Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.VertexCover.Repetition.SelectedJointGame
import OAI.Computability.VertexCover.Games.KernelSampling
import OAI.Computability.VertexCover.Repetition.DistributionMaps
import OAI.Computability.VertexCover.Repetition.SelectedCompletion

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
                                                                                                      
section

namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
open Games
noncomputable section

theorem partialRevealWeight_forget
    {I X Y : Type*} [Fintype I] [DecidableEq I]
    [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]
    (μ : FiniteDistribution (X × Y)) (j : I) (u : I → X × Y) :
    (∑ rest : {i : I // i ≠ j} → X ⊕ Y, partialRevealWeight μ j rest u) =
      ∏ i, μ.weight (u i) := by
  have h := reveal_product_forget μ (fun i : {i : I // i ≠ j} => u i.1)
  simp_rw [reveal_product_factorization] at h
  simp only [partialRevealWeight, ← Finset.mul_sum]
  rw [h]
  exact (Fintype.prod_eq_mul_prod_subtype_ne (fun i => μ.weight (u i)) j).symm

theorem partialReveal_event_forget
    {I X Y : Type*} [Fintype I] [DecidableEq I]
    [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]
    (μ : FiniteDistribution (X × Y)) (j : I) (f : (I → X × Y) → ℝ) :
    (∑ rest : {i : I // i ≠ j} → X ⊕ Y,
      ∑ u : I → X × Y, partialRevealWeight μ j rest u * f u) =
      ∑ u, (∏ i, μ.weight (u i)) * f u := by
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro u _
  rw [← Finset.sum_mul, partialRevealWeight_forget]

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

omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedOutsideLikelihood_event_sum (G : Game Q₁ Q₂ A₁ A₂)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (selected : Finset (Fin n))
    (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
    (∑ t : (selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
      ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
        (∏ i, G.questions.weight (u i)) * selectedOutsideLikelihood G strategy selected t u *
          (if event (selectedQuestionTuple selected t.1 u) then 1 else 0)) =
      (G.repetition n).questions.probability
        (fun q => G.selectedWins strategy selected q && event q) /
          G.selectedSuccess strategy selected := by
  classical
  have hlabel (fixed : selected → Q₁ × Q₂)
      (u : {i : Fin n // i ∉ selected} → Q₁ × Q₂) :
      (∑ label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
        (∏ i, G.questions.weight (u i)) *
          selectedOutsideLikelihood G strategy selected (fixed,label) u *
            (if event (selectedQuestionTuple selected fixed u) then 1 else 0)) =
      ((∏ i : selected, G.questions.weight (fixed i)) * (∏ i, G.questions.weight (u i)) *
        (if G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) &&
          event (selectedQuestionTuple selected fixed u) then 1 else 0)) /
            G.selectedSuccess strategy selected := by
    calc
      _ = (∏ i, G.questions.weight (u i)) *
          (∏ i : selected, G.questions.weight (fixed i)) *
          (∑ label, selectedLikelihood G strategy selected fixed label u) *
          (if event (selectedQuestionTuple selected fixed u) then 1 else 0) /
            G.selectedSuccess strategy selected := by
        simp only [selectedOutsideLikelihood, div_eq_mul_inv, Finset.mul_sum, Finset.sum_mul]
        apply Finset.sum_congr rfl
        intro label _
        ring
      _ = _ := by
        rw [selectedLikelihood_sum]
        cases hw : G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) <;>
          cases he : event (selectedQuestionTuple selected fixed u) <;> simp [] ; ring
  calc
    _ = ∑ fixed : selected → Q₁ × Q₂,
        ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
          ∑ label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
            (∏ i, G.questions.weight (u i)) *
              selectedOutsideLikelihood G strategy selected (fixed,label) u *
                (if event (selectedQuestionTuple selected fixed u) then 1 else 0) := by
      rw [Fintype.sum_prod_type]
      apply Finset.sum_congr rfl
      intro fixed _
      rw [Finset.sum_comm]
    _ = (∑ fixed : selected → Q₁ × Q₂,
        ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
          (∏ i : selected, G.questions.weight (fixed i)) * (∏ i, G.questions.weight (u i)) *
            (if G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) &&
              event (selectedQuestionTuple selected fixed u) then 1 else 0)) /
                G.selectedSuccess strategy selected := by
      simp_rw [hlabel]
      simp only [div_eq_mul_inv, Finset.sum_mul]
    _ = _ := by
      congr 1
      have h := selectedSplit_question_probability G selected
        (fun q => G.selectedWins strategy selected q && event q)
      simpa [selectedSplitLaw, FiniteDistribution.probability, FiniteDistribution.product,
        FiniteDistribution.table, Fintype.sum_prod_type, mul_ite] using h

