Cauchy completeness of the Dedekind real numbers
Content created by Louis Wasserman.
Created on 2025-04-02.
Last modified on 2025-04-02.
{-# OPTIONS --lossy-unification #-} module real-numbers.cauchy-completeness-dedekind-real-numbers where
Imports
open import elementary-number-theory.addition-rational-numbers open import elementary-number-theory.additive-group-of-rational-numbers open import elementary-number-theory.difference-rational-numbers open import elementary-number-theory.positive-rational-numbers open import elementary-number-theory.rational-numbers open import elementary-number-theory.strict-inequality-rational-numbers open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.conjunction open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.empty-types open import foundation.existential-quantification open import foundation.functoriality-disjunction open import foundation.identity-types open import foundation.logical-equivalences open import foundation.negation open import foundation.propositional-truncations open import foundation.subtypes open import foundation.transport-along-identifications open import foundation.universe-levels open import metric-spaces.cauchy-approximations-metric-spaces open import metric-spaces.cauchy-sequences-complete-metric-spaces open import metric-spaces.complete-metric-spaces open import metric-spaces.convergent-cauchy-approximations-metric-spaces open import metric-spaces.limits-of-cauchy-approximations-in-premetric-spaces open import metric-spaces.metric-spaces open import real-numbers.addition-real-numbers open import real-numbers.dedekind-real-numbers open import real-numbers.difference-real-numbers open import real-numbers.lower-dedekind-real-numbers open import real-numbers.metric-space-of-real-numbers open import real-numbers.negation-real-numbers open import real-numbers.positive-real-numbers open import real-numbers.rational-real-numbers open import real-numbers.strict-inequality-real-numbers open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers open import real-numbers.upper-dedekind-real-numbers
Idea
The metric space of the Dedekind real numbers is complete.
Definitions
Cauchy approximations in the real numbers
cauchy-approximation-leq-ℝ : (l : Level) → UU (lsuc l) cauchy-approximation-leq-ℝ l = cauchy-approximation-Metric-Space (metric-space-leq-ℝ l) map-cauchy-approximation-leq-ℝ : {l : Level} → (x : cauchy-approximation-leq-ℝ l) → ℚ⁺ → ℝ l map-cauchy-approximation-leq-ℝ {l} = map-cauchy-approximation-Metric-Space (metric-space-leq-ℝ l) is-cauchy-map-cauchy-approximation-leq-ℝ : {l : Level} → (x : cauchy-approximation-leq-ℝ l) → (ε δ : ℚ⁺) → neighborhood-Metric-Space ( metric-space-leq-ℝ l) ( ε +ℚ⁺ δ) ( map-cauchy-approximation-leq-ℝ x ε) ( map-cauchy-approximation-leq-ℝ x δ) is-cauchy-map-cauchy-approximation-leq-ℝ {l} x = is-cauchy-approximation-map-cauchy-approximation-Metric-Space ( metric-space-leq-ℝ l) ( x)
Properties
