# The decidable total order of natural numbers

Content created by Fredrik Bakke, Egbert Rijke and Victor Blanchi.

Created on 2023-05-10.

module elementary-number-theory.decidable-total-order-natural-numbers where

Imports
open import elementary-number-theory.inequality-natural-numbers

open import foundation.dependent-pair-types
open import foundation.propositional-truncations
open import foundation.universe-levels

open import order-theory.decidable-total-orders
open import order-theory.total-orders


## Idea

The type of natural numbers equipped with its standard ordering relation forms a decidable total order.

## Definition

is-total-leq-ℕ : is-total-Poset ℕ-Poset
is-total-leq-ℕ n m = unit-trunc-Prop (linear-leq-ℕ n m)

ℕ-Total-Order : Total-Order lzero lzero
pr1 ℕ-Total-Order = ℕ-Poset
pr2 ℕ-Total-Order = is-total-leq-ℕ

ℕ-Decidable-Total-Order : Decidable-Total-Order lzero lzero
pr1 ℕ-Decidable-Total-Order = ℕ-Poset
pr1 (pr2 ℕ-Decidable-Total-Order) = is-total-leq-ℕ
pr2 (pr2 ℕ-Decidable-Total-Order) = is-decidable-leq-ℕ