Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.VertexCover.Repetition.Completion
import OAI.Computability.VertexCover.Repetition.SelectedCommon
import OAI.Computability.VertexCover.Repetition.SelectedLikelihood
import OAI.Computability.VertexCover.Repetition.RevealFactorization

namespace OAI

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namespace UniqueGames.Foundations.Repetition

open scoped BigOperators
open Games
noncomputable section

section LocalNormalization

variable {I X Y : Type*} [Fintype I] [DecidableEq I]
  [Fintype X] [Fintype Y]

def completionTupleEquiv : (I → X × Y) ≃ ((I → X) × (I → Y)) where
  toFun u := (fun i => (u i).1, fun i => (u i).2)
  invFun z := fun i => (z.1 i, z.2 i)
  left_inv u := by funext i; rfl
  right_inv z := by cases z; rfl

def updatedTableFallback (μ : FiniteDistribution X) (j : I) (x : X) :
    FiniteDistribution (I → X) :=
  (FiniteDistribution.table (fun _ : I => μ)).pushforward
    (fun l => Function.update l j x)

theorem updatedTableFallback_zero (μ : FiniteDistribution X) (j : I) (x : X)
    (l : I → X) (hne : l j ≠ x) :
    (updatedTableFallback μ j x).weight l = 0 := by
  classical
  simp only [updatedTableFallback, FiniteDistribution.pushforward]
  apply Finset.sum_eq_zero
  intro t _
  apply ite_eq_right
  intro h
  have he := congrFun h j
  exact hne (by simpa only [Function.update_self] using he.symm)

theorem normalizeOr_weight_eq_zero_of_zero
    (w : X → ℝ) (hw : ∀ x, 0 ≤ w x) (fallback : FiniteDistribution X)
    (x : X) (hraw : w x = 0) (hdefault : fallback.weight x = 0) :
    (normalizeOr w hw fallback).weight x = 0 := by
  classical
  unfold normalizeOr
  split
  · change normalizedWeight w x = 0
    simp only [normalizedWeight, hraw, zero_div]
  · exact hdefault

end LocalNormalization

variable {Q₁ Q₂ A₁ A₂ : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
  [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

def selectedLeftRaw (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) : ℝ :=
  (if l j = x then 1 else 0) * partialLeftFactor G.questions j s.2 l *
    selectedLocalTest strategy.1 selected (fun i => (s.1.1 i).1) s.1.2.1 l

def selectedRightRaw (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) : ℝ :=
  (if r j = y then 1 else 0) * partialRightFactor G.questions j s.2 r *
    selectedLocalTest strategy.2 selected (fun i => (s.1.1 i).2) s.1.2.2 r

omit [DecidableEq Q₂] in
theorem selectedLeftRaw_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) :
    0 ≤ selectedLeftRaw G strategy selected j s x l := by
  apply mul_nonneg
  · apply mul_nonneg
    · split <;> norm_num
    · exact Finset.prod_nonneg (fun i _ => revealLeftFactor_nonnegative G.questions (s.2 i) (l i.1))
  · exact selectedLocalTest_nonnegative _ _ _ _ _

omit [DecidableEq Q₁] in
theorem selectedRightRaw_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) :
    0 ≤ selectedRightRaw G strategy selected j s y r := by
  apply mul_nonneg
  · apply mul_nonneg
    · split <;> norm_num
    · exact Finset.prod_nonneg (fun i _ => revealRightFactor_nonnegative G.questions (s.2 i) (r i.1))
  · exact selectedLocalTest_nonnegative _ _ _ _ _

def selectedLeftCompletion (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (x : Q₁) : FiniteDistribution ({i : Fin n // i ∉ selected} → Q₁) :=
  normalizeOr (selectedLeftRaw G strategy selected j s x)
    (selectedLeftRaw_nonnegative G strategy selected j s x)
    (updatedTableFallback (G.questions.pushforward Prod.fst) j x)

