zeroCompositionLaw_forces_cosh

formal source

  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
  75  constructor
  76  · intro h
  77    have hz : zeroDeviation ρ = 0 :=
  78      (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mp h
  79    exact (zeroDeviation_eq_zero_iff_on_critical_line ρ).mp hz
  80  · intro h
  81    have hz : zeroDeviation ρ = 0 :=
  82      (zeroDeviation_eq_zero_iff_on_critical_line ρ).mpr h
  83    exact (zeroCompositionLaw_forces_unique_minimum zc (zeroDeviation ρ)).mpr hz
  84
  85/-- A concrete Vector C witness at a specific complex point. -/
  86structure ZeroCompositionWitness (ρ : ℂ) where
  87  law : ZeroCompositionLaw