Datasets:

roseking
/

openAI-math-dataset

import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Population
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Canonicalization
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityIncidence

namespace OAI

/-! Joint tensor extraction, compatibility and entropy estimates. -/

noncomputable section

namespace MatrixMultiplication.JointPopulationCompatibility

open MatrixMultiplication.Foundation JointPopulation JointTypeCounts InheritedMasks
open JointCompatibilityIncidence
open scoped BigOperators

attribute [local instance] Classical.propDecidable

variable {H : Type*} [Fintype H] [DecidableEq H]
  (counts : H → Shape → ℕ)

abbrev ClassKey := H × Fin 17

abbrev ClassPos (b : Position counts → Fin 17) (c : ClassKey (H := H)) :=
  {i : Positions counts c.1 // b ⟨c.1, i⟩ = c.2}

def classCounts (s : Fin 3) (c : ClassKey (H := H)) (u : Shape) : ℕ :=
  if shapeSide s u = c.2 then counts c.1 u else 0

abbrev FixedSideTargets (s : Fin 3) (b : Position counts → Fin 17) :=
  {e : Target counts // ∀ h i, shapeSide s ((e h).val i) = b ⟨h, i⟩}

def candidateEquiv (s : Fin 3) (b : Position counts → Fin 17) :
    FixedSideTargets counts s b ≃
      FixedClassWords (ClassPos counts b) (fun _ => Shape) (classCounts counts s) := by
  let split : FixedSideTargets counts s b ≃
      (∀ h, FixedCoarseWords (counts h) (shapeSide s) (fun i => b ⟨h, i⟩)) :=
    { toFun := fun e h => ⟨(e.val h).val, (e.val h).property, e.property h⟩
      invFun := fun v =>
        ⟨fun h => ⟨(v h).val, (v h).property.1⟩, fun h => (v h).property.2⟩
      left_inv := by
        intro e
        apply Subtype.ext
        funext h
        apply Subtype.ext
        rfl
      right_inv := by
        intro v
        funext h
        apply Subtype.ext
        rfl }
  let perHistory (h : H) :=
    (fixedCoarseWordsEquiv (counts h) (shapeSide s) (fun i => b ⟨h, i⟩)).trans
      (conditionalWordsEquiv (fun i : Positions counts h => b ⟨h, i⟩)
        (fun a u => if shapeSide s u = a then counts h u else 0))
  refine split.trans ((Equiv.piCongrRight perHistory).trans ?_)
  exact
    { toFun := fun v c => v c.1 c.2
      invFun := fun v h a => v (h, a)
      left_inv := by intro v; rfl
      right_inv := by
        intro v
        funext c
        rcases c with ⟨h, a⟩
        rfl }

omit [Fintype H] [DecidableEq H] in
@[simp] theorem candidateEquiv_apply (s : Fin 3) (b : Position counts → Fin 17)
    (e : FixedSideTargets counts s b) (c : ClassKey (H := H)) (i : ClassPos counts b c) :
    ((candidateEquiv counts s b e) c).val i = (e.val c.1).val i.val := rfl

omit [Fintype H] [DecidableEq H] in
theorem classCounts_size (s : Fin 3) (b : Position counts → Fin 17)
    (e : FixedSideTargets counts s b) (c : ClassKey (H := H)) :
    Fintype.card (ClassPos counts b c) = ∑ u, classCounts counts s c u := by
  calc
    Fintype.card (ClassPos counts b c) =
        ∑ u, wordPopulation ((candidateEquiv counts s b e) c).val u :=
      (wordPopulation_sum _).symm
    _ = ∑ u, classCounts counts s c u :=
      Finset.sum_congr rfl fun u _ => ((candidateEquiv counts s b e) c).property u

theorem fixedSideTargets_card (s : Fin 3) (b : Position counts → Fin 17)
    (e : FixedSideTargets counts s b) :
    Fintype.card (FixedSideTargets counts s b) =
      ∏ c : ClassKey (H := H), Nat.multinomial Finset.univ (classCounts counts s c) := by
  rw [Fintype.card_congr (candidateEquiv counts s b)]
  exact fixedClassWords_card _ _ _ (classCounts_size counts s b e)

def shapeStatisticCount {A : Type*} [Fintype A] [DecidableEq A]
    (e : Target counts) (h : H) (u : Shape) (f : Positions counts h → A) (a : A) : ℕ :=
  wordPopulation (fun i : Class counts e h u => f i.val) a

