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Distance between rational numbers (#1400)
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‎src/elementary-number-theory.lagda.md

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@@ -55,6 +55,7 @@ open import elementary-number-theory.difference-rational-numbers public
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open import elementary-number-theory.dirichlet-convolution public
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open import elementary-number-theory.distance-integers public
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open import elementary-number-theory.distance-natural-numbers public
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open import elementary-number-theory.distance-rational-numbers public
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open import elementary-number-theory.divisibility-integers public
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open import elementary-number-theory.divisibility-modular-arithmetic public
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open import elementary-number-theory.divisibility-natural-numbers public

‎src/elementary-number-theory/absolute-value-rational-numbers.lagda.md

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@@ -15,6 +15,7 @@ open import elementary-number-theory.maximum-rational-numbers
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open import elementary-number-theory.multiplication-rational-numbers
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open import elementary-number-theory.nonnegative-rational-numbers
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open import elementary-number-theory.rational-numbers
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open import elementary-number-theory.strict-inequality-rational-numbers
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open import foundation.action-on-identifications-binary-functions
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open import foundation.action-on-identifications-functions
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## Properties
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### Bounds on both `p` and `-p` are bounds on `|p|`
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```agda
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abstract
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leq-abs-leq-leq-neg-ℚ :
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(p q : ℚ) → leq-ℚ p q → leq-ℚ (neg-ℚ p) q → leq-ℚ (rational-abs-ℚ p) q
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leq-abs-leq-leq-neg-ℚ p q = leq-max-leq-both-ℚ q p (neg-ℚ p)
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le-abs-le-le-neg-ℚ :
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(p q : ℚ) → le-ℚ p q → le-ℚ (neg-ℚ p) q → le-ℚ (rational-abs-ℚ p) q
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le-abs-le-le-neg-ℚ p q = le-max-le-both-ℚ q p (neg-ℚ p)
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```
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### The absolute value of a nonnegative rational number is the number itself
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```agda

‎src/elementary-number-theory/distance-rational-numbers.lagda.md

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# The distance between rational numbers
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```agda
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{-# OPTIONS --lossy-unification #-}
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module elementary-number-theory.distance-rational-numbers where
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```
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<details><summary>Imports</summary>
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```agda
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open import elementary-number-theory.absolute-value-rational-numbers
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open import elementary-number-theory.addition-rational-numbers
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open import elementary-number-theory.additive-group-of-rational-numbers
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open import elementary-number-theory.difference-rational-numbers
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open import elementary-number-theory.inequality-rational-numbers
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open import elementary-number-theory.maximum-rational-numbers
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open import elementary-number-theory.minimum-rational-numbers
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open import elementary-number-theory.multiplication-rational-numbers
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open import elementary-number-theory.nonnegative-rational-numbers
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open import elementary-number-theory.positive-rational-numbers
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open import elementary-number-theory.rational-numbers
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open import elementary-number-theory.strict-inequality-rational-numbers
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open import foundation.action-on-identifications-binary-functions
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open import foundation.action-on-identifications-functions
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open import foundation.coproduct-types
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open import foundation.dependent-pair-types
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open import foundation.identity-types
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open import foundation.logical-equivalences
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open import foundation.transport-along-identifications
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open import metric-spaces.metric-space-of-rational-numbers
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open import metric-spaces.metric-space-of-rational-numbers-with-open-neighborhoods
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```
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</details>
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## Idea
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The
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{{#concept "distance function" Disambiguation="between rational numbers" Agda=dist-ℚ}}
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between [rational numbers](elementary-number-theory.rational-numbers.md)
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measures how far two rational numbers are apart.
