ode_neg_zero_uniqueness

formal source

  73/-- **Zero uniqueness for `f'' = -f`**: if `f(0)=0` and `f'(0)=0`,
  74then `f ≡ 0`. Proof by conservation of the energy
  75`E(t) = f(t)^2 + f'(t)^2`. -/
  76theorem ode_neg_zero_uniqueness (f : ℝ → ℝ)
  77    (h_diff2 : ContDiff ℝ 2 f)
  78    (h_ode : ∀ t, deriv (deriv f) t = -(f t))
  79    (h_f0 : f 0 = 0) (h_f'0 : deriv f 0 = 0) :
  80    ∀ t, f t = 0 := by
  81  have h_d1 : Differentiable ℝ f := h_diff2.differentiable (by decide : (2 : WithTop ℕ∞) ≠ 0)
  82  have hCD1 : ContDiff ℝ 1 (deriv f) := by
  83    rw [show (2 : WithTop ℕ∞) = 1 + 1 from rfl] at h_diff2
  84    rw [contDiff_succ_iff_deriv] at h_diff2
  85    exact h_diff2.2.2
  86  have h_dd : Differentiable ℝ (deriv f) :=
  87    hCD1.differentiable (by decide : (1 : WithTop ℕ∞) ≠ 0)
  88  have hE_deriv_zero : ∀ s, deriv (fun t => f t ^ 2 + deriv f t ^ 2) s = 0 := by
  89    intro s
  90    have h1 : HasDerivAt (fun x => f x ^ 2 + deriv f x ^ 2)
  91        (↑2 * f s ^ (2 - 1) * deriv f s + ↑2 * deriv f s ^ (2 - 1) * deriv (deriv f) s) s :=
  92      ((h_d1 s).hasDerivAt.pow 2).add ((h_dd s).hasDerivAt.pow 2)
  93    have h2 := h1.deriv
  94    rw [h_ode s] at h2
  95    push_cast at h2
  96    simp only [pow_one] at h2
  97    linarith
  98  have hE_eq := is_const_of_deriv_eq_zero
  99    (show Differentiable ℝ (fun t => f t ^ 2 + deriv f t ^ 2) from
 100      (h_d1.pow 2).add (h_dd.pow 2))
 101    hE_deriv_zero
 102  intro t
 103  have hE0 : f 0 ^ 2 + deriv f 0 ^ 2 = 0 := by rw [h_f0, h_f'0]; ring
 104  have hEt := hE_eq t 0
 105  simp only [hE0] at hEt
 106  nlinarith [sq_nonneg (f t), sq_nonneg (deriv f t)]