Exact area of half-open rectangles and unit-cell unions

Mathematical statement

In an equal-unit orthonormal frame, a half-open rectangle [x0,x1)×[y0,y1) with x0≤x1 and y0≤y1 has area (x1−x0)(y1−y0). Zero width or height makes it empty. An integer unit cell [i,i+1)×[j,j+1) has area one, and a finite union of distinct such cells has area equal to their count. For two such rectangles, area(A∪B)=area(A)+area(B)−area(A∩B).

Representative formula (exact TeX)

A(w,h)

Assumptions

• Exercises use nonnegative integer dimensions or ordered integer endpoints, with an explicit half-open convention and equal-unit frame.
• The intersection width and height are each max(0,min(upper endpoints)−max(lower endpoints)). Disjoint or merely touching rectangles have intersection area zero.
• Deduplicate unit cells by coordinates; repeated listings do not increase area. The answer unit is the square of the selected length unit.

Symbol scope

• A — scoped rectangle-area operation
• w — nonnegative width
• h — nonnegative height

Intuition, definition, and misconceptions

Area is not perimeter. Adding two rectangle areas counts their overlap twice, so subtract the intersection once. The union of two identical rectangles has the area of just one rectangle.

Finite unit-cell enumeration gives a separate exact computation for the integer-coordinate cases. An empty rectangle or empty cell list has area zero; repeated cell representations are counted only once.

The candidate is deliberately limited to these finite rectangle and cell constructions. It does not establish general measure or integration theory for arbitrary plane regions.

Common errors include adding overlapping areas without correction, confusing area with perimeter, or counting duplicate cells as additional region.

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

• Euclidean length — Fix the length unit and orthonormal frame.
• Arithmetic operations — Use exact width-times-height products and addition or subtraction.
• Coordinate plane — Specify rectangle endpoints and unit-cell locations.
• Set operations — Use union, intersection, and deduplication of representations.

Verification boundary

The finite scoped areas can be recomputed exactly; general Lebesgue area has no newly attached Evidence.

not_formalized

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

Next obligation

Bind pinned two-dimensional box volume and the relevant finite-union identities.

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.