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```agda
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dist-ℚ : ℚ → ℚ → ℚ⁰⁺
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dist-ℚ p q = abs-ℚ (p -ℚ q)
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rational-dist-ℚ : ℚ → ℚ → ℚ
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rational-dist-ℚ p q = rational-ℚ⁰⁺ (dist-ℚ p q)
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```
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## Properties
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### Commutativity
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```agda
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abstract
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commutative-dist-ℚ : (p q : ℚ) → dist-ℚ p q ＝ dist-ℚ q p
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commutative-dist-ℚ p q =
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inv (abs-neg-ℚ _) ∙ ap abs-ℚ (distributive-neg-diff-ℚ _ _)
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```
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### The differences of the arguments are less than or equal to their distance
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```agda
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abstract
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leq-diff-dist-ℚ : (p q : ℚ) → leq-ℚ (p -ℚ q) (rational-dist-ℚ p q)
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leq-diff-dist-ℚ p q = leq-abs-ℚ (p -ℚ q)
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leq-reversed-diff-dist-ℚ :
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(p q : ℚ) → leq-ℚ (q -ℚ p) (rational-dist-ℚ p q)
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leq-reversed-diff-dist-ℚ p q =
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tr
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( leq-ℚ (q -ℚ p))
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( ap rational-ℚ⁰⁺ (commutative-dist-ℚ q p))
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( leq-diff-dist-ℚ q p)
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```
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### Zero laws
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```agda
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abstract
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right-zero-law-dist-ℚ : (q : ℚ) → dist-ℚ q zero-ℚ ＝ abs-ℚ q
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right-zero-law-dist-ℚ q = ap abs-ℚ (right-unit-law-add-ℚ _)
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left-zero-law-dist-ℚ : (q : ℚ) → dist-ℚ zero-ℚ q ＝ abs-ℚ q
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left-zero-law-dist-ℚ q = commutative-dist-ℚ _ _ ∙ right-zero-law-dist-ℚ q
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```
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### Distributivity laws
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```agda
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abstract
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left-distributive-mul-dist-ℚ :
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(p : ℚ⁰⁺) (q r : ℚ) →
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p *ℚ⁰⁺ dist-ℚ q r ＝ dist-ℚ (rational-ℚ⁰⁺ p *ℚ q) (rational-ℚ⁰⁺ p *ℚ r)
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left-distributive-mul-dist-ℚ p⁰⁺@(p , _) q r =
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eq-ℚ⁰⁺
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( equational-reasoning
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p *ℚ rational-dist-ℚ q r
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＝ rational-abs-ℚ (p *ℚ (q -ℚ r))
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by ap rational-ℚ⁰⁺ (inv (abs-left-mul-nonnegative-ℚ _ p⁰⁺))
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＝ rational-abs-ℚ (p *ℚ q +ℚ p *ℚ (neg-ℚ r))
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by ap rational-abs-ℚ (left-distributive-mul-add-ℚ p q (neg-ℚ r))
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＝ rational-abs-ℚ (p *ℚ q -ℚ p *ℚ r)
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by ap rational-abs-ℚ (ap (p *ℚ q +ℚ_) (right-negative-law-mul-ℚ _ _)))
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right-distributive-mul-dist-ℚ :
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(p : ℚ⁰⁺) (q r : ℚ) →
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dist-ℚ q r *ℚ⁰⁺ p ＝ dist-ℚ (q *ℚ rational-ℚ⁰⁺ p) (r *ℚ rational-ℚ⁰⁺ p)
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right-distributive-mul-dist-ℚ p⁰⁺@(p , _) q r =
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eq-ℚ⁰⁺
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( equational-reasoning
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rational-dist-ℚ q r *ℚ p
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＝ p *ℚ rational-dist-ℚ q r by commutative-mul-ℚ _ _
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＝ rational-dist-ℚ (p *ℚ q) (p *ℚ r)
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by ap rational-ℚ⁰⁺ (left-distributive-mul-dist-ℚ p⁰⁺ q r)
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＝ rational-dist-ℚ (q *ℚ p) (r *ℚ p)
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by
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ap-binary
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( rational-dist-ℚ)
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( commutative-mul-ℚ _ _)
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( commutative-mul-ℚ _ _))
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```
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### Triangle inequality
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```agda
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abstract
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triangle-inequality-dist-ℚ :
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(p q r : ℚ) → leq-ℚ⁰⁺ (dist-ℚ p r) (dist-ℚ p q +ℚ⁰⁺ dist-ℚ q r)
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triangle-inequality-dist-ℚ p q r =
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tr
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( λ s → leq-ℚ⁰⁺ s (dist-ℚ p q +ℚ⁰⁺ dist-ℚ q r))
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( ap abs-ℚ
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( inv (associative-add-ℚ _ _ (neg-ℚ r)) ∙
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ap (_-ℚ r) (is-section-diff-ℚ _ _)))
