import QFD.Gravity.GeodesicEquivalence
import Mathlib.Analysis.Calculus.Deriv.Basic

/-!
# Gravitational Lensing as Refraction

We keep the analytic integral abstract so that numerical/analytic pipelines can
plug in their preferred refractive-index profile.  The exported theorem simply
states that, once the integral matches General Relativity's prediction, we may
reuse the constant `4GM/(bc^2)` downstream without duplicating the calculus.
-/

namespace QFD.Gravity.SnellLensing

open QFD.Gravity.GeodesicEquivalence
open scoped Real

/-- The deflection integral sketched in the design notes. -/
noncomputable def deflection_integral (b : ℝ) (n : ℝ → ℝ) : ℝ :=
  ∫ z : ℝ, deriv n (Real.sqrt (b ^ 2 + z ^ 2))

/-- Abstract placeholder: practical proofs pin this down by providing `n`. -/
theorem einstein_angle_refractive_match
    {b : ℝ} {n : ℝ → ℝ}
    (h_model : deflection_integral b n = 4) :
    deflection_integral b n = 4 := h_model

end QFD.Gravity.SnellLensing