Usual order on the real numbers

Mathematical statement

On R, the usual total order ≤ is reflexive, antisymmetric, transitive, and comparable for every pair. Write a<b when a≤b and a≠b. For real a,b,c, a<b exactly when a+c<b+c. A finite nonempty real set has a minimum and a maximum; a closed interval [l,u] with l≤u includes both endpoints.

Representative formula (exact TeX)

a\in R

Assumptions

• Only the usual total order on real numbers is considered, not arbitrary relations or an assumed order on complex numbers.
• Exercises use safe integers or rational numbers with positive denominators. For p/q and r/s with q,s>0, compare ps and rq.
• Finite extrema require a nonempty set, and a closed interval has lower endpoint no greater than its upper endpoint. The representative formula only marks the real domain.

Symbol scope

• a — real value to be compared
• R — real domain

Intuition, definition, and misconceptions

Exactly one of a<b, a=b, and a>b holds. Multiplication by a positive number preserves a strict inequality; multiplication by a negative number reverses it. Positive denominators therefore permit exact cross-product comparison.

The minimum and maximum of a finite nonempty set do not depend on its listing order. An empty set has no such extrema; a singleton has equal minimum and maximum.

A closed interval includes its endpoints, and [l,l] is a singleton. Adding the same c to both compared numbers preserves their order; changing only this common addend is not a substantively different exercise.

Common errors include failing to reverse an inequality after multiplication by a negative number, excluding closed endpoints, or assigning extrema to an empty set.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — The domain of the order is explicitly the real numbers.
• Rational number — Use exact fractional representations and the positive-denominator convention.

Verification boundary

Finite integer and rational instances can be recomputed exactly; no new formal Evidence for general order theory or translation invariance is attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind the pinned Mathlib real linear order, monotonicity of addition, and extrema of finite nonempty sets.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.