ZeroCompositionLaw

formal source

  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
  65    (zc : ZeroCompositionLaw) (t : ℝ) :
  66    zc.H t = 1 ↔ t = 0 := by
  67  rw [zeroCompositionLaw_forces_cosh zc t]
  68  exact cosh_eq_one_iff t
  69
  70/-- A zero-composition law forces the corresponding point onto the critical
  71line once the observable attains its minimum at that point's deviation. -/
  72theorem zeroCompositionLaw_forces_eta_zero
  73    (zc : ZeroCompositionLaw) (ρ : ℂ) :
  74    zc.H (zeroDeviation ρ) = 1 ↔ OnCriticalLine ρ := by