The limit of a Cauchy sequence
module _ {l : Level} (x : cauchy-approximation-leq-ℝ l) where lower-cut-lim-cauchy-approximation-leq-ℝ : subtype l ℚ lower-cut-lim-cauchy-approximation-leq-ℝ q = ∃ ( ℚ⁺ × ℚ⁺) ( λ (ε , θ) → lower-cut-ℝ ( map-cauchy-approximation-leq-ℝ x ε) ( q +ℚ rational-ℚ⁺ (ε +ℚ⁺ θ))) upper-cut-lim-cauchy-approximation-leq-ℝ : subtype l ℚ upper-cut-lim-cauchy-approximation-leq-ℝ q = ∃ ( ℚ⁺ × ℚ⁺) ( λ (ε , θ) → upper-cut-ℝ ( map-cauchy-approximation-leq-ℝ x ε) ( q -ℚ (rational-ℚ⁺ (ε +ℚ⁺ θ)))) abstract is-inhabited-lower-cut-lim-cauchy-approximation-leq-ℝ : exists ℚ lower-cut-lim-cauchy-approximation-leq-ℝ is-inhabited-lower-cut-lim-cauchy-approximation-leq-ℝ = let open do-syntax-trunc-Prop (∃ ℚ lower-cut-lim-cauchy-approximation-leq-ℝ) x1 = map-cauchy-approximation-leq-ℝ x one-ℚ⁺ two-ℚ = one-ℚ +ℚ one-ℚ in do p , p<x1 ← is-inhabited-lower-cut-ℝ x1 intro-exists ( p -ℚ two-ℚ) ( intro-exists ( one-ℚ⁺ , one-ℚ⁺) ( inv-tr ( is-in-lower-cut-ℝ x1) ( is-section-diff-ℚ two-ℚ p) ( p<x1))) is-inhabited-upper-cut-lim-cauchy-approximation-leq-ℝ : exists ℚ upper-cut-lim-cauchy-approximation-leq-ℝ is-inhabited-upper-cut-lim-cauchy-approximation-leq-ℝ = let open do-syntax-trunc-Prop (∃ ℚ upper-cut-lim-cauchy-approximation-leq-ℝ) x1 = map-cauchy-approximation-leq-ℝ x one-ℚ⁺ two-ℚ = one-ℚ +ℚ one-ℚ in do q , x1<q ← is-inhabited-upper-cut-ℝ x1 intro-exists ( q +ℚ two-ℚ) ( intro-exists ( one-ℚ⁺ , one-ℚ⁺) ( inv-tr ( is-in-upper-cut-ℝ x1) ( is-retraction-diff-ℚ two-ℚ q) ( x1<q))) forward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → is-in-subtype lower-cut-lim-cauchy-approximation-leq-ℝ q → exists ( ℚ) ( λ r → le-ℚ-Prop q r ∧ lower-cut-lim-cauchy-approximation-leq-ℝ r) forward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ q q<lim = let open do-syntax-trunc-Prop ( ∃ ( ℚ) ( λ r → le-ℚ-Prop q r ∧ lower-cut-lim-cauchy-approximation-leq-ℝ r)) in do (ε⁺@(ε , _) , θ⁺@(θ , _)) , q+ε+θ<xε ← q<lim let xε = map-cauchy-approximation-leq-ℝ x ε⁺ r , q+ε+θ<r , r<xε ← forward-implication ( is-rounded-lower-cut-ℝ ( map-cauchy-approximation-leq-ℝ x ε⁺) ( q +ℚ (ε +ℚ θ))) ( q+ε+θ<xε) intro-exists ( r -ℚ (ε +ℚ θ)) ( le-transpose-left-add-ℚ q (ε +ℚ θ) r q+ε+θ<r , intro-exists ( ε⁺ , θ⁺) ( inv-tr ( is-in-lower-cut-ℝ xε) ( is-section-diff-ℚ (ε +ℚ θ) r) ( r<xε))) backward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → exists ( ℚ) ( λ r → le-ℚ-Prop q r ∧ lower-cut-lim-cauchy-approximation-leq-ℝ r) → is-in-subtype lower-cut-lim-cauchy-approximation-leq-ℝ q backward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ q ∃r = let open do-syntax-trunc-Prop (lower-cut-lim-cauchy-approximation-leq-ℝ q) in do r , q<r , r<lim ← ∃r (ε⁺@(ε , _) , θ⁺@(θ , _)) , r+ε+θ<xε ← r<lim let xε = map-cauchy-approximation-leq-ℝ x ε⁺ intro-exists ( ε⁺ , θ⁺) ( le-lower-cut-ℝ ( xε) ( q +ℚ (ε +ℚ θ)) ( r +ℚ (ε +ℚ θ)) ( preserves-le-left-add-ℚ (ε +ℚ θ) q r q<r) ( r+ε+θ<xε)) is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → