commutativeRing

lemma :  n₁ d₁ n₂ d₂ n₃ d₃
n₁ ℤ.* d₂ ℤ.* d₃ ℤ.≤ n₂ ℤ.* d₁ ℤ.* d₃
n₂ ℤ.* d₃ ℤ.* d₁ ℤ.≤ n₃ ℤ.* d₂ ℤ.* d₁
n₁ ℤ.* d₃ ℤ.* d₂ ℤ.≤ n₃ ℤ.* d₁ ℤ.* d₂
lemma n₁ d₁ n₂ d₂ n₃ d₃
rewrite *-assoc n₁ d₂ d₃
| *-comm d₂ d₃
| sym (*-assoc n₁ d₃ d₂)
| *-assoc n₃ d₂ d₁
| *-comm d₂ d₁
| sym (*-assoc n₃ d₁ d₂)
| *-assoc n₂ d₁ d₃
| *-comm d₁ d₃
| sym (*-assoc n₂ d₃ d₁)
= ℤO.trans

antisym : Antisymmetric _≡_ _≤_
antisym (*≤* le₁) (*≤* le₂) = ≃⇒≡ (ℤO.antisym le₁ le₂)

total : Total _≤_
total p q =
[ inj₁ ∘′ *≤* , inj₂ ∘′ *≤* ]′
(ℤO.total (ℚ.numerator p ℤ.* ℚ.denominator q)
(ℚ.numerator q ℤ.* ℚ.denominator p))