Closed intervals in the rational numbers
Content created by Louis Wasserman and Fredrik Bakke.
Created on 2025-03-25.
Last modified on 2025-10-07.
module elementary-number-theory.closed-intervals-rational-numbers where
Imports
open import elementary-number-theory.decidable-total-order-rational-numbers open import elementary-number-theory.inequality-rational-numbers open import elementary-number-theory.maximum-rational-numbers open import elementary-number-theory.minimum-rational-numbers open import elementary-number-theory.multiplication-positive-rational-numbers open import elementary-number-theory.multiplication-rational-numbers open import elementary-number-theory.negative-rational-numbers open import elementary-number-theory.nonnegative-rational-numbers open import elementary-number-theory.positive-and-negative-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.dependent-pair-types open import foundation.identity-types open import foundation.injective-maps open import foundation.propositions open import foundation.sets open import foundation.subtypes open import foundation.universe-levels open import order-theory.closed-interval-preserving-maps-posets open import order-theory.closed-intervals-posets open import order-theory.decidable-total-orders
Idea
An closed interval¶ in the rational numbers is a closed interval in the poset of rational numbers. Note, in particular, that we thus consider closed intervals to be inhabited by convention.
Definition
closed-interval-ℚ : UU lzero closed-interval-ℚ = closed-interval-Poset ℚ-Poset lower-bound-closed-interval-ℚ : closed-interval-ℚ → ℚ lower-bound-closed-interval-ℚ = lower-bound-closed-interval-Poset ℚ-Poset upper-bound-closed-interval-ℚ : closed-interval-ℚ → ℚ upper-bound-closed-interval-ℚ = upper-bound-closed-interval-Poset ℚ-Poset
Properties
The subtype associated with a closed interval
subtype-closed-interval-ℚ : closed-interval-ℚ → subtype lzero ℚ subtype-closed-interval-ℚ = subtype-closed-interval-Poset ℚ-Poset is-in-closed-interval-ℚ : closed-interval-ℚ → ℚ → UU lzero is-in-closed-interval-ℚ [a,b] = is-in-subtype (subtype-closed-interval-ℚ [a,b])
The property of being above or below a closed interval
is-below-prop-closed-interval-ℚ : closed-interval-ℚ → subtype lzero ℚ is-below-prop-closed-interval-ℚ ((a , _) , _) b = le-ℚ-Prop b a is-above-prop-closed-interval-ℚ : closed-interval-ℚ → subtype lzero ℚ is-above-prop-closed-interval-ℚ ((_ , a) , _) b = le-ℚ-Prop a b
The width of a closed interval
nonnegative-width-closed-interval-ℚ : closed-interval-ℚ → ℚ⁰⁺ nonnegative-width-closed-interval-ℚ ((a , b) , a≤b) = nonnegative-diff-leq-ℚ a b a≤b width-closed-interval-ℚ : closed-interval-ℚ → ℚ width-closed-interval-ℚ [a,b] = rational-ℚ⁰⁺ (nonnegative-width-closed-interval-ℚ [a,b])
Important ranges
zero-zero-closed-interval-ℚ : closed-interval-ℚ zero-zero-closed-interval-ℚ = ((zero-ℚ , zero-ℚ) , refl-leq-ℚ zero-ℚ) one-one-closed-interval-ℚ : closed-interval-ℚ one-one-closed-interval-ℚ = ((one-ℚ , one-ℚ) , refl-leq-ℚ one-ℚ)
The map from closed intervals to their subtypes is injective
is-injective-subtype-closed-interval-ℚ : is-injective subtype-closed-interval-ℚ is-injective-subtype-closed-interval-ℚ = is-injective-subtype-closed-interval-Poset ℚ-Poset
Characterization of equality
eq-closed-interval-ℚ : ([a,b] [c,d] : closed-interval-ℚ) → ( lower-bound-closed-interval-ℚ [a,b] = lower-bound-closed-interval-ℚ [c,d]) → ( upper-bound-closed-interval-ℚ [a,b] = upper-bound-closed-interval-ℚ [c,d]) → [a,b] = [c,d] eq-closed-interval-ℚ = eq-closed-interval-Poset ℚ-Poset set-closed-interval-ℚ : Set lzero set-closed-interval-ℚ = set-closed-interval-Poset ℚ-Poset
Unordered intervals
unordered-closed-interval-ℚ : ℚ → ℚ → closed-interval-ℚ unordered-closed-interval-ℚ a b = ( (min-ℚ a b , max-ℚ a b) , min-leq-max-Decidable-Total-Order ℚ-Decidable-Total-Order a b) abstract unordered-closed-interval-leq-ℚ : (p q : ℚ) → (p≤q : leq-ℚ p q) → unordered-closed-interval-ℚ p q = ((p , q) , p≤q) unordered-closed-interval-leq-ℚ p q p≤q = eq-closed-interval-ℚ _ _ ( left-leq-right-min-ℚ p q p≤q) ( left-leq-right-max-ℚ p q p≤q) unordered-closed-interval-leq-ℚ' : (p q : ℚ) → (q≤p : leq-ℚ q p) → unordered-closed-interval-ℚ p q = ((q , p) , q≤p) unordered-closed-interval-leq-ℚ' p q q≤p = eq-closed-interval-ℚ _ _ ( right-leq-left-min-ℚ p q q≤p) ( right-leq-left-max-ℚ p q q≤p)
The bounds of a closed interval are elements
lower-bound-is-in-closed-interval-ℚ : ([a,b] : closed-interval-ℚ) → is-in-closed-interval-ℚ [a,b] (lower-bound-closed-interval-ℚ [a,b]) lower-bound-is-in-closed-interval-ℚ = lower-bound-is-in-closed-interval-Poset ℚ-Poset upper-bound-is-in-closed-interval-ℚ : ([a,b] : closed-interval-ℚ) → is-in-closed-interval-ℚ [a,b] (upper-bound-closed-interval-ℚ [a,b]) upper-bound-is-in-closed-interval-ℚ = upper-bound-is-in-closed-interval-Poset ℚ-Poset
Recent changes
- 2025-10-07. Louis Wasserman. Multiplication of interior intervals in the rational numbers (#1561).
- 2025-10-04. Louis Wasserman. Split out operations on positive rational numbers (#1562).
- 2025-10-02. Louis Wasserman. Intervals and interval arithmetic (#1497).
- 2025-04-25. Louis Wasserman. Arithmetic-geometric mean inequality on the rational numbers (#1406).
- 2025-03-25. Louis Wasserman and Fredrik Bakke. Multiplication on closed intervals of rational numbers (#1351).