is-in-subtype lower-cut-lim-cauchy-approximation-leq-ℝ q ↔ exists ( ℚ) ( λ r → le-ℚ-Prop q r ∧ lower-cut-lim-cauchy-approximation-leq-ℝ r) is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ q = ( forward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ ( q) , backward-implication-is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ ( q)) forward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → is-in-subtype upper-cut-lim-cauchy-approximation-leq-ℝ q → exists ( ℚ) ( λ p → le-ℚ-Prop p q ∧ upper-cut-lim-cauchy-approximation-leq-ℝ p) forward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ q lim<q = let open do-syntax-trunc-Prop ( ∃ ( ℚ) ( λ p → le-ℚ-Prop p q ∧ upper-cut-lim-cauchy-approximation-leq-ℝ p)) in do (ε⁺@(ε , _) , θ⁺@(θ , _)) , xε<q-ε-θ ← lim<q let xε = map-cauchy-approximation-leq-ℝ x ε⁺ p , p<q-ε-θ , xε<p ← forward-implication ( is-rounded-upper-cut-ℝ xε (q -ℚ (ε +ℚ θ))) ( xε<q-ε-θ) intro-exists ( p +ℚ (ε +ℚ θ)) ( le-transpose-right-diff-ℚ p q (ε +ℚ θ) p<q-ε-θ , intro-exists ( ε⁺ , θ⁺) ( inv-tr ( is-in-upper-cut-ℝ xε) ( is-retraction-diff-ℚ (ε +ℚ θ) p) ( xε<p))) backward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → exists ( ℚ) ( λ p → le-ℚ-Prop p q ∧ upper-cut-lim-cauchy-approximation-leq-ℝ p) → is-in-subtype upper-cut-lim-cauchy-approximation-leq-ℝ q backward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ q ∃p = let open do-syntax-trunc-Prop (upper-cut-lim-cauchy-approximation-leq-ℝ q) in do p , p<q , lim<p ← ∃p (ε⁺@(ε , _) , θ⁺@(θ , _)) , xε<p-ε-θ ← lim<p let xε = map-cauchy-approximation-leq-ℝ x ε⁺ intro-exists ( ε⁺ , θ⁺) ( le-upper-cut-ℝ ( xε) ( p -ℚ (ε +ℚ θ)) ( q -ℚ (ε +ℚ θ)) ( preserves-le-left-add-ℚ (neg-ℚ (ε +ℚ θ)) p q p<q) ( xε<p-ε-θ)) is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → is-in-subtype upper-cut-lim-cauchy-approximation-leq-ℝ q ↔ exists ( ℚ) ( λ p → le-ℚ-Prop p q ∧ upper-cut-lim-cauchy-approximation-leq-ℝ p) is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ q = ( forward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ ( q) , backward-implication-is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ ( q)) lower-real-lim-cauchy-approximation-leq-ℝ : lower-ℝ l lower-real-lim-cauchy-approximation-leq-ℝ = lower-cut-lim-cauchy-approximation-leq-ℝ , is-inhabited-lower-cut-lim-cauchy-approximation-leq-ℝ , is-rounded-lower-cut-lim-cauchy-approximation-leq-ℝ upper-real-lim-cauchy-approximation-leq-ℝ : upper-ℝ l upper-real-lim-cauchy-approximation-leq-ℝ = upper-cut-lim-cauchy-approximation-leq-ℝ , is-inhabited-upper-cut-lim-cauchy-approximation-leq-ℝ , is-rounded-upper-cut-lim-cauchy-approximation-leq-ℝ abstract is-disjoint-cut-lim-cauchy-approximation-leq-ℝ : (q : ℚ) → ¬ ( is-in-subtype lower-cut-lim-cauchy-approximation-leq-ℝ q × is-in-subtype upper-cut-lim-cauchy-approximation-leq-ℝ q) is-disjoint-cut-lim-cauchy-approximation-leq-ℝ q (q<lim , lim<q) = let open do-syntax-trunc-Prop empty-Prop in do (εl⁺@(εl , _) , θl⁺@(θl , _)) , q+εl+θl<xεl ← q<lim (εu⁺@(εu , _) , θu⁺@(θu , _)) , xεu<q-εu-θu ← lim<q let xεl = map-cauchy-approximation-leq-ℝ x εl⁺ xεu = map-cauchy-approximation-leq-ℝ x εu⁺ q-εu+θl<xεu : is-in-lower-cut-ℝ xεu ((q -ℚ εu) +ℚ θl) q-εu+θl<xεu = pr2 ( is-cauchy-map-cauchy-approximation-leq-ℝ x εl⁺ εu⁺) ( (q -ℚ εu) +ℚ θl) ( inv-tr ( is-in-lower-cut-ℝ xεl) ( equational-reasoning ((q -ℚ εu) +ℚ θl) +ℚ (εl +ℚ εu) ＝ ((q +ℚ θl) -ℚ εu) +ℚ (εu +ℚ εl) by ap-add-ℚ ( right-swap-add-ℚ q (neg-ℚ εu) θl) ( commutative-add-ℚ εl εu) ＝ ((q +ℚ θl) -ℚ εu) +ℚ εu +ℚ εl by inv (associative-add-ℚ _ _ _) ＝ (q +ℚ θl) +ℚ εl by ap (_+ℚ εl) (is-section-diff-ℚ _ _) ＝ q +ℚ (θl +ℚ εl) by associative-add-ℚ _ _ _ ＝ q +ℚ (εl +ℚ θl) by ap (q +ℚ_) (commutative-add-ℚ _ _)) ( q+εl+θl<xεl)) is-disjoint-cut-ℝ ( xεu) ( q -ℚ εu) ( le-lower-cut-ℝ ( xεu) ( q -ℚ εu) ( (q -ℚ εu) +ℚ θl) ( le-right-add-rational-ℚ⁺ (q -ℚ εu) θl⁺) ( q-εu+θl<xεu) , tr ( is-in-upper-cut-ℝ xεu) ( equational-reasoning (q -ℚ (εu +ℚ θu)) +ℚ θu ＝ (q +ℚ (neg-ℚ εu +ℚ neg-ℚ θu)) +ℚ θu by ap (λ r → (q +ℚ r) +ℚ θu) (distributive-neg-add-ℚ εu θu) ＝ ((q -ℚ εu) -ℚ θu) +ℚ θu by ap (_+ℚ θu) (inv (associative-add-ℚ _ _ _)) ＝ q -ℚ εu by is-section-diff-ℚ θu _) ( le-upper-cut-ℝ ( xεu) ( q -ℚ (εu +ℚ θu)) ( (q -ℚ (εu +ℚ θu)) +ℚ θu) ( le-right-add-rational-ℚ⁺ (q -ℚ (εu +ℚ θu)) θu⁺) ( xεu<q-εu-θu))) is-located-lower-upper-cut-lim-cauchy-approximation-leq-ℝ : is-located-lower-upper-ℝ ( lower-real-lim-cauchy-approximation-leq-ℝ) ( upper-real-lim-cauchy-approximation-leq-ℝ) is-located-lower-upper-cut-lim-cauchy-approximation-leq-ℝ p q p<q = let open do-syntax-trunc-Prop ( lower-cut-lim-cauchy-approximation-leq-ℝ p ∨ upper-cut-lim-cauchy-approximation-leq-ℝ q) in do ε'⁺@(ε' , _) , 2ε'⁺<q-p ← double-le-ℚ⁺ (positive-diff-le-ℚ p q p<q) ε⁺@(ε , _) , 2ε⁺<ε'⁺ ← double-le-ℚ⁺ ε'⁺ let 2ε' = ε' +ℚ ε' 2ε = ε +ℚ ε 4ε = 2ε +ℚ 2ε xε = map-cauchy-approximation-leq-ℝ x ε⁺ map-disjunction ( lower-cut-ℝ xε (p +ℚ 2ε)) ( lower-cut-lim-cauchy-approximation-leq-ℝ p) ( upper-cut-ℝ xε (q -ℚ 2ε)) ( upper-cut-lim-cauchy-approximation-leq-ℝ q) ( intro-exists (ε⁺ , ε⁺)) ( intro-exists (ε⁺ , ε⁺)) ( is-located-lower-upper-cut-ℝ ( map-cauchy-approximation-leq-ℝ x ε⁺) ( p +ℚ 2ε) ( q -ℚ 2ε) ( le-transpose-left-add-ℚ ( p +ℚ 2ε) ( 2ε) ( q) ( inv-tr ( λ r → le-ℚ r q) ( associative-add-ℚ p 2ε 2ε ∙ commutative-add-ℚ p 4ε) ( le-transpose-right-diff-ℚ ( 4ε) ( q) ( p) ( transitive-le-ℚ ( 4ε) ( 2ε') ( q -ℚ p) ( 2ε'⁺<q-p) ( preserves-le-add-ℚ {2ε} {ε'} {2ε} {ε'} ( 2ε⁺<ε'⁺) ( 2ε⁺<ε'⁺))))))) lim-cauchy-approximation-leq-ℝ : ℝ l lim-cauchy-approximation-leq-ℝ = real-lower-upper-ℝ ( lower-real-lim-cauchy-approximation-leq-ℝ) ( upper-real-lim-cauchy-approximation-leq-ℝ) ( is-disjoint-cut-lim-cauchy-approximation-leq-ℝ) ( is-located-lower-upper-cut-lim-cauchy-approximation-leq-ℝ)
The limit satisfies the definition of a limit of a Cauchy approximation