theorem scaled_count_window {x n p χ : ℝ} (hn : 0 < n)
    (hw : |x / n - p| ≤ χ) : |x - n * p| ≤ n * χ := by
  calc
    |x - n * p| = |(x / n - p) * n| := by
      rw [sub_mul, div_mul_cancel₀ _ hn.ne', mul_comm p n]
    _ = |x / n - p| * n := by rw [abs_mul, abs_of_pos hn]
    _ ≤ χ * n := mul_le_mul_of_nonneg_right hw hn.le
    _ = n * χ := mul_comm _ _
omit [Fintype H] [DecidableEq H] in
theorem shape_count_window_of_typeWindow {A : Type*} [Fintype A] [DecidableEq A]
    (e : Target counts) (h : H) (u : Shape) (f : Positions counts h → A)
    (law : A → ℝ) (χ : ℝ)
    (hw : 0 < counts h u → typeWindow law χ (fun i : Class counts e h u => f i.val)) :
    ∀ a, |(shapeStatisticCount counts e h u f a : ℝ) -
      (counts h u : ℝ) * law a| ≤ (counts h u : ℝ) * χ := by
  intro a
  by_cases hn : 0 < counts h u
  · apply scaled_count_window (Nat.cast_pos.mpr hn)
    simpa only [empiricalLaw, class_card, shapeStatisticCount] using hw hn a
  · have hn0 : counts h u = 0 := Nat.eq_zero_of_not_pos hn
    have hc : shapeStatisticCount counts e h u f a = 0 := by
      apply Nat.eq_zero_of_le_zero
      let forget : {i : Class counts e h u // f i.val = a} → Class counts e h u :=
        Subtype.val
      have hinj : Function.Injective forget := Subtype.val_injective
      have hbound := Fintype.card_le_of_injective forget hinj
      calc
        shapeStatisticCount counts e h u f a ≤ Fintype.card (Class counts e h u) :=
          hbound
        _ = 0 := (class_card counts e h u).trans hn0
    simp only [hc, hn0, Nat.cast_zero, zero_mul, sub_zero, abs_zero, le_refl]
theorem jointCount_le_population {P U Z A : Type*}
    [Fintype P] [Fintype U] [DecidableEq U] [DecidableEq A]
    (w : P → U) (z : P → Z) (part : Z → A) (u : U) (a : A) :
    jointCount w z part u a ≤ wordPopulation w u := by
  exact Fintype.card_le_of_injective
    (fun i : {i : P // w i = u ∧ part (z i) = a} =>
      (⟨i.val, i.property.1⟩ : {i : P // w i = u}))
    (fun _ _ h => Subtype.ext
      (congrArg (fun j : {i : P // w i = u} => j.val) h))
omit [Fintype H] [DecidableEq H] in
theorem class_jointCount_eq {A : Type*} [Fintype A] [DecidableEq A]
    (s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
    (c : ClassKey (H := H)) (u : Shape) (hu : shapeSide s u = c.2)
    (f : Positions counts c.1 → A) (a : A) :
    jointCount ((candidateEquiv counts s b e) c).val
      (fun i => f i.val) id u a = shapeStatisticCount counts (A := A) e.val c.1 u f a := by
  let eFiber : {i : ClassPos counts b c //
      ((candidateEquiv counts s b e) c).val i = u ∧ f i.val = a} ≃
      {i : Class counts e.val c.1 u // f i.val = a} :=
    { toFun := fun i => ⟨⟨i.val.val, by
        simpa only [candidateEquiv_apply] using i.property.1⟩, i.property.2⟩
      invFun := fun i =>
        ⟨⟨i.val.val, by
          calc
            b ⟨c.1, i.val.val⟩ = shapeSide s ((e.val c.1).val i.val.val) :=
              (e.property c.1 i.val.val).symm
            _ = shapeSide s u := congrArg (shapeSide s) i.val.property
            _ = c.2 := hu⟩,
          by simpa only [candidateEquiv_apply] using i.val.property, i.property⟩
      left_inv := by
        intro i
        apply Subtype.ext
        apply Subtype.ext
        rfl
      right_inv := by
        intro i
        apply Subtype.ext
        apply Subtype.ext
        rfl }
  unfold jointCount shapeStatisticCount wordPopulation
  exact Fintype.card_congr eFiber
omit [Fintype H] [DecidableEq H] in
theorem class_jointCount_zero {A : Type*} [DecidableEq A]
    (s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
    (c : ClassKey (H := H)) (u : Shape) (hu : shapeSide s u ≠ c.2)
    (f : ClassPos counts b c → A) (a : A) :
    jointCount ((candidateEquiv counts s b e) c).val f id u a = 0 := by
  apply Nat.eq_zero_of_le_zero
  calc
    jointCount ((candidateEquiv counts s b e) c).val f id u a ≤
        wordPopulation ((candidateEquiv counts s b e) c).val u :=
      jointCount_le_population _ _ _ _ _
    _ = classCounts counts s c u := ((candidateEquiv counts s b e) c).property u
    _ = 0 := ite_eq_right hu
omit [Fintype H] [DecidableEq H] in
theorem side_compatibility_iff_class_counts
    (A : ClassKey (H := H) → Type*)
    [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)]
    (s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
    (f : ∀ h k, Positions counts h → A (h, k))
    (designated : ClassKey (H := H) → Shape → Prop)
    (law : ∀ c, Shape → A c → ℝ) (χ : ℝ) :
    (∀ h u, designated (h, shapeSide s u) u → ∀ a : A (h, shapeSide s u),
      |(shapeStatisticCount counts (A := A (h, shapeSide s u))
          e.val h u (f h (shapeSide s u)) a : ℝ) -
        (counts h u : ℝ) * law (h, shapeSide s u) u a| ≤ (counts h u : ℝ) * χ) ↔
    (∀ c u, designated c u → ∀ a : A c,
      |(jointCount ((candidateEquiv counts s b e) c).val
          (fun i => f c.1 c.2 i.val) id u a : ℝ) -
        (classCounts counts s c u : ℝ) * law c u a| ≤
          (classCounts counts s c u : ℝ) * χ) := by
  constructor
  · intro h c u hd a
    by_cases hu : shapeSide s u = c.2
    · rw [class_jointCount_eq counts s b e c u hu]
      rcases c with ⟨h₀, k⟩
      dsimp at hu
      subst k
      simpa only [classCounts, ite_eq_left rfl, ite_true] using h h₀ u hd a
    · rw [class_jointCount_zero counts s b e c u hu]
      simp only [classCounts, ite_eq_right hu, Nat.cast_zero, zero_mul, sub_zero, abs_zero, le_refl]
  · intro h h₀ u hd a
    have hh := h (h₀, shapeSide s u) u hd a
    rw [class_jointCount_eq counts s b e (h₀, shapeSide s u) u rfl] at hh
    simpa only [classCounts, ite_eq_left rfl, ite_true] using hh
section Statistics
variable (L R : H → Type*)
  (SL SR : ClassKey (H := H) → Type*)
  [∀ c, Fintype (SL c)] [∀ c, DecidableEq (SL c)]
  [∀ c, Fintype (SR c)] [∀ c, DecidableEq (SR c)]
  (leftStatistic : ∀ c, L c.1 → SL c)
  (rightStatistic : ∀ c, R c.1 → SR c)
def statisticWord (b : Position counts → Fin 17)
    (w : JointCanonicalization.RawPairs counts L R) (c : ClassKey (H := H)) :
    ClassPos counts b c → SL c × SR c :=
  fun i => (leftStatistic c (w c.1 i.val).1, rightStatistic c (w c.1 i.val).2)
def statisticCounts (b : Position counts → Fin 17)
    (w : JointCanonicalization.RawPairs counts L R) (c : ClassKey (H := H)) :
    SL c × SR c → ℕ :=
  wordPopulation (statisticWord counts L R SL SR leftStatistic rightStatistic b w c)
def statisticElement (b : Position counts → Fin 17)
    (w : JointCanonicalization.RawPairs counts L R) :
    FixedClassWords (ClassPos counts b) (fun c => SL c × SR c)
      (statisticCounts counts L R SL SR leftStatistic rightStatistic b w) :=
  fun c => ⟨statisticWord counts L R SL SR leftStatistic rightStatistic b w c, fun _ => rfl⟩
variable (designatedLeft designatedRight : ClassKey (H := H) → Shape → Prop)
  (leftLaw : ∀ c, Shape → SL c → ℝ) (rightLaw : ∀ c, Shape → SR c → ℝ) (χ : ℝ)