theorem selected_partial_event_total (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})
    (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
    (∑ s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j,
      ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
        partialRevealWeight G.questions j s.2 u *
          selectedOutsideLikelihood G strategy selected s.1 u *
            (if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0)) =
      (G.repetition n).questions.probability
        (fun q => G.selectedWins strategy selected q && event q) /
          G.selectedSuccess strategy selected := by
  have hrest (t : (selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) :=
    partialReveal_event_forget G.questions j
      (fun u => selectedOutsideLikelihood G strategy selected t u *
        (if event (selectedQuestionTuple selected t.1 u) then 1 else 0))
  simp only [← mul_assoc] at hrest
  rw [Fintype.sum_prod_type]
  simp_rw [hrest]
  exact selectedOutsideLikelihood_event_sum G strategy selected event

def selectedFullLeftCompletion (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})
    (q : Q₁ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j) :
    FiniteDistribution (Fin n → Q₁) :=
  (selectedLeftCompletion G strategy selected j q.2 q.1).pushforward
    (mergeCoordinates selected (fun i => (q.2.1.1 i).1))

def selectedFullRightCompletion (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})
    (q : Q₂ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j) :
    FiniteDistribution (Fin n → Q₂) :=
  (selectedRightCompletion G strategy selected j q.2 q.1).pushforward
    (mergeCoordinates selected (fun i => (q.2.1.1 i).2))

omit [DecidableEq Q₂] in
theorem selectedFullLeftCompletion_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})
    (q : Q₁ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (l : Fin n → Q₁) (h : (selectedFullLeftCompletion G strategy selected j q).weight l ≠ 0) :
    l j.1 = q.1 := by
  classical
  by_contra hne
  apply h
  simp only [selectedFullLeftCompletion, FiniteDistribution.pushforward]
  apply Finset.sum_eq_zero
  intro t _
  split
  · rename_i he
    apply selectedLeftCompletion_zero
    intro ht
    apply hne
    have hcoord := congrFun he j.1
    simpa [mergeCoordinates, j.property, ht] using hcoord.symm
  · rfl

omit [DecidableEq Q₁] in
theorem selectedFullRightCompletion_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})
    (q : Q₂ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
    (r : Fin n → Q₂) (h : (selectedFullRightCompletion G strategy selected j q).weight r ≠ 0) :
    r j.1 = q.1 := by
  classical
  by_contra hne
  apply h
  simp only [selectedFullRightCompletion, FiniteDistribution.pushforward]
  apply Finset.sum_eq_zero
  intro t _
  split
  · rename_i he
    apply selectedRightCompletion_zero
    intro ht
    apply hne
    have hcoord := congrFun he j.1
    simpa [mergeCoordinates, j.property, ht] using hcoord.symm
  · rfl

theorem selected_full_completion_probability (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) (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
    ((selectedFullLeftCompletion G strategy selected j (xy.1,s)).product
      (selectedFullRightCompletion G strategy selected j (xy.2,s))).probability event =
    ∑ lr : ({i : Fin n // i ∉ selected} → Q₁) × ({i : Fin n // i ∉ selected} → Q₂),
      ((selectedLeftCompletion G strategy selected j s xy.1).product
        (selectedRightCompletion G strategy selected j s xy.2)).weight lr *
          (if event (selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i)))
            then 1 else 0) := by
  rw [selectedFullLeftCompletion, selectedFullRightCompletion,
    ← FiniteDistribution.product_pushforward, FiniteDistribution.probability_pushforward]
  unfold FiniteDistribution.probability
  apply Finset.sum_congr rfl
  intro lr _
  have he : selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i)) =
      (mergeCoordinates selected (fun i => (s.1.1 i).1) lr.1,
       mergeCoordinates selected (fun i => (s.1.1 i).2) lr.2) :=
    Prod.ext (selectedQuestionTuple_left selected s.1.1 _)
      (selectedQuestionTuple_right selected s.1.1 _)
  rw [he]
  cases hevent : event (mergeCoordinates selected (fun i => (s.1.1 i).1) lr.1,
    mergeCoordinates selected (fun i => (s.1.1 i).2) lr.2) <;> simp [hevent]

