`symm`

proved

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module: IndisputableMonolith.RRF.Core.DisplayChannel
domain: RRF
line: 61 · github
papers citing: none yet

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IndisputableMonolith.RRF.Core.DisplayChannel on GitHub at line 61.

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formal source

  58theorem refl (C : DisplayChannel State Obs₁) : QualityEquiv C C :=
  59  fun _ _ => Iff.rfl
  60
  61theorem symm {C₁ : DisplayChannel State Obs₁} {C₂ : DisplayChannel State Obs₂}
  62    (h : QualityEquiv C₁ C₂) : QualityEquiv C₂ C₁ :=
  63  fun x y => (h x y).symm
  64
  65theorem trans {C₁ : DisplayChannel State Obs₁}
  66    {C₂ : DisplayChannel State Obs₂}
  67    {C₃ : DisplayChannel State Obs₃}
  68    (h₁₂ : QualityEquiv C₁ C₂) (h₂₃ : QualityEquiv C₂ C₃) : QualityEquiv C₁ C₃ :=
  69  fun x y => (h₁₂ x y).trans (h₂₃ x y)
  70
  71/-- Quality-equivalent channels have the same optimal states. -/
  72theorem optimal_iff {C₁ : DisplayChannel State Obs₁} {C₂ : DisplayChannel State Obs₂}
  73    (heq : QualityEquiv C₁ C₂) (x : State) :
  74    C₁.isOptimal x ↔ C₂.isOptimal x :=
  75  ⟨fun h1 y => (heq x y).mp (h1 y), fun h2 y => (heq x y).mpr (h2 y)⟩
  76
  77end QualityEquiv
  78
  79end RRF.Core
  80end IndisputableMonolith

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