def Compatible (s : Fin 3) (e : Target counts)
    (w : JointCanonicalization.RawPairs counts L R) : Prop :=
  (∀ h u, designatedLeft (h, shapeSide s u) u → ∀ a,
    |(shapeStatisticCount counts e h u
        (fun i => leftStatistic (h, shapeSide s u) (w h i).1) a : ℝ) -
      (counts h u : ℝ) * leftLaw (h, shapeSide s u) u a| ≤ (counts h u : ℝ) * χ) ∧
  (∀ h u, designatedRight (h, shapeSide s u) u → ∀ a,
    |(shapeStatisticCount counts e h u
        (fun i => rightStatistic (h, shapeSide s u) (w h i).2) a : ℝ) -
      (counts h u : ℝ) * rightLaw (h, shapeSide s u) u a| ≤ (counts h u : ℝ) * χ)
omit [Fintype H] [DecidableEq H] in
theorem compatible_of_designated_windows (s : Fin 3) (e : Target counts)
    (w : JointCanonicalization.RawPairs counts L R)
    (hl : ∀ h u, 0 < counts h u → designatedLeft (h, shapeSide s u) u →
      typeWindow (leftLaw (h, shapeSide s u) u) χ
        (fun i : Class counts e h u => leftStatistic (h, shapeSide s u) (w h i.val).1))
    (hr : ∀ h u, 0 < counts h u → designatedRight (h, shapeSide s u) u →
      typeWindow (rightLaw (h, shapeSide s u) u) χ
        (fun i : Class counts e h u => rightStatistic (h, shapeSide s u) (w h i.val).2)) :
    Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
      leftLaw rightLaw χ s e w := by
  constructor
  · intro h u hd
    exact shape_count_window_of_typeWindow counts e h u _ _ χ (fun hn => hl h u hn hd)
  · intro h u hd
    exact shape_count_window_of_typeWindow counts e h u _ _ χ (fun hn => hr h u hn hd)
omit [Fintype H] [DecidableEq H] in
theorem compatible_iff_kernel (s : Fin 3) (b : Position counts → Fin 17)
    (e : FixedSideTargets counts s b) (w : JointCanonicalization.RawPairs counts L R) :
    Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
      leftLaw rightLaw χ s e.val w ↔
    DesignatedCompatibility (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
      SL SR (classCounts counts s)
      (statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
      (fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
      (candidateEquiv counts s b e)
      (statisticElement counts L R SL SR leftStatistic rightStatistic b w) := by
  exact and_congr
    (side_compatibility_iff_class_counts counts SL s b e
      (fun h k i => leftStatistic (h, k) (w h i).1) designatedLeft leftLaw χ)
    (side_compatibility_iff_class_counts counts SR s b e
      (fun h k i => rightStatistic (h, k) (w h i).2) designatedRight rightLaw χ)
def compatibleTargetCount (s : Fin 3) (b : Position counts → Fin 17)
    (w : JointCanonicalization.RawPairs counts L R) : ℕ :=
  Fintype.card {e : FixedSideTargets counts s b //
    Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
      leftLaw rightLaw χ s e.val w}
theorem compatibleTargetCount_eq_kernel (s : Fin 3) (b : Position counts → Fin 17)
    (w : JointCanonicalization.RawPairs counts L R) :
    compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ s b w =
    compatibleCandidateCount (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
      SL SR (classCounts counts s)
      (statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
      (fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
      (statisticElement counts L R SL SR leftStatistic rightStatistic b w) := by
  exact Fintype.card_congr ((candidateEquiv counts s b).subtypeEquiv
    (fun e => compatible_iff_kernel counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ s b e w))
theorem compatibleTarget_incidence (s : Fin 3) (b : Position counts → Fin 17)
    (e : FixedSideTargets counts s b) (w : JointCanonicalization.RawPairs counts L R) :
    Fintype.card (FixedSideTargets counts s b) *
      compatibleStatisticCount (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
        SL SR (classCounts counts s)
        (statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
        (fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
        (candidateEquiv counts s b e) =
    Fintype.card (FixedClassWords (ClassPos counts b) (fun c => SL c × SR c)
      (statisticCounts counts L R SL SR leftStatistic rightStatistic b w)) *
      compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
        designatedLeft designatedRight leftLaw rightLaw χ s b w := by
  rw [compatibleTargetCount_eq_kernel,
    Fintype.card_congr (candidateEquiv counts s b)]
  exact designated_incidence_count _ _ _ _ _ _ _ _ _ _ _ _ _ _
    (candidateEquiv counts s b e)
    (statisticElement counts L R SL SR leftStatistic rightStatistic b w)
def compatibleByTriple (s : Fin 3)
    (t : JointCoarseHashing.Triple (Position counts))
    (w : JointCanonicalization.RawPairs counts L R) : Prop :=
  ∃ e : Target counts, triple counts e = t ∧
    Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
      leftLaw rightLaw χ s e w
omit [Fintype H] [DecidableEq H] in
theorem compatibleByTriple_target (s : Fin 3) (e : Target counts)
    (w : JointCanonicalization.RawPairs counts L R) :
    compatibleByTriple counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ s (triple counts e) w ↔
    Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
      leftLaw rightLaw χ s e w := by
  constructor
  · rintro ⟨f, hf, hw⟩
    have he : f = e := triple_injective counts hf
    subst f
    exact hw
  · intro hw
    exact ⟨e, rfl, hw⟩
def compatibleWithCoarse
    (ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
    (s : Fin 3) (t : JointCoarseHashing.Triple (Position counts))
    (w : JointCanonicalization.RawPairs counts L R) : Prop :=
  (∀ h i, sideWord counts s t h i = ownCoarse w ⟨h, i⟩) ∧
    compatibleByTriple counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ s t w
omit [Fintype H] [DecidableEq H] in
theorem compatibleWithCoarse_target
    (ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
    (s : Fin 3) (e : Target counts) (w : JointCanonicalization.RawPairs counts L R) :
    compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s (triple counts e) w ↔
    (∀ h i, shapeSide s ((e h).val i) = ownCoarse w ⟨h, i⟩) ∧
      Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
        leftLaw rightLaw χ s e w := by
  exact and_congr Iff.rfl (compatibleByTriple_target counts L R SL SR leftStatistic
    rightStatistic designatedLeft designatedRight leftLaw rightLaw χ s e w)
omit [Fintype H] [DecidableEq H] in
theorem compatibleWithCoarse_shares_side
    (ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
    (s : Fin 3) (t t' : JointCoarseHashing.Triple (Position counts))
    (w : JointCanonicalization.RawPairs counts L R)
    (ht : compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s t w)
    (ht' : compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s t' w) :
    ∀ h, sideWord counts s t h = sideWord counts s t' h := by
  intro h
  funext i
  exact (ht.1 h i).trans (ht'.1 h i).symm

