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Rational+.agda
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Rational+.agda
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Rational numbers
------------------------------------------------------------------------
module Rational+ where
import Algebra
import Data.Sign as S
open import Data.Empty using (⊥)
open import Data.Unit using (⊤; tt)
import Data.Bool.Properties as Bool
open import Function
open import Data.Product
open import Data.Integer as ℤ using (ℤ; ∣_∣; +_; -[1+_]; _◃_; sign)
open import Data.Integer.Divisibility as ℤDiv using (Coprime)
import Data.Integer.Properties as ℤ
open import Data.Nat.GCD
open import Data.Nat.Divisibility as ℕDiv using (_∣_; divides)
import Data.Nat.Coprimality as C
open import Data.Nat as ℕ using (ℕ; zero; suc; _≥_)
open import Data.Nat.Show renaming (show to ℕshow)
open import Data.Sum
open import Data.String using (String; _++_)
import Level
open import Relation.Nullary.Decidable
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P
using (_≡_; refl; subst; cong; cong₂)
open P.≡-Reasoning
infix 8 -_ 1/_
infixl 7 _*_ _/_
infixl 6 _-_ _+_
------------------------------------------------------------------------
-- 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 ℚ : Set where
field
numerator : ℤ
denominator-1 : ℕ
isCoprime : True (C.coprime? ∣ numerator ∣ (suc denominator-1))
denominator : ℤ
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 : ℤ) (denominator : ℕ)
{coprime : True (C.coprime? ∣ numerator ∣ denominator)}
{≢0 : False (ℕ._≟_ denominator 0)} →
ℚ
(n ÷ zero) {≢0 = ()}
(n ÷ suc d) {c} =
record { numerator = n; denominator-1 = d; isCoprime = c }
-- n/1 from n
_÷1 : ℕ → ℚ
_÷1 n = record { numerator = + n
; denominator-1 = 0
; isCoprime = fromWitness (λ {_} → C.sym (C.1-coprimeTo n))
}
private
-- Note that the implicit arguments do not need to be given for
-- concrete inputs:
0/1 : ℚ
0/1 = + 0 ÷ 1
-½ : ℚ
-½ = (ℤ.- + 1) ÷ 2
------------------------------------------------------------------------
-- Two useful lemmas to help with operations on rationals
NonZero : ℕ → Set
NonZero 0 = ⊥
NonZero (suc _) = ⊤
-- normalize takes two natural numbers, say 6 and 21 and their gcd 3, and
-- returns them normalized as 2 and 7 and a proof that they are coprime
normalize : ∀ {m n g} → {n≢0 : NonZero n} → {g≢0 : NonZero g} →
GCD m n g → Σ[ p ∈ ℕ ] Σ[ q ∈ ℕ ] (False (q ℕ.≟ 0) × C.Coprime p q)
normalize {m} {n} {0} {_} {()} _
normalize {m} {n} {ℕ.suc g} {_} {_} G with Bézout.identity G
normalize {m} {.0} {ℕ.suc g} {()} {_}
(GCD.is (divides p m≡pg' , divides 0 refl) _) | _
normalize {m} {n} {ℕ.suc g} {_} {_}
(GCD.is (divides p m≡pg' , divides (ℕ.suc q) n≡qg') _) | Bézout.+- x y eq =
(p , ℕ.suc q , tt , C.Bézout-coprime {p} {ℕ.suc q} {g} (Bézout.+- x y
(begin
ℕ.suc g ℕ.+ y ℕ.* (ℕ.suc q ℕ.* ℕ.suc g)
≡⟨ cong (λ h → ℕ.suc g ℕ.+ y ℕ.* h) (P.sym n≡qg') ⟩
ℕ.suc g ℕ.+ y ℕ.* n
≡⟨ eq ⟩
x ℕ.* m
≡⟨ cong (λ h → x ℕ.* h) m≡pg' ⟩
x ℕ.* (p ℕ.* ℕ.suc g) ∎)))
normalize {m} {n} {ℕ.suc g} {_} {_}
(GCD.is (divides p m≡pg' , divides (ℕ.suc q) n≡qg') _) | Bézout.-+ x y eq =
