IndisputableMonolith.RRF.Theorems.OrderPreservation

   1import IndisputableMonolith.RRF.Core.DisplayChannel
   2import IndisputableMonolith.RRF.Core.Strain
   3import Mathlib.Data.Finset.Basic
   4import Mathlib.Order.Basic
   5import Mathlib.Tactic
   6
   7/-!
   8# RRF Theorems: Order Preservation
   9
  10Theorems about how ordering on states is preserved under various operations.
  11
  12Key insight: RRF cares about *ordering* of states by strain, not absolute values.
  13Any operation that preserves this ordering preserves "meaning."
  14
  15## Main Results
  16
  17- `rank_invariant`: Monotone transforms preserve rank ordering
  18- `channel_transfer`: If channels are quality-equivalent, optimization transfers
  19-/
  20
  21
  22namespace IndisputableMonolith
  23namespace RRF.Theorems
  24
  25variable {State : Type*}
  26
  27/-! ## Rank Preservation -/
  28
  29/-- The rank of a state x in a finite set S: number of states with lower J. -/
  30noncomputable def rank [Fintype State] [DecidableEq State]
  31    (J : State → ℝ) (x : State) : ℕ :=
  32  Finset.card { y ∈ Finset.univ | J y < J x }
  33
  34/-- Strictly monotone transforms preserve strict ordering, hence ranks. -/
  35theorem strictMono_preserves_strict_order
  36    {J : State → ℝ} {g : ℝ → ℝ} (hg : StrictMono g)
  37    {x y : State} (hxy : J x < J y) :
  38    (g ∘ J) x < (g ∘ J) y :=
  39  hg hxy
  40
  41/-- For finite state spaces, strictly monotone transforms preserve ranks. -/
  42theorem rank_invariant [Fintype State] [DecidableEq State]
  43    {J : State → ℝ} {g : ℝ → ℝ} (hg : StrictMono g) (x : State) :
  44    rank J x = rank (g ∘ J) x := by
  45  simp only [rank]
  46  congr 1
  47  ext y
  48  simp only [Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_apply]
  49  exact ⟨fun h => hg h, fun h => hg.lt_iff_lt.mp h⟩
  50
  51/-! ## Channel Transfer -/
  52
  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. -/
  87theorem strain_strict_order_trans {S : StrainFunctional State}
  88    {x y z : State}
  89    (hxy : S.strictlyBetter x y) (hyz : S.strictlyBetter y z) :
  90    S.strictlyBetter x z :=
  91  lt_trans hxy hyz
  92
  93/-- Equilibria are minimal elements. -/
  94theorem equilibria_minimal [StrainFunctional.NonNeg S]
  95    {x : State} (hx : S.isBalanced x) :
  96    S.isMinimizer x :=
  97  StrainFunctional.equilibria_are_minimizers x hx
  98
  99end RRF.Theorems
 100end IndisputableMonolith
 101