theorem compatibleWithCoarse_target_count
    (ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
    (s : Fin 3) (w : JointCanonicalization.RawPairs counts L R) :
    Fintype.card {e : Target counts //
      compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
        designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s (triple counts e) w} =
    compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
      designatedLeft designatedRight leftLaw rightLaw χ s (ownCoarse w) w := by
  apply Fintype.card_congr
  exact (Equiv.subtypeEquivRight (fun e => compatibleWithCoarse_target counts L R SL SR
    leftStatistic rightStatistic designatedLeft designatedRight leftLaw rightLaw χ
      ownCoarse s e w)).trans (Equiv.subtypeSubtypeEquivSubtypeInter
        (fun e : Target counts => ∀ h i, shapeSide s ((e h).val i) = ownCoarse w ⟨h, i⟩)
        (fun e => Compatible counts L R SL SR leftStatistic rightStatistic
          designatedLeft designatedRight leftLaw rightLaw χ s e w)).symm

end Statistics

def designatedYLeft (_ : ClassKey (H := H)) (u : Shape) : Prop :=
  (shapeSide 2 u).val = 0

def designatedYRight (parentShape : H → Fin 3 → ℕ)
    (c : ClassKey (H := H)) (u : Shape) : Prop :=
  parentShape c.1 2 - (shapeSide 2 u).val = 0

def designatedZLeft (_ : ClassKey (H := H)) (u : Shape) : Prop :=
  (shapeSide 0 u).val * (shapeSide 1 u).val = 0

def designatedZRight (parentShape : H → Fin 3 → ℕ)
    (c : ClassKey (H := H)) (u : Shape) : Prop :=
  (parentShape c.1 0 - (shapeSide 0 u).val) *
    (parentShape c.1 1 - (shapeSide 1 u).val) = 0

end MatrixMultiplication.JointPopulationCompatibility

end

end OAI

	import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Population
	import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Canonicalization
	import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityIncidence

	namespace OAI

	/-! Joint tensor extraction, compatibility and entropy estimates. -/

	noncomputable section

	namespace MatrixMultiplication.JointPopulationCompatibility

	open MatrixMultiplication.Foundation JointPopulation JointTypeCounts InheritedMasks
	open JointCompatibilityIncidence
	open scoped BigOperators

	attribute [local instance] Classical.propDecidable

	variable {H : Type*} [Fintype H] [DecidableEq H]
	(counts : H → Shape → ℕ)

	abbrev ClassKey := H × Fin 17

	abbrev ClassPos (b : Position counts → Fin 17) (c : ClassKey (H := H)) :=
	{i : Positions counts c.1 // b ⟨c.1, i⟩ = c.2}

	def classCounts (s : Fin 3) (c : ClassKey (H := H)) (u : Shape) : ℕ :=
	if shapeSide s u = c.2 then counts c.1 u else 0

	abbrev FixedSideTargets (s : Fin 3) (b : Position counts → Fin 17) :=
	{e : Target counts // ∀ h i, shapeSide s ((e h).val i) = b ⟨h, i⟩}

	def candidateEquiv (s : Fin 3) (b : Position counts → Fin 17) :
	FixedSideTargets counts s b ≃
	FixedClassWords (ClassPos counts b) (fun _ => Shape) (classCounts counts s) := by
	let split : FixedSideTargets counts s b ≃
	(∀ h, FixedCoarseWords (counts h) (shapeSide s) (fun i => b ⟨h, i⟩)) :=
	{ toFun := fun e h => ⟨(e.val h).val, (e.val h).property, e.property h⟩
	invFun := fun v =>
	⟨fun h => ⟨(v h).val, (v h).property.1⟩, fun h => (v h).property.2⟩
	left_inv := by
	intro e
	apply Subtype.ext
	funext h
	apply Subtype.ext
	rfl
	right_inv := by
	intro v
	funext h
	apply Subtype.ext
	rfl }
	let perHistory (h : H) :=
	(fixedCoarseWordsEquiv (counts h) (shapeSide s) (fun i => b ⟨h, i⟩)).trans
	(conditionalWordsEquiv (fun i : Positions counts h => b ⟨h, i⟩)
	(fun a u => if shapeSide s u = a then counts h u else 0))
	refine split.trans ((Equiv.piCongrRight perHistory).trans ?_)
	exact
	{ toFun := fun v c => v c.1 c.2
	invFun := fun v h a => v (h, a)
	left_inv := by intro v; rfl
	right_inv := by
	intro v
	funext c
	rcases c with ⟨h, a⟩
	rfl }

	omit [Fintype H] [DecidableEq H] in
	@[simp] theorem candidateEquiv_apply (s : Fin 3) (b : Position counts → Fin 17)
	(e : FixedSideTargets counts s b) (c : ClassKey (H := H)) (i : ClassPos counts b c) :
	((candidateEquiv counts s b e) c).val i = (e.val c.1).val i.val := rfl

	omit [Fintype H] [DecidableEq H] in
	theorem classCounts_size (s : Fin 3) (b : Position counts → Fin 17)
	(e : FixedSideTargets counts s b) (c : ClassKey (H := H)) :
	Fintype.card (ClassPos counts b c) = ∑ u, classCounts counts s c u := by
	calc
	Fintype.card (ClassPos counts b c) =
	∑ u, wordPopulation ((candidateEquiv counts s b e) c).val u :=
	(wordPopulation_sum _).symm
	_ = ∑ u, classCounts counts s c u :=
	Finset.sum_congr rfl fun u _ => ((candidateEquiv counts s b e) c).property u

	theorem fixedSideTargets_card (s : Fin 3) (b : Position counts → Fin 17)
	(e : FixedSideTargets counts s b) :
	Fintype.card (FixedSideTargets counts s b) =
	∏ c : ClassKey (H := H), Nat.multinomial Finset.univ (classCounts counts s c) := by
	rw [Fintype.card_congr (candidateEquiv counts s b)]
	exact fixedClassWords_card _ _ _ (classCounts_size counts s b e)

	def shapeStatisticCount {A : Type*} [Fintype A] [DecidableEq A]
	(e : Target counts) (h : H) (u : Shape) (f : Positions counts h → A) (a : A) : ℕ :=
	wordPopulation (fun i : Class counts e h u => f i.val) a

	theorem scaled_count_window {x n p χ : ℝ} (hn : 0 < n)
	(hw : \|x / n - p\| ≤ χ) : \|x - n * p\| ≤ n * χ := by
	calc
	\|x - n * p\| = \|(x / n - p) * n\| := by
	rw [sub_mul, div_mul_cancel₀ _ hn.ne', mul_comm p n]
	_ = \|x / n - p\| * n := by rw [abs_mul, abs_of_pos hn]
	_ ≤ χ * n := mul_le_mul_of_nonneg_right hw hn.le
	_ = n * χ := mul_comm _ _