(p , ℕ.suc q , tt , C.Bézout-coprime {p} {ℕ.suc q} {g} (Bézout.-+ x y
(begin
ℕ.suc g ℕ.+ x ℕ.* (p ℕ.* ℕ.suc g)
≡⟨ cong (λ h → ℕ.suc g ℕ.+ x ℕ.* h) (P.sym m≡pg') ⟩
ℕ.suc g ℕ.+ x ℕ.* m
≡⟨ eq ⟩
y ℕ.* n
≡⟨ cong (λ h → y ℕ.* h) n≡qg' ⟩
y ℕ.* (ℕ.suc q ℕ.* ℕ.suc g) ∎)))
-- a version of gcd that returns a proof that the result is non-zero given
-- that one of the inputs is non-zero
gcd≢0 : (m n : ℕ) → {m≢0 : NonZero m} → ∃ λ d → GCD m n d × NonZero d
gcd≢0 m n {m≢0} with gcd m n
gcd≢0 m n {m≢0} | (0 , GCD.is (0|m , _) _) with ℕDiv.0∣⇒≡0 0|m
gcd≢0 .0 n {()} | (0 , GCD.is (0|m , _) _) | refl
gcd≢0 m n {_} | (ℕ.suc d , G) = (ℕ.suc d , G , tt)
------------------------------------------------------------------------------
-- Convenient way of constructing the rational number m / suc n
helper+ : (n : ℤ) → (d : ℕ) → {d≢0 : NonZero d} → ℚ
helper+ (+ 0) d {d≢0} = + 0 ÷ 1
helper+ (+ ℕ.suc n) d {d≢0} =
let (g , G , g≢0) = gcd≢0 ∣ + ℕ.suc n ∣ d {tt}
(nn , nd , nd≢0 , nc) = normalize {∣ + ℕ.suc n ∣} {d} {g} {d≢0} {g≢0} G
in ((S.+ ◃ nn) ÷ nd)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h nd) (P.sym (ℤ.abs-◃ S.+ nn)) nc)}
{nd≢0}
helper+ -[1+ n ] d {d≢0} =
let (g , G , g≢0) = gcd≢0 ∣ -[1+ n ] ∣ d {tt}
(nn , nd , nd≢0 , nc) = normalize {∣ -[1+ n ] ∣} {d} {g} {d≢0} {g≢0} G
in ((S.- ◃ nn) ÷ nd)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h nd) (P.sym (ℤ.abs-◃ S.- nn)) nc)}
{nd≢0}
mkRational : (m n : ℕ) {n≠0 : NonZero n} → ℚ
mkRational m n {n≠0} = helper+ (+ m) n {n≠0}
-- 1/n from n
1÷_ : (n : ℕ) → {n≥1 : n ≥ 1} → ℚ
(1÷ (suc n)) {ℕ.s≤s n≥1} = mkRational 1 (ℕ.suc n)
------------------------------------------------------------------------------
-- Operations on rationals: unary -, reciprocal, multiplication, addition
-- unary negation
--
-- Andreas Abel says: Agda's type-checker is incomplete when it has to handle
-- types with leading hidden quantification, such as the ones of Coprime m n
-- and c. A work around is to use hidden abstraction explicitly. In your
-- case, giving λ {i} -> c works. Not pretty, but unavoidable until we
-- improve on the current heuristics. I recorded this as a bug
-- http://code.google.com/p/agda/issues/detail?id=1079
-_ : ℚ → ℚ
-_ p with ℚ.numerator p | ℚ.denominator-1 p | toWitness (ℚ.isCoprime p)
... | -[1+ n ] | d | c = (+ ℕ.suc n ÷ ℕ.suc d) {fromWitness (λ {i} → c)}
... | + 0 | d | _ = p
... | + ℕ.suc n | d | c = (-[1+ n ] ÷ ℕ.suc d) {fromWitness (λ {i} → c)}
-- reciprocal: requires a proof that the numerator is not zero
1/_ : (p : ℚ) → {n≢0 : NonZero ∣ ℚ.numerator p ∣} → ℚ
1/_ p {n≢0} with ℚ.numerator p | ℚ.denominator-1 p | toWitness (ℚ.isCoprime p)
1/_ p {()} | + 0 | d | c
... | + (ℕ.suc n) | d | c =
((S.+ ◃ ℕ.suc d) ÷ ℕ.suc n)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h (ℕ.suc n))
(P.sym (ℤ.abs-◃ S.+ (ℕ.suc d)))
(C.sym c))}
... | -[1+ n ] | d | c =
((S.- ◃ ℕ.suc d) ÷ ℕ.suc n)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h (ℕ.suc n))
(P.sym (ℤ.abs-◃ S.- (ℕ.suc d)))
(C.sym c))}
-- multiplication
private
helper* : (n₁ : ℤ) → (d₁ : ℕ) → (n₂ : ℤ) → (d₂ : ℕ) →
{n≢0 : NonZero ∣ n₁ ℤ.* n₂ ∣} →