theorem selected_completion_event_row (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)
    (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
    (∑ xy : Q₁ × Q₂, (selectedCommonLaw G strategy selected positive j).weight (s,xy) *
      ((selectedFullLeftCompletion G strategy selected j (xy.1,s)).product
        (selectedFullRightCompletion G strategy selected j (xy.2,s))).probability event) =
    ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
      partialRevealWeight G.questions j s.2 u *
        selectedOutsideLikelihood G strategy selected s.1 u *
          (if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
  classical
  have hpoint (xy : Q₁ × Q₂)
      (lr : ({i : Fin n // i ∉ selected} → Q₁) × ({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 lr =
      if (lr.1 j,lr.2 j) = xy then
        partialRevealWeight G.questions j s.2 (fun i => (lr.1 i,lr.2 i)) *
          selectedOutsideLikelihood G strategy selected s.1 (fun i => (lr.1 i,lr.2 i)) else 0 :=
    selected_completion_row_recombine G strategy selected positive j s xy lr.1 lr.2
  simp_rw [selected_full_completion_probability, Finset.mul_sum, ← mul_assoc,
    hpoint]
  have hsum (xy : Q₁ × Q₂) :
      (∑ lr : ({i : Fin n // i ∉ selected} → Q₁) × ({i : Fin n // i ∉ selected} → Q₂),
        (if (lr.1 j,lr.2 j) = xy then
          partialRevealWeight G.questions j s.2 (fun i => (lr.1 i,lr.2 i)) *
            selectedOutsideLikelihood G strategy selected s.1 (fun i => (lr.1 i,lr.2 i)) else 0) *
          (if event (selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i))) then 1 else 0)) =
      ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
        (if u j = xy then partialRevealWeight G.questions j s.2 u *
          selectedOutsideLikelihood G strategy selected s.1 u else 0) *
            (if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
    exact ((completionTupleEquiv (I := {i : Fin n // i ∉ selected})
      (X := Q₁) (Y := Q₂)).sum_comp _).symm
  simp_rw [hsum]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro u _
  rw [← Finset.sum_mul]
  simp

def selectedFullCompletionMixture (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}) :
    FiniteDistribution ((Fin n → Q₁) × (Fin n → Q₂)) :=
  ((selectedCommonLaw G strategy selected positive j).transport (Equiv.prodComm _ _)).mixture
    (fun z => (selectedFullLeftCompletion G strategy selected j (z.1.1,z.2)).product
      (selectedFullRightCompletion G strategy selected j (z.1.2,z.2)))

theorem selectedFullCompletionMixture_probability (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})
    (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
    (selectedFullCompletionMixture G strategy selected positive j).probability event =
      ((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).probability event := by
  erw [FiniteDistribution.probability_condition]
  rw [selectedFullCompletionMixture, FiniteDistribution.probability_mixture]
  simp only [FiniteDistribution.transport]
  rw [Fintype.sum_prod_type, Finset.sum_comm]
  calc
    _ = ∑ s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j,
        ∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
          partialRevealWeight G.questions j s.2 u *
            selectedOutsideLikelihood G strategy selected s.1 u *
              (if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
      apply Finset.sum_congr rfl
      intro s _
      exact selected_completion_event_row G strategy selected positive j s event
    _ = _ := selected_partial_event_total G strategy selected j event

theorem selectedFullCompletionMixture_eq_conditionedQuestions (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}) :
    selectedFullCompletionMixture G strategy selected positive j =
      (G.repetition n).questions.condition (G.selectedWins strategy selected) positive := by
  classical
  apply FiniteDistribution.eq_of_weight_eq
  intro q
  rw [FiniteDistribution.weight_eq_probability_singleton,
    FiniteDistribution.weight_eq_probability_singleton]
  exact selectedFullCompletionMixture_probability G strategy selected positive j _

