cosh_eq_one_iff

formal source

  31
  32noncomputable section
  33
  34private theorem cosh_eq_one_iff (t : ℝ) : Real.cosh t = 1 ↔ t = 0 := by
  35  constructor
  36  · intro h
  37    by_contra hne
  38    have hgt : 1 < Real.cosh t := Real.one_lt_cosh.mpr hne
  39    linarith
  40  · intro h
  41    simp [h]
  42
  43/-- The abstract zero-location composition law needed by Vector C. -/
  44structure ZeroCompositionLaw where
  45  H : ℝ → ℝ
  46  H_zero : H 0 = 1
  47  continuous : Continuous H
  48  dAlembert : ∀ t u : ℝ, H (t + u) + H (t - u) = 2 * H t * H u
  49  curvature : deriv (deriv H) 0 = 1
  50  smooth_hyp : dAlembert_continuous_implies_smooth_hypothesis H
  51  ode_hyp : dAlembert_to_ODE_hypothesis H
  52  cont_hyp : ode_regularity_continuous_hypothesis H
  53  diff_hyp : ode_regularity_differentiable_hypothesis H
  54  bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H
  55
  56/-- Any instantiated zero-composition law is forced to be `cosh`. -/
  57theorem zeroCompositionLaw_forces_cosh (zc : ZeroCompositionLaw) :
  58    ∀ t : ℝ, zc.H t = Real.cosh t :=
  59  dAlembert_cosh_solution zc.H zc.H_zero zc.continuous zc.dAlembert
  60    zc.curvature zc.smooth_hyp zc.ode_hyp zc.cont_hyp zc.diff_hyp
  61    zc.bootstrap_hyp
  62
  63/-- Consequently, the minimum value `1` occurs exactly at `t = 0`. -/
  64theorem zeroCompositionLaw_forces_unique_minimum