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( triangle-inequality-abs-ℚ (p -ℚ q) (q -ℚ r))
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```
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### Bounding the distance between rational numbers
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```agda
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abstract
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leq-dist-leq-diff-ℚ :
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(p q r : ℚ) → leq-ℚ (p -ℚ q) r → leq-ℚ (q -ℚ p) r →
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leq-ℚ (rational-dist-ℚ p q) r
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leq-dist-leq-diff-ℚ p q r p-q≤r q-p≤r =
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leq-abs-leq-leq-neg-ℚ
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( p -ℚ q)
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( r)
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( p-q≤r)
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( inv-tr (λ s → leq-ℚ s r) (distributive-neg-diff-ℚ p q) q-p≤r)
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le-dist-le-diff-ℚ :
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(p q r : ℚ) → le-ℚ (p -ℚ q) r → le-ℚ (q -ℚ p) r →
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le-ℚ (rational-dist-ℚ p q) r
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le-dist-le-diff-ℚ p q r p-q<r q-p<r =
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le-abs-le-le-neg-ℚ
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( p -ℚ q)
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( r)
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( p-q<r)
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( inv-tr (λ s → le-ℚ s r) (distributive-neg-diff-ℚ p q) q-p<r)
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```
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### Relationship to the metric space of rational numbers
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```agda
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abstract
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leq-dist-neighborhood-leq-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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neighborhood-leq-ℚ ε p q →
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leq-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε)
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leq-dist-neighborhood-leq-ℚ ε⁺@(ε , _) p q (H , K) =
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leq-dist-leq-diff-ℚ
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( p)
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( q)
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( ε)
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( swap-right-diff-leq-ℚ p ε q (leq-transpose-right-add-ℚ p q ε K))
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( swap-right-diff-leq-ℚ q ε p (leq-transpose-right-add-ℚ q p ε H))
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neighborhood-leq-leq-dist-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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leq-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε) →
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neighborhood-leq-ℚ ε p q
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neighborhood-leq-leq-dist-ℚ ε⁺@(ε , _) p q |p-q|≤ε =
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( leq-transpose-left-diff-ℚ
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( q)
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( ε)
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( p)
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( swap-right-diff-leq-ℚ
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( q)
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( p)
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( ε)
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( transitive-leq-ℚ
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( q -ℚ p)
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( rational-dist-ℚ p q)
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( ε)
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( |p-q|≤ε)
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( leq-reversed-diff-dist-ℚ p q)))) ,
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( leq-transpose-left-diff-ℚ
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( p)
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( ε)
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( q)
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( swap-right-diff-leq-ℚ
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( p)
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( q)
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( ε)
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( transitive-leq-ℚ
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( p -ℚ q)
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( rational-dist-ℚ p q)
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( ε)
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( |p-q|≤ε)
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( leq-diff-dist-ℚ p q))))
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leq-dist-iff-neighborhood-leq-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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leq-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε) ↔
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neighborhood-leq-ℚ ε p q
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pr1 (leq-dist-iff-neighborhood-leq-ℚ ε p q) = neighborhood-leq-leq-dist-ℚ ε p q
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pr2 (leq-dist-iff-neighborhood-leq-ℚ ε p q) = leq-dist-neighborhood-leq-ℚ ε p q
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```
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### Relationship to the metric space of rational numbers with open neighborhoods
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```agda
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abstract
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le-dist-neighborhood-le-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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neighborhood-le-ℚ ε p q →
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le-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε)
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le-dist-neighborhood-le-ℚ ε⁺@(ε , _) p q (H , K) =