	omit [Fintype H] [DecidableEq H] in
	theorem shape_count_window_of_typeWindow {A : Type*} [Fintype A] [DecidableEq A]
	(e : Target counts) (h : H) (u : Shape) (f : Positions counts h → A)
	(law : A → ℝ) (χ : ℝ)
	(hw : 0 < counts h u → typeWindow law χ (fun i : Class counts e h u => f i.val)) :
	∀ a, \|(shapeStatisticCount counts e h u f a : ℝ) -
	(counts h u : ℝ) * law a\| ≤ (counts h u : ℝ) * χ := by
	intro a
	by_cases hn : 0 < counts h u
	· apply scaled_count_window (Nat.cast_pos.mpr hn)
	simpa only [empiricalLaw, class_card, shapeStatisticCount] using hw hn a
	· have hn0 : counts h u = 0 := Nat.eq_zero_of_not_pos hn
	have hc : shapeStatisticCount counts e h u f a = 0 := by
	apply Nat.eq_zero_of_le_zero
	let forget : {i : Class counts e h u // f i.val = a} → Class counts e h u :=
	Subtype.val
	have hinj : Function.Injective forget := Subtype.val_injective
	have hbound := Fintype.card_le_of_injective forget hinj
	calc
	shapeStatisticCount counts e h u f a ≤ Fintype.card (Class counts e h u) :=
	hbound
	_ = 0 := (class_card counts e h u).trans hn0
	simp only [hc, hn0, Nat.cast_zero, zero_mul, sub_zero, abs_zero, le_refl]

	theorem jointCount_le_population {P U Z A : Type*}
	[Fintype P] [Fintype U] [DecidableEq U] [DecidableEq A]
	(w : P → U) (z : P → Z) (part : Z → A) (u : U) (a : A) :
	jointCount w z part u a ≤ wordPopulation w u := by
	exact Fintype.card_le_of_injective
	(fun i : {i : P // w i = u ∧ part (z i) = a} =>
	(⟨i.val, i.property.1⟩ : {i : P // w i = u}))
	(fun _ _ h => Subtype.ext
	(congrArg (fun j : {i : P // w i = u} => j.val) h))

	omit [Fintype H] [DecidableEq H] in
	theorem class_jointCount_eq {A : Type*} [Fintype A] [DecidableEq A]
	(s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
	(c : ClassKey (H := H)) (u : Shape) (hu : shapeSide s u = c.2)
	(f : Positions counts c.1 → A) (a : A) :
	jointCount ((candidateEquiv counts s b e) c).val
	(fun i => f i.val) id u a = shapeStatisticCount counts (A := A) e.val c.1 u f a := by
	let eFiber : {i : ClassPos counts b c //
	((candidateEquiv counts s b e) c).val i = u ∧ f i.val = a} ≃
	{i : Class counts e.val c.1 u // f i.val = a} :=
	{ toFun := fun i => ⟨⟨i.val.val, by
	simpa only [candidateEquiv_apply] using i.property.1⟩, i.property.2⟩
	invFun := fun i =>
	⟨⟨i.val.val, by
	calc
	b ⟨c.1, i.val.val⟩ = shapeSide s ((e.val c.1).val i.val.val) :=
	(e.property c.1 i.val.val).symm
	_ = shapeSide s u := congrArg (shapeSide s) i.val.property
	_ = c.2 := hu⟩,
	by simpa only [candidateEquiv_apply] using i.val.property, i.property⟩
	left_inv := by
	intro i
	apply Subtype.ext
	apply Subtype.ext
	rfl
	right_inv := by
	intro i
	apply Subtype.ext
	apply Subtype.ext
	rfl }
	unfold jointCount shapeStatisticCount wordPopulation
	exact Fintype.card_congr eFiber

	omit [Fintype H] [DecidableEq H] in
	theorem class_jointCount_zero {A : Type*} [DecidableEq A]
	(s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
	(c : ClassKey (H := H)) (u : Shape) (hu : shapeSide s u ≠ c.2)
	(f : ClassPos counts b c → A) (a : A) :
	jointCount ((candidateEquiv counts s b e) c).val f id u a = 0 := by
	apply Nat.eq_zero_of_le_zero
	calc
	jointCount ((candidateEquiv counts s b e) c).val f id u a ≤
	wordPopulation ((candidateEquiv counts s b e) c).val u :=
	jointCount_le_population _ _ _ _ _
	_ = classCounts counts s c u := ((candidateEquiv counts s b e) c).property u
	_ = 0 := ite_eq_right hu

	omit [Fintype H] [DecidableEq H] in
	theorem side_compatibility_iff_class_counts
	(A : ClassKey (H := H) → Type*)
	[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)]
	(s : Fin 3) (b : Position counts → Fin 17) (e : FixedSideTargets counts s b)
	(f : ∀ h k, Positions counts h → A (h, k))
	(designated : ClassKey (H := H) → Shape → Prop)
	(law : ∀ c, Shape → A c → ℝ) (χ : ℝ) :
	(∀ h u, designated (h, shapeSide s u) u → ∀ a : A (h, shapeSide s u),
	\|(shapeStatisticCount counts (A := A (h, shapeSide s u))
	e.val h u (f h (shapeSide s u)) a : ℝ) -
	(counts h u : ℝ) * law (h, shapeSide s u) u a\| ≤ (counts h u : ℝ) * χ) ↔
	(∀ c u, designated c u → ∀ a : A c,
	\|(jointCount ((candidateEquiv counts s b e) c).val
	(fun i => f c.1 c.2 i.val) id u a : ℝ) -
	(classCounts counts s c u : ℝ) * law c u a\| ≤
	(classCounts counts s c u : ℝ) * χ) := by
	constructor
	· intro h c u hd a
	by_cases hu : shapeSide s u = c.2
	· rw [class_jointCount_eq counts s b e c u hu]
	rcases c with ⟨h₀, k⟩
	dsimp at hu
	subst k
	simpa only [classCounts, ite_eq_left rfl, ite_true] using h h₀ u hd a
	· rw [class_jointCount_zero counts s b e c u hu]
	simp only [classCounts, ite_eq_right hu, Nat.cast_zero, zero_mul, sub_zero, abs_zero, le_refl]
	· intro h h₀ u hd a
	have hh := h (h₀, shapeSide s u) u hd a
	rw [class_jointCount_eq counts s b e (h₀, shapeSide s u) u rfl] at hh
	simpa only [classCounts, ite_eq_left rfl, ite_true] using hh