module _ {l : Level} (x : cauchy-approximation-leq-ℝ l) where is-limit-lim-cauchy-approximation-leq-ℝ : is-limit-cauchy-approximation-Premetric-Space ( premetric-space-leq-ℝ l) ( x) ( lim-cauchy-approximation-leq-ℝ x) is-limit-lim-cauchy-approximation-leq-ℝ ε⁺@(ε , _) θ⁺@(θ , _) = let open do-syntax-trunc-Prop ( premetric-leq-ℝ ( l) ( ε⁺ +ℚ⁺ θ⁺) ( map-cauchy-approximation-leq-ℝ x ε⁺) ( lim-cauchy-approximation-leq-ℝ x)) lim = lim-cauchy-approximation-leq-ℝ x xε = map-cauchy-approximation-leq-ℝ x ε⁺ θ'⁺@(θ' , _) = left-summand-split-ℚ⁺ θ⁺ θ''⁺@(θ'' , _) = right-summand-split-ℚ⁺ θ⁺ ε+θ'+θ''=ε+θ = associative-add-ℚ _ _ _ ∙ ap (ε +ℚ_) (ap rational-ℚ⁺ (eq-add-split-ℚ⁺ θ⁺)) in do ( r , xε+ε+θ'<r , r<xε+ε+θ) ← tr ( le-ℝ (xε +ℝ real-ℚ (ε +ℚ θ'))) ( associative-add-ℝ _ _ _ ∙ ap ( xε +ℝ_) (add-real-ℚ _ _ ∙ ap real-ℚ ε+θ'+θ''=ε+θ)) ( le-left-add-real-ℝ⁺ ( xε +ℝ (real-ℚ (ε +ℚ θ'))) ( positive-real-ℚ⁺ θ''⁺)) ( q , xε-ε-θ<q , q<xε-ε-θ') ← tr ( λ y → le-ℝ y (xε -ℝ real-ℚ (ε +ℚ θ'))) ( associative-add-ℝ _ _ _ ∙ ap ( xε +ℝ_) ( inv (distributive-neg-add-ℝ _ _) ∙ ap neg-ℝ (add-real-ℚ _ _ ∙ ap real-ℚ ε+θ'+θ''=ε+θ))) ( le-diff-real-ℝ⁺ (xε -ℝ real-ℚ (ε +ℚ θ')) (positive-real-ℚ⁺ θ''⁺)) neighborhood-real-bound-each-leq-ℝ ( ε⁺ +ℚ⁺ θ⁺) ( xε) ( lim) ( leq-le-ℝ ( xε) ( lim +ℝ real-ℚ (ε +ℚ θ)) ( le-transpose-left-diff-ℝ ( xε) ( real-ℚ (ε +ℚ θ)) ( lim) ( transitive-le-ℝ ( xε -ℝ real-ℚ (ε +ℚ θ)) ( real-ℚ q) ( lim) ( le-real-is-in-lower-cut-ℚ ( q) ( lim) ( intro-exists ( ε⁺ , θ'⁺) ( transpose-is-in-lower-cut-diff-ℝ xε (ε +ℚ θ') q q<xε-ε-θ'))) ( le-real-is-in-upper-cut-ℚ q (xε -ℝ real-ℚ (ε +ℚ θ)) xε-ε-θ<q)))) ( leq-le-ℝ ( lim) ( xε +ℝ real-ℚ (ε +ℚ θ)) ( transitive-le-ℝ ( lim) ( real-ℚ r) ( xε +ℝ real-ℚ (ε +ℚ θ)) ( le-real-is-in-lower-cut-ℚ r (xε +ℝ real-ℚ (ε +ℚ θ)) r<xε+ε+θ) ( le-real-is-in-upper-cut-ℚ ( r) ( lim) ( intro-exists ( ε⁺ , θ'⁺) ( transpose-is-in-upper-cut-add-ℝ xε (ε +ℚ θ') r xε+ε+θ'<r))))) is-convergent-cauchy-approximation-leq-ℝ : is-convergent-cauchy-approximation-Metric-Space ( metric-space-leq-ℝ l) ( x) is-convergent-cauchy-approximation-leq-ℝ = ( lim-cauchy-approximation-leq-ℝ x , is-limit-lim-cauchy-approximation-leq-ℝ)
The standard metric space of real numbers is complete
is-complete-metric-space-leq-ℝ : (l : Level) → is-complete-Metric-Space (metric-space-leq-ℝ l) is-complete-metric-space-leq-ℝ _ = is-convergent-cauchy-approximation-leq-ℝ complete-metric-space-leq-ℝ : (l : Level) → Complete-Metric-Space (lsuc l) l pr1 (complete-metric-space-leq-ℝ l) = metric-space-leq-ℝ l pr2 (complete-metric-space-leq-ℝ l) = is-complete-metric-space-leq-ℝ l
Limits of Cauchy sequences in ℝ
cauchy-sequence-ℝ : (l : Level) → UU (lsuc l) cauchy-sequence-ℝ l = cauchy-sequence-Complete-Metric-Space (complete-metric-space-leq-ℝ l) lim-cauchy-sequence-ℝ : {l : Level} → cauchy-sequence-ℝ l → ℝ l lim-cauchy-sequence-ℝ {l} = limit-cauchy-sequence-Complete-Metric-Space (complete-metric-space-leq-ℝ l)
Recent changes
- 2025-04-02. Louis Wasserman. Cauchy completeness of the Dedekind real numbers (#1343).