IndisputableMonolith.Verification.DAlembertSymmetryCert

   1import Mathlib
   2import IndisputableMonolith.Cost.FunctionalEquation
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace DAlembertSymmetry
   7
   8open IndisputableMonolith.Cost.FunctionalEquation
   9
  10/-!
  11# D'Alembert Symmetry Certificate
  12
  13This certificate packages the proof that the d'Alembert functional equation
  14implies even symmetry:
  15
  16If H satisfies:
  17- H(0) = 1
  18- H(t+u) + H(t-u) = 2·H(t)·H(u) for all t, u
  19
  20Then H is an even function: H(-t) = H(t) for all t.
  21
  22## Why this matters for the certificate chain
  23
  24The d'Alembert functional equation is the characteristic equation of cosh and cos.
  25This certificate proves:
  26
  271. **Symmetry is forced**: The d'Alembert equation alone (with H(0) = 1) implies
  28   H is even - no additional symmetry assumption is needed
  29
  302. **Foundation for ODE analysis**: Even symmetry at differentiable points implies
  31   H'(0) = 0, which combined with H(0) = 1 gives the correct initial conditions
  32   for the ODE H'' = H
  33
  343. **Cost uniqueness connection**: This is a key step in proving Jlog (and hence J)
  35   is uniquely determined
  36
  37## Proven Results
  38
  391. `dAlembert_even`: d'Alembert equation with H(0) = 1 implies H is even
  402. `even_deriv_at_zero`: even differentiable functions have H'(0) = 0
  41-/
  42
  43structure DAlembertSymmetryCert where
  44  deriving Repr
  45
  46/-- Verification predicate: d'Alembert functional equation implies symmetry.
  47
  48This certifies:
  491. d'Alembert with H(0) = 1 implies H is even (H(-t) = H(t))
  502. Even + differentiable at 0 implies H'(0) = 0
  51-/
  52@[simp] def DAlembertSymmetryCert.verified (_c : DAlembertSymmetryCert) : Prop :=
  53  -- 1) d'Alembert functional equation implies even symmetry
  54  (∀ H : ℝ → ℝ,
  55    H 0 = 1 →
  56    (∀ t u, H (t+u) + H (t-u) = 2 * H t * H u) →
  57    Function.Even H) ∧
  58  -- 2) Even + differentiable at 0 implies derivative is 0
  59  (∀ H : ℝ → ℝ,
  60    Function.Even H →
  61    DifferentiableAt ℝ H 0 →
  62    deriv H 0 = 0)
  63
  64/-- Top-level theorem: the certificate verifies. -/
  65@[simp] theorem DAlembertSymmetryCert.verified_any (c : DAlembertSymmetryCert) :
  66    DAlembertSymmetryCert.verified c := by
  67  constructor
  68  · -- d'Alembert implies even
  69    intro H h_one h_dAlem
  70    exact dAlembert_even H h_one h_dAlem
  71  · -- Even + differentiable implies H'(0) = 0
  72    intro H h_even h_diff
  73    exact even_deriv_at_zero H h_even h_diff
  74
  75end DAlembertSymmetry
  76end Verification
  77end IndisputableMonolith
  78