	section Statistics

	variable (L R : H → Type*)
	(SL SR : ClassKey (H := H) → Type*)
	[∀ c, Fintype (SL c)] [∀ c, DecidableEq (SL c)]
	[∀ c, Fintype (SR c)] [∀ c, DecidableEq (SR c)]
	(leftStatistic : ∀ c, L c.1 → SL c)
	(rightStatistic : ∀ c, R c.1 → SR c)

	def statisticWord (b : Position counts → Fin 17)
	(w : JointCanonicalization.RawPairs counts L R) (c : ClassKey (H := H)) :
	ClassPos counts b c → SL c × SR c :=
	fun i => (leftStatistic c (w c.1 i.val).1, rightStatistic c (w c.1 i.val).2)

	def statisticCounts (b : Position counts → Fin 17)
	(w : JointCanonicalization.RawPairs counts L R) (c : ClassKey (H := H)) :
	SL c × SR c → ℕ :=
	wordPopulation (statisticWord counts L R SL SR leftStatistic rightStatistic b w c)

	def statisticElement (b : Position counts → Fin 17)
	(w : JointCanonicalization.RawPairs counts L R) :
	FixedClassWords (ClassPos counts b) (fun c => SL c × SR c)
	(statisticCounts counts L R SL SR leftStatistic rightStatistic b w) :=
	fun c => ⟨statisticWord counts L R SL SR leftStatistic rightStatistic b w c, fun _ => rfl⟩

	variable (designatedLeft designatedRight : ClassKey (H := H) → Shape → Prop)
	(leftLaw : ∀ c, Shape → SL c → ℝ) (rightLaw : ∀ c, Shape → SR c → ℝ) (χ : ℝ)

	def Compatible (s : Fin 3) (e : Target counts)
	(w : JointCanonicalization.RawPairs counts L R) : Prop :=
	(∀ h u, designatedLeft (h, shapeSide s u) u → ∀ a,
	\|(shapeStatisticCount counts e h u
	(fun i => leftStatistic (h, shapeSide s u) (w h i).1) a : ℝ) -
	(counts h u : ℝ) * leftLaw (h, shapeSide s u) u a\| ≤ (counts h u : ℝ) * χ) ∧
	(∀ h u, designatedRight (h, shapeSide s u) u → ∀ a,
	\|(shapeStatisticCount counts e h u
	(fun i => rightStatistic (h, shapeSide s u) (w h i).2) a : ℝ) -
	(counts h u : ℝ) * rightLaw (h, shapeSide s u) u a\| ≤ (counts h u : ℝ) * χ)

	omit [Fintype H] [DecidableEq H] in
	theorem compatible_of_designated_windows (s : Fin 3) (e : Target counts)
	(w : JointCanonicalization.RawPairs counts L R)
	(hl : ∀ h u, 0 < counts h u → designatedLeft (h, shapeSide s u) u →
	typeWindow (leftLaw (h, shapeSide s u) u) χ
	(fun i : Class counts e h u => leftStatistic (h, shapeSide s u) (w h i.val).1))
	(hr : ∀ h u, 0 < counts h u → designatedRight (h, shapeSide s u) u →
	typeWindow (rightLaw (h, shapeSide s u) u) χ
	(fun i : Class counts e h u => rightStatistic (h, shapeSide s u) (w h i.val).2)) :
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e w := by
	constructor
	· intro h u hd
	exact shape_count_window_of_typeWindow counts e h u _ _ χ (fun hn => hl h u hn hd)
	· intro h u hd
	exact shape_count_window_of_typeWindow counts e h u _ _ χ (fun hn => hr h u hn hd)

	omit [Fintype H] [DecidableEq H] in
	theorem compatible_iff_kernel (s : Fin 3) (b : Position counts → Fin 17)
	(e : FixedSideTargets counts s b) (w : JointCanonicalization.RawPairs counts L R) :
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e.val w ↔
	DesignatedCompatibility (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
	SL SR (classCounts counts s)
	(statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
	(fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
	(candidateEquiv counts s b e)
	(statisticElement counts L R SL SR leftStatistic rightStatistic b w) := by
	exact and_congr
	(side_compatibility_iff_class_counts counts SL s b e
	(fun h k i => leftStatistic (h, k) (w h i).1) designatedLeft leftLaw χ)
	(side_compatibility_iff_class_counts counts SR s b e
	(fun h k i => rightStatistic (h, k) (w h i).2) designatedRight rightLaw χ)

	def compatibleTargetCount (s : Fin 3) (b : Position counts → Fin 17)
	(w : JointCanonicalization.RawPairs counts L R) : ℕ :=
	Fintype.card {e : FixedSideTargets counts s b //
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e.val w}

	theorem compatibleTargetCount_eq_kernel (s : Fin 3) (b : Position counts → Fin 17)
	(w : JointCanonicalization.RawPairs counts L R) :
	compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s b w =
	compatibleCandidateCount (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
	SL SR (classCounts counts s)
	(statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
	(fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
	(statisticElement counts L R SL SR leftStatistic rightStatistic b w) := by
	exact Fintype.card_congr ((candidateEquiv counts s b).subtypeEquiv
	(fun e => compatible_iff_kernel counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s b e w))

