Positive rational numbers

Content created by Louis Wasserman, malarbol, Fredrik Bakke and Garrett Figueroa.

Created on 2024-04-25.
Last modified on 2025-10-07.

{-# OPTIONS --lossy-unification #-}

module elementary-number-theory.positive-rational-numbers where
Imports 
open import elementary-number-theory.addition-rational-numbers
open import elementary-number-theory.cross-multiplication-difference-integer-fractions
open import elementary-number-theory.difference-rational-numbers
open import elementary-number-theory.inequality-rational-numbers
open import elementary-number-theory.integer-fractions
open import elementary-number-theory.integers
open import elementary-number-theory.negative-integers
open import elementary-number-theory.nonzero-natural-numbers
open import elementary-number-theory.nonzero-rational-numbers
open import elementary-number-theory.positive-and-negative-integers
open import elementary-number-theory.positive-integer-fractions
open import elementary-number-theory.positive-integers
open import elementary-number-theory.rational-numbers
open import elementary-number-theory.strict-inequality-rational-numbers

open import foundation.cartesian-product-types
open import foundation.coproduct-types
open import foundation.decidable-subtypes
open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.identity-types
open import foundation.logical-equivalences
open import foundation.negation
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.sets
open import foundation.subtypes
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import set-theory.countable-sets

Idea

A rational number x is said to be positive if its underlying fraction is positive.

Positive rational numbers are a subsemigroup of the additive monoid of rational numbers and a submonoid of the multiplicative monoid of rational numbers.

Definitions

The property of being a positive rational number

module _
  (x : ℚ)
  where

  is-positive-ℚ : UU lzero
  is-positive-ℚ = is-positive-fraction-ℤ (fraction-ℚ x)

  is-prop-is-positive-ℚ : is-prop is-positive-ℚ
  is-prop-is-positive-ℚ = is-prop-is-positive-fraction-ℤ (fraction-ℚ x)

  is-positive-prop-ℚ : Prop lzero
  pr1 is-positive-prop-ℚ = is-positive-ℚ
  pr2 is-positive-prop-ℚ = is-prop-is-positive-ℚ

decidable-subtype-positive-ℚ : decidable-subtype lzero ℚ
decidable-subtype-positive-ℚ x =
  decidable-subtype-positive-fraction-ℤ (fraction-ℚ x)

The type of positive rational numbers

positive-ℚ : UU lzero
positive-ℚ = type-subtype is-positive-prop-ℚ

ℚ⁺ : UU lzero
ℚ⁺ = positive-ℚ

module _
  (x : positive-ℚ)
  where

  rational-ℚ⁺ : ℚ
  rational-ℚ⁺ = pr1 x

  fraction-ℚ⁺ : fraction-ℤ
  fraction-ℚ⁺ = fraction-ℚ rational-ℚ⁺

  numerator-ℚ⁺ : ℤ
  numerator-ℚ⁺ = numerator-ℚ rational-ℚ⁺

  denominator-ℚ⁺ : ℤ
  denominator-ℚ⁺ = denominator-ℚ rational-ℚ⁺

  is-positive-rational-ℚ⁺ : is-positive-ℚ rational-ℚ⁺
  is-positive-rational-ℚ⁺ = pr2 x

  is-positive-fraction-ℚ⁺ : is-positive-fraction-ℤ fraction-ℚ⁺
  is-positive-fraction-ℚ⁺ = is-positive-rational-ℚ⁺

  is-positive-numerator-ℚ⁺ : is-positive-ℤ numerator-ℚ⁺
  is-positive-numerator-ℚ⁺ = is-positive-rational-ℚ⁺

  is-positive-denominator-ℚ⁺ : is-positive-ℤ denominator-ℚ⁺
  is-positive-denominator-ℚ⁺ = is-positive-denominator-ℚ rational-ℚ⁺

abstract
  eq-ℚ⁺ : {x y : ℚ⁺} → rational-ℚ⁺ x ＝ rational-ℚ⁺ y → x ＝ y
  eq-ℚ⁺ {x} {y} = eq-type-subtype is-positive-prop-ℚ

Properties

The positive rational numbers form a set

set-ℚ⁺ : Set lzero
set-ℚ⁺ = set-subset ℚ-Set is-positive-prop-ℚ

is-set-ℚ⁺ : is-set ℚ⁺
is-set-ℚ⁺ = is-set-type-Set set-ℚ⁺

The set of positive rational numbers is countable

abstract
  is-countable-set-ℚ⁺ : is-countable set-ℚ⁺
  is-countable-set-ℚ⁺ =
    is-countable-decidable-subset-is-countable
      ( ℚ-Set)
      ( decidable-subtype-positive-ℚ)
      ( is-countable-ℚ)

The rational image of a positive integer is positive

abstract
  is-positive-rational-ℤ :
    (x : ℤ) → is-positive-ℤ x → is-positive-ℚ (rational-ℤ x)
  is-positive-rational-ℤ x P = P

positive-rational-positive-ℤ : positive-ℤ → ℚ⁺
positive-rational-positive-ℤ (z , pos-z) = rational-ℤ z , pos-z

positive-rational-ℤ⁺ : ℤ⁺ → ℚ⁺
positive-rational-ℤ⁺ = positive-rational-positive-ℤ

one-ℚ⁺ : ℚ⁺
one-ℚ⁺ = (one-ℚ , is-positive-int-positive-ℤ one-positive-ℤ)