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

	import Mathlib
	import OAI.Computability.VertexCover.Repetition.SelectedJointGame
	import OAI.Computability.VertexCover.Games.KernelSampling
	import OAI.Computability.VertexCover.Repetition.DistributionMaps
	import OAI.Computability.VertexCover.Repetition.SelectedCompletion

	namespace OAI

	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section

	section

	namespace UniqueGames.Foundations.Repetition
	open scoped BigOperators
	open Games
	noncomputable section

	theorem partialRevealWeight_forget
	{I X Y : Type*} [Fintype I] [DecidableEq I]
	[Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]
	(μ : FiniteDistribution (X × Y)) (j : I) (u : I → X × Y) :
	(∑ rest : {i : I // i ≠ j} → X ⊕ Y, partialRevealWeight μ j rest u) =
	∏ i, μ.weight (u i) := by
	have h := reveal_product_forget μ (fun i : {i : I // i ≠ j} => u i.1)
	simp_rw [reveal_product_factorization] at h
	simp only [partialRevealWeight, ← Finset.mul_sum]
	rw [h]
	exact (Fintype.prod_eq_mul_prod_subtype_ne (fun i => μ.weight (u i)) j).symm

	theorem partialReveal_event_forget
	{I X Y : Type*} [Fintype I] [DecidableEq I]
	[Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]
	(μ : FiniteDistribution (X × Y)) (j : I) (f : (I → X × Y) → ℝ) :
	(∑ rest : {i : I // i ≠ j} → X ⊕ Y,
	∑ u : I → X × Y, partialRevealWeight μ j rest u * f u) =
	∑ u, (∏ i, μ.weight (u i)) * f u := by
	rw [Finset.sum_comm]
	apply Finset.sum_congr rfl
	intro u _
	rw [← Finset.sum_mul, partialRevealWeight_forget]

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

	omit [DecidableEq Q₁] [DecidableEq Q₂] in
	theorem selectedOutsideLikelihood_event_sum (G : Game Q₁ Q₂ A₁ A₂)
	(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
	(selected : Finset (Fin n))
	(event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
	(∑ t : (selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	(∏ i, G.questions.weight (u i)) * selectedOutsideLikelihood G strategy selected t u *
	(if event (selectedQuestionTuple selected t.1 u) then 1 else 0)) =
	(G.repetition n).questions.probability
	(fun q => G.selectedWins strategy selected q && event q) /
	G.selectedSuccess strategy selected := by
	classical
	have hlabel (fixed : selected → Q₁ × Q₂)
	(u : {i : Fin n // i ∉ selected} → Q₁ × Q₂) :
	(∑ label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
	(∏ i, G.questions.weight (u i)) *
	selectedOutsideLikelihood G strategy selected (fixed,label) u *
	(if event (selectedQuestionTuple selected fixed u) then 1 else 0)) =
	((∏ i : selected, G.questions.weight (fixed i)) * (∏ i, G.questions.weight (u i)) *
	(if G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) &&
	event (selectedQuestionTuple selected fixed u) then 1 else 0)) /
	G.selectedSuccess strategy selected := by
	calc
	_ = (∏ i, G.questions.weight (u i)) *
	(∏ i : selected, G.questions.weight (fixed i)) *
	(∑ label, selectedLikelihood G strategy selected fixed label u) *
	(if event (selectedQuestionTuple selected fixed u) then 1 else 0) /
	G.selectedSuccess strategy selected := by
	simp only [selectedOutsideLikelihood, div_eq_mul_inv, Finset.mul_sum, Finset.sum_mul]
	apply Finset.sum_congr rfl
	intro label _
	ring
	_ = _ := by
	rw [selectedLikelihood_sum]
	cases hw : G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) <;>
	cases he : event (selectedQuestionTuple selected fixed u) <;> simp [] ; ring
	calc
	_ = ∑ fixed : selected → Q₁ × Q₂,
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	∑ label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected,
	(∏ i, G.questions.weight (u i)) *
	selectedOutsideLikelihood G strategy selected (fixed,label) u *
	(if event (selectedQuestionTuple selected fixed u) then 1 else 0) := by
	rw [Fintype.sum_prod_type]
	apply Finset.sum_congr rfl
	intro fixed _
	rw [Finset.sum_comm]
	_ = (∑ fixed : selected → Q₁ × Q₂,
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	(∏ i : selected, G.questions.weight (fixed i)) * (∏ i, G.questions.weight (u i)) *
	(if G.selectedWins strategy selected (selectedQuestionTuple selected fixed u) &&
	event (selectedQuestionTuple selected fixed u) then 1 else 0)) /
	G.selectedSuccess strategy selected := by
	simp_rw [hlabel]
	simp only [div_eq_mul_inv, Finset.sum_mul]
	_ = _ := by
	congr 1
	have h := selectedSplit_question_probability G selected
	(fun q => G.selectedWins strategy selected q && event q)
	simpa [selectedSplitLaw, FiniteDistribution.probability, FiniteDistribution.product,
	FiniteDistribution.table, Fintype.sum_prod_type, mul_ite] using h

