optimal_transfer

formal source

  53open RRF.Core
  54
  55/-- If two channels are quality-equivalent, their optimal states coincide. -/
  56theorem optimal_transfer {Obs₁ Obs₂ : Type*}
  57    {C₁ : DisplayChannel State Obs₁}
  58    {C₂ : DisplayChannel State Obs₂}
  59    (heq : QualityEquiv C₁ C₂)
  60    (x : State) :
  61    C₁.isOptimal x ↔ C₂.isOptimal x :=
  62  QualityEquiv.optimal_iff heq x
  63
  64/-- If C₂.quality = g ∘ C₁.quality for monotone g, they induce the same order. -/
  65theorem channel_mono_transfer {Obs : Type*}
  66    {C₁ C₂ : DisplayChannel State Obs}
  67    {g : ℝ → ℝ} (hg : Monotone g)
  68    (hqual : ∀ o, C₂.quality o = g (C₁.quality o))
  69    (heq : C₁.observe = C₂.observe) :
  70    ∀ x y, C₁.stateQuality x ≤ C₁.stateQuality y →
  71           C₂.stateQuality x ≤ C₂.stateQuality y := by
  72  intro x y h
  73  simp only [DisplayChannel.stateQuality, Function.comp_apply] at *
  74  rw [← heq, hqual, hqual]
  75  exact hg h
  76
  77/-! ## Strain Ordering Lemmas -/
  78
  79/-- Strain ordering is transitive. -/
  80theorem strain_order_trans {S : StrainFunctional State}
  81    {x y z : State}
  82    (hxy : S.weaklyBetter x y) (hyz : S.weaklyBetter y z) :
  83    S.weaklyBetter x z :=
  84  le_trans hxy hyz
  85
  86/-- Strict strain ordering is transitive. -/