{d≢0 : NonZero (d₁ ℕ.* d₂)} →
ℚ
helper* n₁ d₁ n₂ d₂ {n≢0} {d≢0} =
let n = n₁ ℤ.* n₂
d = d₁ ℕ.* d₂
(g , G , g≢0) = gcd≢0 ∣ n ∣ d {n≢0}
(nn , nd , nd≢0 , nc) = normalize {∣ n ∣} {d} {g} {d≢0} {g≢0} G
in ((sign n ◃ nn) ÷ nd)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h nd) (P.sym (ℤ.abs-◃ (sign n) nn)) nc)}
{nd≢0}
_*_ : ℚ → ℚ → ℚ
p₁ * p₂ with ℚ.numerator p₁ | ℚ.numerator p₂
... | + 0 | _ = + 0 ÷ 1
... | _ | + 0 = + 0 ÷ 1
... | + ℕ.suc n₁ | + ℕ.suc n₂ =
helper* (+ ℕ.suc n₁) (ℕ.suc (ℚ.denominator-1 p₁))
(+ ℕ.suc n₂) (ℕ.suc (ℚ.denominator-1 p₂))
... | + ℕ.suc n₁ | -[1+ n₂ ] =
helper* (+ ℕ.suc n₁) (ℕ.suc (ℚ.denominator-1 p₁))
-[1+ n₂ ] (ℕ.suc (ℚ.denominator-1 p₂))
... | -[1+ n₁ ] | + ℕ.suc n₂ =
helper* -[1+ n₁ ] (ℕ.suc (ℚ.denominator-1 p₁))
(+ ℕ.suc n₂) (ℕ.suc (ℚ.denominator-1 p₂))
... | -[1+ n₁ ] | -[1+ n₂ ] =
helper* -[1+ n₁ ] (ℕ.suc (ℚ.denominator-1 p₁))
-[1+ n₂ ] (ℕ.suc (ℚ.denominator-1 p₂))
-- addition
{--
private
helper+ : (n : ℤ) → (d : ℕ) → {d≢0 : NonZero d} → ℚ
helper+ (+ 0) d {d≢0} = + 0 ÷ 1
helper+ (+ ℕ.suc n) d {d≢0} =
let (g , G , g≢0) = gcd≢0 ∣ + ℕ.suc n ∣ d {tt}
(nn , nd , nd≢0 , nc) = normalize {∣ + ℕ.suc n ∣} {d} {g} {d≢0} {g≢0} G
in ((S.+ ◃ nn) ÷ nd)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h nd) (P.sym (ℤ.abs-◃ S.+ nn)) nc)}
{nd≢0}
helper+ -[1+ n ] d {d≢0} =
let (g , G , g≢0) = gcd≢0 ∣ -[1+ n ] ∣ d {tt}
(nn , nd , nd≢0 , nc) = normalize {∣ -[1+ n ] ∣} {d} {g} {d≢0} {g≢0} G
in ((S.- ◃ nn) ÷ nd)
{fromWitness (λ {i} →
subst (λ h → C.Coprime h nd) (P.sym (ℤ.abs-◃ S.- nn)) nc)}
{nd≢0}
--}
_+_ : ℚ → ℚ → ℚ
p₁ + p₂ =
let n₁ = ℚ.numerator p₁
d₁ = ℕ.suc (ℚ.denominator-1 p₁)
n₂ = ℚ.numerator p₂
d₂ = ℕ.suc (ℚ.denominator-1 p₂)
n = (n₁ ℤ.* + d₂) ℤ.+ (n₂ ℤ.* + d₁)
d = d₁ ℕ.* d₂
in helper+ n d
-- subtraction and division
_-_ : ℚ → ℚ → ℚ
p₁ - p₂ = p₁ + (- p₂)
_/_ : (p₁ p₂ : ℚ) → {n≢0 : NonZero ∣ ℚ.numerator p₂ ∣} → ℚ
_/_ p₁ p₂ {n≢0} = p₁ * (1/_ p₂ {n≢0})
-- conventional printed representation
show : ℚ → String
show p = ℤ.show (ℚ.numerator p) ++ "/" ++ ℕshow (ℕ.suc (ℚ.denominator-1 p))
------------------------------------------------------------------------
-- Equality
-- Equality of rational numbers.
infix 4 _≃_
_≃_ : Rel ℚ Level.zero
p ≃ q = numerator p ℤ.* denominator q ≡
numerator q ℤ.* denominator p
where open ℚ
-- _≃_ 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 ℚ
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) (λ ()) 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 = ℚ} _≡_
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 _≤_ : ℚ → ℚ → 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 (λ { (*≤* pq≤qp) → ¬pq≤qp pq≤qp })
decTotalOrder : DecTotalOrder _ _ _
decTotalOrder = record
{ Carrier = ℚ
; _≈_ = _≡_
; _≤_ = _≤_
; 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))
------------------------------------------------------------------------------
-- A few constants and some small tests
0ℚ 1ℚ : ℚ
0ℚ = + 0 ÷ 1
1ℚ = + 1 ÷ 1
private
p₀ p₁ p₂ p₃ : ℚ
p₀ = + 1 ÷ 2
p₁ = + 1 ÷ 3
p₂ = -[1+ 2 ] ÷ 4
p₃ = + 3 ÷ 4
test₀ = show p₂
test₁ = show (- p₂)
test₂ = show (1/ p₂)
test₃ = show (p₀ + p₀)
test₄ = show (p₁ * p₂)