def selectedRightCompletion (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (y : Q₂) : FiniteDistribution ({i : Fin n // i ∉ selected} → Q₂) :=
  normalizeOr (selectedRightRaw G strategy selected j s y)
    (selectedRightRaw_nonnegative G strategy selected j s y)
    (updatedTableFallback (G.questions.pushforward Prod.snd) j y)

omit [DecidableEq Q₂] in
theorem selectedLeftCompletion_zero (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) (hne : l j ≠ x) :
    (selectedLeftCompletion G strategy selected j s x).weight l = 0 := by
  apply normalizeOr_weight_eq_zero_of_zero
  · simp only [selectedLeftRaw, ite_eq_right hne, zero_mul]
  · exact updatedTableFallback_zero _ j x l hne

omit [DecidableEq Q₁] in
theorem selectedRightCompletion_zero (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) (hne : r j ≠ y) :
    (selectedRightCompletion G strategy selected j s y).weight r = 0 := by
  apply normalizeOr_weight_eq_zero_of_zero
  · simp only [selectedRightRaw, ite_eq_right hne, zero_mul]
  · exact updatedTableFallback_zero _ j y r hne

omit [DecidableEq Q₂] in
theorem selectedLeftCompletion_support (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁)
    (h : (selectedLeftCompletion G strategy selected j s x).weight l ≠ 0) : l j = x := by
  by_contra hne
  exact h (selectedLeftCompletion_zero G strategy selected j s x l hne)

omit [DecidableEq Q₁] in
theorem selectedRightCompletion_support (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂)
    (h : (selectedRightCompletion G strategy selected j s y).weight r ≠ 0) : r j = y := by
  by_contra hne
  exact h (selectedRightCompletion_zero G strategy selected j s y r hne)

def selectedCompletionScale (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (xy : Q₁ × Q₂) : ℝ :=
  G.questions.weight xy * (∏ i : selected, G.questions.weight (s.1.1 i)) *
    (if selectedLabelAccepts G selected s.1.1 s.1.2 then 1 else 0) /
      G.selectedSuccess strategy selected

theorem selected_completion_factorization (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (xy : Q₁ × Q₂)
    (l : {i : Fin n // i ∉ selected} → Q₁)
    (r : {i : Fin n // i ∉ selected} → Q₂) :
    selectedCompletionScale G strategy selected j s xy *
        selectedLeftRaw G strategy selected j s xy.1 l *
        selectedRightRaw G strategy selected j s xy.2 r =
      if (l j, r j) = xy then
        partialRevealWeight G.questions j s.2 (fun i => (l i,r i)) *
          selectedOutsideLikelihood G strategy selected s.1 (fun i => (l i,r i))
      else 0 := by
  rcases xy with ⟨x,y⟩
  by_cases hl : l j = x <;> by_cases hr : r j = y
  · rw [ite_eq_left (Prod.ext hl hr), partialRevealWeight_factorization]
    cases hacc : selectedLabelAccepts G selected s.1.1 s.1.2 <;>
      simp [selectedOutsideLikelihood, selectedLikelihood_factorization,
        selectedCompletionScale, selectedLeftRaw, selectedRightRaw, hl, hr, hacc] ; ring
  · simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]
  · simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]
  · simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]

theorem selectedCommonLaw_completion_row_mass (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (xy : Q₁ × Q₂) :
    (selectedCommonLaw G strategy selected positive j).weight (s,xy) =
      ∑ lr : ({i : Fin n // i ∉ selected} → Q₁) ×
          ({i : Fin n // i ∉ selected} → Q₂),
        selectedCompletionScale G strategy selected j s xy *
          selectedLeftRaw G strategy selected j s xy.1 lr.1 *
          selectedRightRaw G strategy selected j s xy.2 lr.2 := by
  classical
  change (∑ u, if u j = xy then partialRevealWeight G.questions j s.2 u *
    selectedOutsideLikelihood G strategy selected s.1 u else 0) = _
  rw [← (completionTupleEquiv (I := {i : Fin n // i ∉ selected})
    (X := Q₁) (Y := Q₂)).sum_comp]
  apply Finset.sum_congr rfl
  intro u _
  exact (selected_completion_factorization G strategy selected j s xy
    (fun i => (u i).1) (fun i => (u i).2)).symm