	theorem selected_partial_event_total (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})
	(event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
	(∑ s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j,
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	partialRevealWeight G.questions j s.2 u *
	selectedOutsideLikelihood G strategy selected s.1 u *
	(if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0)) =
	(G.repetition n).questions.probability
	(fun q => G.selectedWins strategy selected q && event q) /
	G.selectedSuccess strategy selected := by
	have hrest (t : (selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) :=
	partialReveal_event_forget G.questions j
	(fun u => selectedOutsideLikelihood G strategy selected t u *
	(if event (selectedQuestionTuple selected t.1 u) then 1 else 0))
	simp only [← mul_assoc] at hrest
	rw [Fintype.sum_prod_type]
	simp_rw [hrest]
	exact selectedOutsideLikelihood_event_sum G strategy selected event

	def selectedFullLeftCompletion (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})
	(q : Q₁ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j) :
	FiniteDistribution (Fin n → Q₁) :=
	(selectedLeftCompletion G strategy selected j q.2 q.1).pushforward
	(mergeCoordinates selected (fun i => (q.2.1.1 i).1))

	def selectedFullRightCompletion (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})
	(q : Q₂ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j) :
	FiniteDistribution (Fin n → Q₂) :=
	(selectedRightCompletion G strategy selected j q.2 q.1).pushforward
	(mergeCoordinates selected (fun i => (q.2.1.1 i).2))

	omit [DecidableEq Q₂] in
	theorem selectedFullLeftCompletion_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})
	(q : Q₁ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(l : Fin n → Q₁) (h : (selectedFullLeftCompletion G strategy selected j q).weight l ≠ 0) :
	l j.1 = q.1 := by
	classical
	by_contra hne
	apply h
	simp only [selectedFullLeftCompletion, FiniteDistribution.pushforward]
	apply Finset.sum_eq_zero
	intro t _
	split
	· rename_i he
	apply selectedLeftCompletion_zero
	intro ht
	apply hne
	have hcoord := congrFun he j.1
	simpa [mergeCoordinates, j.property, ht] using hcoord.symm
	· rfl

	omit [DecidableEq Q₁] in
	theorem selectedFullRightCompletion_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})
	(q : Q₂ × SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j)
	(r : Fin n → Q₂) (h : (selectedFullRightCompletion G strategy selected j q).weight r ≠ 0) :
	r j.1 = q.1 := by
	classical
	by_contra hne
	apply h
	simp only [selectedFullRightCompletion, FiniteDistribution.pushforward]
	apply Finset.sum_eq_zero
	intro t _
	split
	· rename_i he
	apply selectedRightCompletion_zero
	intro ht
	apply hne
	have hcoord := congrFun he j.1
	simpa [mergeCoordinates, j.property, ht] using hcoord.symm
	· rfl

