`dAlembert_contDiff_top`

proved

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module: IndisputableMonolith.Cost.AczelProof
domain: Cost
line: 393 · github
papers citing: none yet

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IndisputableMonolith.Cost.AczelProof on GitHub at line 393.

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formal source

 390      have := is_const_of_deriv_eq_zero hDiff h_H'_zero t 0
 391      simp [h_one] at this; exact this
 392
 393private theorem dAlembert_contDiff_top (H : ℝ → ℝ)
 394    (h_one : H 0 = 1) (h_cont : Continuous H)
 395    (h_dAl : ∀ t u, H (t + u) + H (t - u) = 2 * H t * H u) :
 396    ContDiff ℝ ⊤ H := by
 397  rcases dAlembert_classification H h_one h_cont h_dAl with h | ⟨α, h⟩ | ⟨α, h⟩
 398  · rw [show H = fun _ => (1 : ℝ) from funext h]
 399    exact contDiff_const
 400  · rw [show H = fun t => Real.cosh (α * t) from funext h]
 401    exact Real.contDiff_cosh.comp (contDiff_const.mul contDiff_id)
 402  · rw [show H = fun t => Real.cos (α * t) from funext h]
 403    exact Real.contDiff_cos.comp (contDiff_const.mul contDiff_id)
 404
 405/-! ## Instance -/
 406
 407instance : AczelSmoothnessPackage where
 408  smooth_of_dAlembert H h_one h_cont h_dAl := dAlembert_contDiff_top H h_one h_cont h_dAl
 409
 410end
 411
 412end FunctionalEquation
 413end Cost
 414end IndisputableMonolith

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