theorem selected_completion_row_recombine (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
    (j : {i : Fin n // i ∉ selected})
    (s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (xy : Q₁ × Q₂)
    (l : {i : Fin n // i ∉ selected} → Q₁)
    (r : {i : Fin n // i ∉ selected} → Q₂) :
    (selectedCommonLaw G strategy selected positive j).weight (s,xy) *
      ((selectedLeftCompletion G strategy selected j s xy.1).product
        (selectedRightCompletion G strategy selected j s xy.2)).weight (l,r) =
      if (l j,r j) = xy then
        partialRevealWeight G.questions j s.2 (fun i => (l i,r i)) *
          selectedOutsideLikelihood G strategy selected s.1 (fun i => (l i,r i))
      else 0 := by
  rw [selectedCommonLaw_completion_row_mass]
  calc
    _ = selectedCompletionScale G strategy selected j s xy *
        selectedLeftRaw G strategy selected j s xy.1 l *
        selectedRightRaw G strategy selected j s xy.2 r :=
      factorized_completion_row
        (selectedLeftRaw G strategy selected j s xy.1)
        (selectedRightRaw G strategy selected j s xy.2)
        (selectedLeftRaw_nonnegative G strategy selected j s xy.1)
        (selectedRightRaw_nonnegative G strategy selected j s xy.2)
        (updatedTableFallback (G.questions.pushforward Prod.fst) j xy.1)
        (updatedTableFallback (G.questions.pushforward Prod.snd) j xy.2)
        (selectedCompletionScale G strategy selected j s xy) (l,r)
    _ = _ := selected_completion_factorization G strategy selected j s xy l r

end
end UniqueGames.Foundations.Repetition
end


end
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end

end OAI

	import Mathlib
	import OAI.Computability.VertexCover.Repetition.Completion
	import OAI.Computability.VertexCover.Repetition.SelectedCommon
	import OAI.Computability.VertexCover.Repetition.SelectedLikelihood
	import OAI.Computability.VertexCover.Repetition.RevealFactorization

	namespace OAI

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	section

	namespace UniqueGames.Foundations.Repetition

	open scoped BigOperators
	open Games
	noncomputable section

	section LocalNormalization

	variable {I X Y : Type*} [Fintype I] [DecidableEq I]
	[Fintype X] [Fintype Y]

	def completionTupleEquiv : (I → X × Y) ≃ ((I → X) × (I → Y)) where
	toFun u := (fun i => (u i).1, fun i => (u i).2)
	invFun z := fun i => (z.1 i, z.2 i)
	left_inv u := by funext i; rfl
	right_inv z := by cases z; rfl

	def updatedTableFallback (μ : FiniteDistribution X) (j : I) (x : X) :
	FiniteDistribution (I → X) :=
	(FiniteDistribution.table (fun _ : I => μ)).pushforward
	(fun l => Function.update l j x)

	theorem updatedTableFallback_zero (μ : FiniteDistribution X) (j : I) (x : X)
	(l : I → X) (hne : l j ≠ x) :
	(updatedTableFallback μ j x).weight l = 0 := by
	classical
	simp only [updatedTableFallback, FiniteDistribution.pushforward]
	apply Finset.sum_eq_zero
	intro t _
	apply ite_eq_right
	intro h
	have he := congrFun h j
	exact hne (by simpa only [Function.update_self] using he.symm)

	theorem normalizeOr_weight_eq_zero_of_zero
	(w : X → ℝ) (hw : ∀ x, 0 ≤ w x) (fallback : FiniteDistribution X)
	(x : X) (hraw : w x = 0) (hdefault : fallback.weight x = 0) :
	(normalizeOr w hw fallback).weight x = 0 := by
	classical
	unfold normalizeOr
	split
	· change normalizedWeight w x = 0
	simp only [normalizedWeight, hraw, zero_div]
	· exact hdefault

	end LocalNormalization

	variable {Q₁ Q₂ A₁ A₂ : Type*}
	[Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
	[DecidableEq Q₁] [DecidableEq Q₂] {n : Nat}

