Geometry Tetrahedron Realization Sq Edge Of Points Nonneg

Nonnegative squared edges

A tetrahedron is a solid with four triangular faces, six edges, and four vertices. In ordinary Euclidean three-dimensional space, each edge is a straight segment between two vertices. Its squared length is the square of the distance between those two points. Since a square of a real number is always zero or positive, the squared length of every edge is nonnegative. This is a basic fact of geometry, not a deep theorem.

The framework's machine-checked library of formal theorems records this fact as a named declaration, sqEdgeOfPoints_nonneg. The declaration states that for any realized tetrahedron, meaning four points in Euclidean 3-space that are affinely independent, and for any of the six edges, the squared edge length is greater than or equal to zero. The proof is one line: it unfolds the definition of squared distance and applies the standard fact that a square is nonnegative.

The declaration does not claim that the squared edge lengths are positive. A degenerate tetrahedron, where four points lie in a plane or a line, can have a zero squared edge length. The declaration also does not claim that any six nonnegative numbers form a tetrahedron; that would require the Cayley-Menger conditions, which are separate. It only establishes the lower bound for edges that come from actual points.

This result is a building block, not a destination. It feeds into later proofs about dihedral angles and positive volume, where the nonnegativity of squared edges is a necessary hypothesis. The declaration makes explicit what a reader might take for granted, so that subsequent arguments can rely on it without re-deriving it each time.

/-- Squared edge lengths from points are nonnegative. -/
theorem sqEdgeOfPoints_nonneg (T : RealizedTet) (e : Fin 6) :
    0 ≤ sqEdgeOfPoints T e := by
  unfold sqEdgeOfPoints vertexSqDist
  exact sq_nonneg _

/-- Squared edge lengths from points are nonnegative. -/
theorem sqEdgeOfPoints_nonneg (T : RealizedTet) (e : Fin 6) :
    0 ≤ sqEdgeOfPoints T e := by
  unfold sqEdgeOfPoints vertexSqDist
  exact sq_nonneg _

What this page does not claim

The squared edge lengths are strictly positive for all realized tetrahedra. Any six nonnegative numbers are realizable as squared edge lengths of a tetrahedron. The declaration establishes anything about the volume or dihedral angles of the tetrahedron.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Geometry/TetrahedronRealization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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