	theorem compatibleTarget_incidence (s : Fin 3) (b : Position counts → Fin 17)
	(e : FixedSideTargets counts s b) (w : JointCanonicalization.RawPairs counts L R) :
	Fintype.card (FixedSideTargets counts s b) *
	compatibleStatisticCount (ClassPos counts b) (fun _ => Shape) (fun c => SL c × SR c)
	SL SR (classCounts counts s)
	(statisticCounts counts L R SL SR leftStatistic rightStatistic b w)
	(fun _ p => p.1) (fun _ p => p.2) designatedLeft designatedRight leftLaw rightLaw χ
	(candidateEquiv counts s b e) =
	Fintype.card (FixedClassWords (ClassPos counts b) (fun c => SL c × SR c)
	(statisticCounts counts L R SL SR leftStatistic rightStatistic b w)) *
	compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s b w := by
	rw [compatibleTargetCount_eq_kernel,
	Fintype.card_congr (candidateEquiv counts s b)]
	exact designated_incidence_count _ _ _ _ _ _ _ _ _ _ _ _ _ _
	(candidateEquiv counts s b e)
	(statisticElement counts L R SL SR leftStatistic rightStatistic b w)

	def compatibleByTriple (s : Fin 3)
	(t : JointCoarseHashing.Triple (Position counts))
	(w : JointCanonicalization.RawPairs counts L R) : Prop :=
	∃ e : Target counts, triple counts e = t ∧
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e w

	omit [Fintype H] [DecidableEq H] in
	theorem compatibleByTriple_target (s : Fin 3) (e : Target counts)
	(w : JointCanonicalization.RawPairs counts L R) :
	compatibleByTriple counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s (triple counts e) w ↔
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e w := by
	constructor
	· rintro ⟨f, hf, hw⟩
	have he : f = e := triple_injective counts hf
	subst f
	exact hw
	· intro hw
	exact ⟨e, rfl, hw⟩

	def compatibleWithCoarse
	(ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
	(s : Fin 3) (t : JointCoarseHashing.Triple (Position counts))
	(w : JointCanonicalization.RawPairs counts L R) : Prop :=
	(∀ h i, sideWord counts s t h i = ownCoarse w ⟨h, i⟩) ∧
	compatibleByTriple counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s t w

	omit [Fintype H] [DecidableEq H] in
	theorem compatibleWithCoarse_target
	(ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
	(s : Fin 3) (e : Target counts) (w : JointCanonicalization.RawPairs counts L R) :
	compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s (triple counts e) w ↔
	(∀ h i, shapeSide s ((e h).val i) = ownCoarse w ⟨h, i⟩) ∧
	Compatible counts L R SL SR leftStatistic rightStatistic designatedLeft designatedRight
	leftLaw rightLaw χ s e w := by
	exact and_congr Iff.rfl (compatibleByTriple_target counts L R SL SR leftStatistic
	rightStatistic designatedLeft designatedRight leftLaw rightLaw χ s e w)

	omit [Fintype H] [DecidableEq H] in
	theorem compatibleWithCoarse_shares_side
	(ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
	(s : Fin 3) (t t' : JointCoarseHashing.Triple (Position counts))
	(w : JointCanonicalization.RawPairs counts L R)
	(ht : compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s t w)
	(ht' : compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s t' w) :
	∀ h, sideWord counts s t h = sideWord counts s t' h := by
	intro h
	funext i
	exact (ht.1 h i).trans (ht'.1 h i).symm

	theorem compatibleWithCoarse_target_count
	(ownCoarse : JointCanonicalization.RawPairs counts L R → Position counts → Fin 17)
	(s : Fin 3) (w : JointCanonicalization.RawPairs counts L R) :
	Fintype.card {e : Target counts //
	compatibleWithCoarse counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ ownCoarse s (triple counts e) w} =
	compatibleTargetCount counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s (ownCoarse w) w := by
	apply Fintype.card_congr
	exact (Equiv.subtypeEquivRight (fun e => compatibleWithCoarse_target counts L R SL SR
	leftStatistic rightStatistic designatedLeft designatedRight leftLaw rightLaw χ
	ownCoarse s e w)).trans (Equiv.subtypeSubtypeEquivSubtypeInter
	(fun e : Target counts => ∀ h i, shapeSide s ((e h).val i) = ownCoarse w ⟨h, i⟩)
	(fun e => Compatible counts L R SL SR leftStatistic rightStatistic
	designatedLeft designatedRight leftLaw rightLaw χ s e w)).symm

	end Statistics

	def designatedYLeft (_ : ClassKey (H := H)) (u : Shape) : Prop :=
	(shapeSide 2 u).val = 0

	def designatedYRight (parentShape : H → Fin 3 → ℕ)
	(c : ClassKey (H := H)) (u : Shape) : Prop :=
	parentShape c.1 2 - (shapeSide 2 u).val = 0

	def designatedZLeft (_ : ClassKey (H := H)) (u : Shape) : Prop :=
	(shapeSide 0 u).val * (shapeSide 1 u).val = 0

	def designatedZRight (parentShape : H → Fin 3 → ℕ)
	(c : ClassKey (H := H)) (u : Shape) : Prop :=
	(parentShape c.1 0 - (shapeSide 0 u).val) *
	(parentShape c.1 1 - (shapeSide 1 u).val) = 0

	end MatrixMultiplication.JointPopulationCompatibility

	end

	end OAI