	def selectedLeftRaw (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) : ℝ :=
	(if l j = x then 1 else 0) * partialLeftFactor G.questions j s.2 l *
	selectedLocalTest strategy.1 selected (fun i => (s.1.1 i).1) s.1.2.1 l

	def selectedRightRaw (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) : ℝ :=
	(if r j = y then 1 else 0) * partialRightFactor G.questions j s.2 r *
	selectedLocalTest strategy.2 selected (fun i => (s.1.1 i).2) s.1.2.2 r

	omit [DecidableEq Q₂] in
	theorem selectedLeftRaw_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) :
	0 ≤ selectedLeftRaw G strategy selected j s x l := by
	apply mul_nonneg
	· apply mul_nonneg
	· split <;> norm_num
	· exact Finset.prod_nonneg (fun i _ => revealLeftFactor_nonnegative G.questions (s.2 i) (l i.1))
	· exact selectedLocalTest_nonnegative _ _ _ _ _

	omit [DecidableEq Q₁] in
	theorem selectedRightRaw_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) :
	0 ≤ selectedRightRaw G strategy selected j s y r := by
	apply mul_nonneg
	· apply mul_nonneg
	· split <;> norm_num
	· exact Finset.prod_nonneg (fun i _ => revealRightFactor_nonnegative G.questions (s.2 i) (r i.1))
	· exact selectedLocalTest_nonnegative _ _ _ _ _

	def selectedLeftCompletion (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(x : Q₁) : FiniteDistribution ({i : Fin n // i ∉ selected} → Q₁) :=
	normalizeOr (selectedLeftRaw G strategy selected j s x)
	(selectedLeftRaw_nonnegative G strategy selected j s x)
	(updatedTableFallback (G.questions.pushforward Prod.fst) j x)

	def selectedRightCompletion (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(y : Q₂) : FiniteDistribution ({i : Fin n // i ∉ selected} → Q₂) :=
	normalizeOr (selectedRightRaw G strategy selected j s y)
	(selectedRightRaw_nonnegative G strategy selected j s y)
	(updatedTableFallback (G.questions.pushforward Prod.snd) j y)

	omit [DecidableEq Q₂] in
	theorem selectedLeftCompletion_zero (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁) (hne : l j ≠ x) :
	(selectedLeftCompletion G strategy selected j s x).weight l = 0 := by
	apply normalizeOr_weight_eq_zero_of_zero
	· simp only [selectedLeftRaw, ite_eq_right hne, zero_mul]
	· exact updatedTableFallback_zero _ j x l hne

	omit [DecidableEq Q₁] in
	theorem selectedRightCompletion_zero (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂) (hne : r j ≠ y) :
	(selectedRightCompletion G strategy selected j s y).weight r = 0 := by
	apply normalizeOr_weight_eq_zero_of_zero
	· simp only [selectedRightRaw, ite_eq_right hne, zero_mul]
	· exact updatedTableFallback_zero _ j y r hne

	omit [DecidableEq Q₂] in
	theorem selectedLeftCompletion_support (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(x : Q₁) (l : {i : Fin n // i ∉ selected} → Q₁)
	(h : (selectedLeftCompletion G strategy selected j s x).weight l ≠ 0) : l j = x := by
	by_contra hne
	exact h (selectedLeftCompletion_zero G strategy selected j s x l hne)

	omit [DecidableEq Q₁] in
	theorem selectedRightCompletion_support (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(y : Q₂) (r : {i : Fin n // i ∉ selected} → Q₂)
	(h : (selectedRightCompletion G strategy selected j s y).weight r ≠ 0) : r j = y := by
	by_contra hne
	exact h (selectedRightCompletion_zero G strategy selected j s y r hne)

	def selectedCompletionScale (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(xy : Q₁ × Q₂) : ℝ :=
	G.questions.weight xy * (∏ i : selected, G.questions.weight (s.1.1 i)) *
	(if selectedLabelAccepts G selected s.1.1 s.1.2 then 1 else 0) /
	G.selectedSuccess strategy selected

