Geometry Dihedral Derivatives Has Deriv At Arccos Comp

The arccos chain rule

In calculus, the arccosine function, written arccos, is the inverse of the cosine function on the interval from 0 to π. For a number c between -1 and 1, arccos(c) is the angle whose cosine is c. The derivative of arccos is -1 / sqrt(1 - c^2), a formula found in any standard calculus text. The theorem hasDerivAt_arccos_comp in the framework's machine-checked library of formal theorems states the chain rule version of this fact: if a function f has derivative f' at a point x, and f(x) is not equal to -1 or 1, then the composite function arccos(f(t)) has derivative -(1 / sqrt(1 - f(x)^2)) * f' at x.

The condition that f(x) is not -1 or 1 is not a technicality. At those two values, the cosine is at its extremes, and the arccos graph has a vertical tangent, meaning the derivative is infinite. The theorem explicitly excludes those endpoints, so it applies only when the cosine value is strictly inside the open interval (-1, 1). This is the standard regularity condition for the arccos chain rule to hold as a finite derivative.

In Recognition Science, this theorem is the analytic interface for a geometric computation. The framework models a tetrahedron's dihedral angle as arccos of a cosine value derived from the Cayley-Menger cofactor formula. The theorem hasDerivAt_arccos_comp (a machine-checked proof that the chain rule applies) isolates the calculus step: once the derivative of the cosine is known, the derivative of the angle follows by this rule. The remaining work, computing the cosine derivative from edge lengths, belongs to a separate algebraic layer.

The theorem does not claim that any particular cosine derivative exists. It takes that derivative as a hypothesis. It also does not handle the endpoint cases where the cosine equals 1 or -1, which correspond to degenerate or flat tetrahedra. The framework provides separate theorems for those endpoint hypotheses, but the chain rule itself stays within the interior range.

/-- Generic derivative of `arccos ∘ f`. -/
theorem hasDerivAt_arccos_comp
    {f : ℝ → ℝ} {f' x : ℝ}
    (hf : HasDerivAt f f' x)
    (hm : f x ≠ -1) (hp : f x ≠ 1) :
    HasDerivAt (fun t : ℝ => Real.arccos (f t))
      (-(1 / Real.sqrt (1 - (f x) ^ 2)) * f') x := by
  have hacos := Real.hasDerivAt_arccos hm hp
  simpa [mul_comm, mul_left_comm, mul_assoc] using hacos.comp x hf

/-- Generic derivative of `arccos ∘ f`. -/
theorem hasDerivAt_arccos_comp
    {f : ℝ → ℝ} {f' x : ℝ}
    (hf : HasDerivAt f f' x)
    (hm : f x ≠ -1) (hp : f x ≠ 1) :
    HasDerivAt (fun t : ℝ => Real.arccos (f t))
      (-(1 / Real.sqrt (1 - (f x) ^ 2)) * f') x := by
  have hacos := Real.hasDerivAt_arccos hm hp
  simpa [mul_comm, mul_left_comm, mul_assoc] using hacos.comp x hf

What this page does not claim

The theorem does not prove that any particular cosine derivative exists. The theorem does not cover the endpoint cases where the cosine equals 1 or -1. The theorem does not derive the Cayley-Menger cofactor formula itself.

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

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