The type of positive rational numbers is inhabited

abstract
  is-inhabited-ℚ⁺ : ║ ℚ⁺ ║₋₁
  is-inhabited-ℚ⁺ = unit-trunc-Prop one-ℚ⁺

The rational image of a positive natural number is positive

positive-rational-ℕ⁺ : ℕ⁺ → ℚ⁺
positive-rational-ℕ⁺ n = positive-rational-positive-ℤ (positive-int-ℕ⁺ n)

The rational image of a positive integer fraction is positive

opaque
  unfolding rational-fraction-ℤ

  is-positive-rational-fraction-ℤ :
    {x : fraction-ℤ} (P : is-positive-fraction-ℤ x) →
    is-positive-ℚ (rational-fraction-ℤ x)
  is-positive-rational-fraction-ℤ = is-positive-reduce-fraction-ℤ

A rational number x is positive if and only if 0 < x

module _
  (x : ℚ)
  where

  opaque
    unfolding le-ℚ-Prop

    le-zero-is-positive-ℚ : is-positive-ℚ x → le-ℚ zero-ℚ x
    le-zero-is-positive-ℚ =
      is-positive-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))

    is-positive-le-zero-ℚ : le-ℚ zero-ℚ x → is-positive-ℚ x
    is-positive-le-zero-ℚ =
      is-positive-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))

Zero is not a positive rational number

abstract
  is-not-positive-zero-ℚ : ¬ (is-positive-ℚ zero-ℚ)
  is-not-positive-zero-ℚ pos-0 =
    irreflexive-le-ℚ zero-ℚ (le-zero-is-positive-ℚ zero-ℚ pos-0)

The difference of a rational number with a lesser rational number is positive

module _
  (x y : ℚ) (H : le-ℚ x y)
  where

  abstract
    is-positive-diff-le-ℚ : is-positive-ℚ (y -ℚ x)
    is-positive-diff-le-ℚ =
      is-positive-le-zero-ℚ
        ( y -ℚ x)
        ( backward-implication
          ( iff-translate-diff-le-zero-ℚ x y)
          ( H))

  positive-diff-le-ℚ : ℚ⁺
  positive-diff-le-ℚ = y -ℚ x , is-positive-diff-le-ℚ

  left-law-positive-diff-le-ℚ : (rational-ℚ⁺ positive-diff-le-ℚ) +ℚ x ＝ y
  left-law-positive-diff-le-ℚ =
    ( associative-add-ℚ y (neg-ℚ x) x) ∙
    ( inv-tr
      ( λ u → y +ℚ u ＝ y)
      ( left-inverse-law-add-ℚ x)
      ( right-unit-law-add-ℚ y))

  right-law-positive-diff-le-ℚ : x +ℚ (rational-ℚ⁺ positive-diff-le-ℚ) ＝ y
  right-law-positive-diff-le-ℚ =
    commutative-add-ℚ x (y -ℚ x) ∙ left-law-positive-diff-le-ℚ

A nonzero rational number or its negative is positive

opaque
  unfolding neg-ℚ

  decide-is-negative-is-positive-is-nonzero-ℚ :
    {x : ℚ} → is-nonzero-ℚ x → is-positive-ℚ (neg-ℚ x) + is-positive-ℚ x
  decide-is-negative-is-positive-is-nonzero-ℚ {x} H =
    rec-coproduct
      ( inl ∘ is-positive-neg-is-negative-ℤ)
      ( inr)
      ( decide-sign-nonzero-ℤ
        { numerator-ℚ x}
        ( is-nonzero-numerator-is-nonzero-ℚ x H))

A rational and its negative are not both positive

opaque
  unfolding neg-ℚ

  not-is-negative-is-positive-ℚ :
    (x : ℚ) → ¬ (is-positive-ℚ (neg-ℚ x) × is-positive-ℚ x)
  not-is-negative-is-positive-ℚ x (N , P) =
    is-not-negative-and-positive-ℤ
      ( numerator-ℚ x)
      ( ( is-negative-eq-ℤ
          (neg-neg-ℤ (numerator-ℚ x))
          (is-negative-neg-is-positive-ℤ {numerator-ℚ (neg-ℚ x)} N)) ,
        ( P))

Positive rational numbers are nonzero

abstract
  is-nonzero-is-positive-ℚ : {x : ℚ} → is-positive-ℚ x → is-nonzero-ℚ x
  is-nonzero-is-positive-ℚ {x} H =
    is-nonzero-is-nonzero-numerator-ℚ x
      ( is-nonzero-is-positive-ℤ
        { numerator-ℚ x}
        ( H))

nonzero-ℚ⁺ : positive-ℚ → nonzero-ℚ
nonzero-ℚ⁺ (x , P) = (x , is-nonzero-is-positive-ℚ P)

If p ≤ q for positive p, then q is positive

abstract
  is-positive-leq-ℚ⁺ :
    (p : ℚ⁺) (q : ℚ) → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
  is-positive-leq-ℚ⁺ (p , pos-p) q p≤q =
    is-positive-le-zero-ℚ
      ( q)
      ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ p pos-p) p≤q)

  is-positive-le-ℚ⁺ :
    (p : ℚ⁺) (q : ℚ) → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
  is-positive-le-ℚ⁺ p q p<q =
    is-positive-leq-ℚ⁺ p q (leq-le-ℚ p<q)

Recent changes