	theorem selected_completion_factorization (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(xy : Q₁ × Q₂)
	(l : {i : Fin n // i ∉ selected} → Q₁)
	(r : {i : Fin n // i ∉ selected} → Q₂) :
	selectedCompletionScale G strategy selected j s xy *
	selectedLeftRaw G strategy selected j s xy.1 l *
	selectedRightRaw G strategy selected j s xy.2 r =
	if (l j, r j) = xy then
	partialRevealWeight G.questions j s.2 (fun i => (l i,r i)) *
	selectedOutsideLikelihood G strategy selected s.1 (fun i => (l i,r i))
	else 0 := by
	rcases xy with ⟨x,y⟩
	by_cases hl : l j = x <;> by_cases hr : r j = y
	· rw [ite_eq_left (Prod.ext hl hr), partialRevealWeight_factorization]
	cases hacc : selectedLabelAccepts G selected s.1.1 s.1.2 <;>
	simp [selectedOutsideLikelihood, selectedLikelihood_factorization,
	selectedCompletionScale, selectedLeftRaw, selectedRightRaw, hl, hr, hacc] ; ring
	· simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]
	· simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]
	· simp [selectedLeftRaw, selectedRightRaw, hl, hr, Prod.mk.injEq]

	theorem selectedCommonLaw_completion_row_mass (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
	(j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(xy : Q₁ × Q₂) :
	(selectedCommonLaw G strategy selected positive j).weight (s,xy) =
	∑ lr : ({i : Fin n // i ∉ selected} → Q₁) ×
	({i : Fin n // i ∉ selected} → Q₂),
	selectedCompletionScale G strategy selected j s xy *
	selectedLeftRaw G strategy selected j s xy.1 lr.1 *
	selectedRightRaw G strategy selected j s xy.2 lr.2 := by
	classical
	change (∑ u, if u j = xy then partialRevealWeight G.questions j s.2 u *
	selectedOutsideLikelihood G strategy selected s.1 u else 0) = _
	rw [← (completionTupleEquiv (I := {i : Fin n // i ∉ selected})
	(X := Q₁) (Y := Q₂)).sum_comp]
	apply Finset.sum_congr rfl
	intro u _
	exact (selected_completion_factorization G strategy selected j s xy
	(fun i => (u i).1) (fun i => (u i).2)).symm

	theorem selected_completion_row_recombine (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
	(j : {i : Fin n // i ∉ selected})
	(s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(xy : Q₁ × Q₂)
	(l : {i : Fin n // i ∉ selected} → Q₁)
	(r : {i : Fin n // i ∉ selected} → Q₂) :
	(selectedCommonLaw G strategy selected positive j).weight (s,xy) *
	((selectedLeftCompletion G strategy selected j s xy.1).product
	(selectedRightCompletion G strategy selected j s xy.2)).weight (l,r) =
	if (l j,r j) = xy then
	partialRevealWeight G.questions j s.2 (fun i => (l i,r i)) *
	selectedOutsideLikelihood G strategy selected s.1 (fun i => (l i,r i))
	else 0 := by
	rw [selectedCommonLaw_completion_row_mass]
	calc
	_ = selectedCompletionScale G strategy selected j s xy *
	selectedLeftRaw G strategy selected j s xy.1 l *
	selectedRightRaw G strategy selected j s xy.2 r :=
	factorized_completion_row
	(selectedLeftRaw G strategy selected j s xy.1)
	(selectedRightRaw G strategy selected j s xy.2)
	(selectedLeftRaw_nonnegative G strategy selected j s xy.1)
	(selectedRightRaw_nonnegative G strategy selected j s xy.2)
	(updatedTableFallback (G.questions.pushforward Prod.fst) j xy.1)
	(updatedTableFallback (G.questions.pushforward Prod.snd) j xy.2)
	(selectedCompletionScale G strategy selected j s xy) (l,r)
	_ = _ := selected_completion_factorization G strategy selected j s xy l r

	end
	end UniqueGames.Foundations.Repetition
	end


	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end

	end OAI