Geometry Dihedral Derivatives Arccos Endpoint Hypotheses Of Interior

The endpoint condition

The arccos function, the inverse of the cosine, is not differentiable at its endpoints. At an input of exactly 1, where the angle is 0 degrees, and at an input of exactly -1, where the angle is 180 degrees, the function's graph has a vertical tangent, so its derivative is infinite. A derivative of a dihedral angle, the angle between two faces of a tetrahedron, therefore needs a condition: the cosine of that angle must stay strictly between -1 and 1.

The machine-checked theorem arccos_endpoint_hypotheses_of_interior states this condition in a precise form. It says that if a real number c is greater than -1 and less than 1, then c is not equal to -1 and not equal to 1. This is a simple logical fact, but it is the exact hypothesis the chain rule for arccos requires. The theorem is proved in the framework's machine-checked library of formal theorems, and it is used to justify that the derivative of a dihedral angle along a path is valid whenever the angle stays in the interior of its range.

The theorem does not compute any derivative itself. It does not say how the cosine of a dihedral angle changes, nor does it provide the formula for the angle's derivative. It only supplies the endpoint condition that the derivative formula needs. The actual derivative, which involves a square root in the denominator, is established by a separate theorem that assumes this condition as a hypothesis.

In the framework's geometry, this lemma is a small but necessary step. It lets a downstream module, which computes how a dihedral angle changes as the edge lengths of a tetrahedron vary, assume that the angle is not degenerate. Without this condition, the derivative formula would be undefined. The theorem is a guardrail that keeps the calculus valid.

/-- Interior range is enough to satisfy the endpoint hypotheses for arccos. -/
theorem arccos_endpoint_hypotheses_of_interior {c : ℝ}
    (hlo : -1 < c) (hhi : c < 1) : c ≠ -1 ∧ c ≠ 1 := by
  constructor
  · intro h
    linarith
  · intro h
    linarith

What this page does not claim

This theorem does not compute the derivative of a dihedral angle. This theorem does not prove that a dihedral angle is never 0 or 180 degrees for a realized tetrahedron.

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

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