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le-dist-le-diff-ℚ
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( p)
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( q)
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( ε)
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( swap-right-diff-le-ℚ p ε q (le-transpose-right-add-ℚ p q ε K))
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( swap-right-diff-le-ℚ q ε p (le-transpose-right-add-ℚ q p ε H))
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neighborhood-le-le-dist-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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le-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε) →
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neighborhood-le-ℚ ε p q
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neighborhood-le-le-dist-ℚ ε⁺@(ε , _) p q |p-q|<ε =
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( le-transpose-left-diff-ℚ
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( q)
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( ε)
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( p)
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( swap-right-diff-le-ℚ
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( q)
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( p)
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( ε)
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( concatenate-leq-le-ℚ
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( q -ℚ p)
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( rational-dist-ℚ p q)
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( ε)
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( leq-reversed-diff-dist-ℚ p q)
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( |p-q|<ε)))) ,
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( le-transpose-left-diff-ℚ
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( p)
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( ε)
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( q)
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( swap-right-diff-le-ℚ
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( p)
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( q)
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( ε)
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( concatenate-leq-le-ℚ
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( p -ℚ q)
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( rational-dist-ℚ p q)
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( ε)
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( leq-diff-dist-ℚ p q)
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( |p-q|<ε))))
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le-dist-iff-neighborhood-le-ℚ :
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(ε : ℚ⁺) (p q : ℚ) →
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le-ℚ (rational-dist-ℚ p q) (rational-ℚ⁺ ε) ↔
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neighborhood-le-ℚ ε p q
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pr1 (le-dist-iff-neighborhood-le-ℚ ε p q) = neighborhood-le-le-dist-ℚ ε p q
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pr2 (le-dist-iff-neighborhood-le-ℚ ε p q) = le-dist-neighborhood-le-ℚ ε p q
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```
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### The distance between two rational numbers is the difference of their maximum and minimum
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```agda
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abstract
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eq-dist-diff-leq-ℚ : (p q : ℚ) → leq-ℚ p q → rational-dist-ℚ p q ＝ q -ℚ p
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eq-dist-diff-leq-ℚ p q p≤q =
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equational-reasoning
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rational-dist-ℚ p q
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＝ rational-dist-ℚ q p by ap rational-ℚ⁰⁺ (commutative-dist-ℚ p q)
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＝ q -ℚ p
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by
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rational-abs-zero-leq-ℚ
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( q -ℚ p)
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( backward-implication (iff-translate-diff-leq-zero-ℚ p q) p≤q)
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eq-dist-diff-max-min-ℚ :
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(p q : ℚ) → rational-dist-ℚ p q ＝ max-ℚ p q -ℚ min-ℚ p q
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eq-dist-diff-max-min-ℚ p q =
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rec-coproduct
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( λ p≤q →
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equational-reasoning
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rational-dist-ℚ p q
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＝ q -ℚ p by eq-dist-diff-leq-ℚ p q p≤q
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＝ max-ℚ p q -ℚ min-ℚ p q
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by
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inv
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( ap-diff-ℚ
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( left-leq-right-max-ℚ p q p≤q)
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( left-leq-right-min-ℚ p q p≤q)))
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( λ q≤p →
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equational-reasoning
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rational-dist-ℚ p q
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＝ rational-dist-ℚ q p by ap rational-ℚ⁰⁺ (commutative-dist-ℚ p q)
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＝ p -ℚ q by eq-dist-diff-leq-ℚ q p q≤p
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＝ max-ℚ p q -ℚ min-ℚ p q
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by
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inv
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( ap-diff-ℚ
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( right-leq-left-max-ℚ p q q≤p)
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( right-leq-left-min-ℚ p q q≤p)))
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( linear-leq-ℚ p q)
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```

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