`id`

definition

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module: IndisputableMonolith.RRF.Core.Octave
domain: RRF
line: 62 · github
papers citing: none yet

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IndisputableMonolith.RRF.Core.Octave on GitHub at line 62.

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formal source

  59namespace OctaveMorphism
  60
  61/-- Identity morphism. -/
  62def id (O : Octave) : OctaveMorphism O O where
  63  map := fun x => x
  64  preserves_order := fun _ _ h => h
  65
  66/-- Composition of morphisms. -/
  67def comp {O₁ O₂ O₃ : Octave}
  68    (g : OctaveMorphism O₂ O₃) (f : OctaveMorphism O₁ O₂) : OctaveMorphism O₁ O₃ where
  69  map := g.map ∘ f.map
  70  preserves_order := fun x y h => g.preserves_order _ _ (f.preserves_order x y h)
  71
  72/-- Morphisms preserve equilibria (if target has NonNeg strain). -/
  73theorem preserves_equilibria {O₁ O₂ : Octave}
  74    [inst : StrainFunctional.NonNeg O₂.strain]
  75    (f : OctaveMorphism O₁ O₂)
  76    (x : O₁.State) (hx : O₁.strain.isBalanced x)
  77    (hf : O₂.strain.J (f.map x) ≤ O₁.strain.J x) :
  78    O₂.strain.isBalanced (f.map x) := by
  79  simp only [StrainFunctional.isBalanced] at *
  80  rw [hx] at hf
  81  have h₁ := @StrainFunctional.NonNeg.nonneg _ O₂.strain inst (f.map x)
  82  linarith
  83
  84end OctaveMorphism
  85
  86/-- Two octaves are equivalent if there exist mutually inverse morphisms
  87    that both preserve strain exactly (isometric in both directions). -/
  88structure OctaveEquiv (O₁ O₂ : Octave) where
  89  /-- Forward morphism. -/
  90  toFun : OctaveMorphism O₁ O₂
  91  /-- Backward morphism. -/
  92  invFun : OctaveMorphism O₂ O₁

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