	theorem selected_full_completion_probability (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) (event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
	((selectedFullLeftCompletion G strategy selected j (xy.1,s)).product
	(selectedFullRightCompletion G strategy selected j (xy.2,s))).probability event =
	∑ lr : ({i : Fin n // i ∉ selected} → Q₁) × ({i : Fin n // i ∉ selected} → Q₂),
	((selectedLeftCompletion G strategy selected j s xy.1).product
	(selectedRightCompletion G strategy selected j s xy.2)).weight lr *
	(if event (selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i)))
	then 1 else 0) := by
	rw [selectedFullLeftCompletion, selectedFullRightCompletion,
	← FiniteDistribution.product_pushforward, FiniteDistribution.probability_pushforward]
	unfold FiniteDistribution.probability
	apply Finset.sum_congr rfl
	intro lr _
	have he : selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i)) =
	(mergeCoordinates selected (fun i => (s.1.1 i).1) lr.1,
	mergeCoordinates selected (fun i => (s.1.1 i).2) lr.2) :=
	Prod.ext (selectedQuestionTuple_left selected s.1.1 _)
	(selectedQuestionTuple_right selected s.1.1 _)
	rw [he]
	cases hevent : event (mergeCoordinates selected (fun i => (s.1.1 i).1) lr.1,
	mergeCoordinates selected (fun i => (s.1.1 i).2) lr.2) <;> simp [hevent]

	theorem selected_completion_event_row (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)
	(event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
	(∑ xy : Q₁ × Q₂, (selectedCommonLaw G strategy selected positive j).weight (s,xy) *
	((selectedFullLeftCompletion G strategy selected j (xy.1,s)).product
	(selectedFullRightCompletion G strategy selected j (xy.2,s))).probability event) =
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	partialRevealWeight G.questions j s.2 u *
	selectedOutsideLikelihood G strategy selected s.1 u *
	(if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
	classical
	have hpoint (xy : Q₁ × Q₂)
	(lr : ({i : Fin n // i ∉ selected} → Q₁) × ({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 lr =
	if (lr.1 j,lr.2 j) = xy then
	partialRevealWeight G.questions j s.2 (fun i => (lr.1 i,lr.2 i)) *
	selectedOutsideLikelihood G strategy selected s.1 (fun i => (lr.1 i,lr.2 i)) else 0 :=
	selected_completion_row_recombine G strategy selected positive j s xy lr.1 lr.2
	simp_rw [selected_full_completion_probability, Finset.mul_sum, ← mul_assoc,
	hpoint]
	have hsum (xy : Q₁ × Q₂) :
	(∑ lr : ({i : Fin n // i ∉ selected} → Q₁) × ({i : Fin n // i ∉ selected} → Q₂),
	(if (lr.1 j,lr.2 j) = xy then
	partialRevealWeight G.questions j s.2 (fun i => (lr.1 i,lr.2 i)) *
	selectedOutsideLikelihood G strategy selected s.1 (fun i => (lr.1 i,lr.2 i)) else 0) *
	(if event (selectedQuestionTuple selected s.1.1 (fun i => (lr.1 i,lr.2 i))) then 1 else 0)) =
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	(if u j = xy then partialRevealWeight G.questions j s.2 u *
	selectedOutsideLikelihood G strategy selected s.1 u else 0) *
	(if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
	exact ((completionTupleEquiv (I := {i : Fin n // i ∉ selected})
	(X := Q₁) (Y := Q₂)).sum_comp _).symm
	simp_rw [hsum]
	rw [Finset.sum_comm]
	apply Finset.sum_congr rfl
	intro u _
	rw [← Finset.sum_mul]
	simp

	def selectedFullCompletionMixture (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}) :
	FiniteDistribution ((Fin n → Q₁) × (Fin n → Q₂)) :=
	((selectedCommonLaw G strategy selected positive j).transport (Equiv.prodComm _ _)).mixture
	(fun z => (selectedFullLeftCompletion G strategy selected j (z.1.1,z.2)).product
	(selectedFullRightCompletion G strategy selected j (z.1.2,z.2)))

	theorem selectedFullCompletionMixture_probability (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})
	(event : ((Fin n → Q₁) × (Fin n → Q₂)) → Bool) :
	(selectedFullCompletionMixture G strategy selected positive j).probability event =
	((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).probability event := by
	erw [FiniteDistribution.probability_condition]
	rw [selectedFullCompletionMixture, FiniteDistribution.probability_mixture]
	simp only [FiniteDistribution.transport]
	rw [Fintype.sum_prod_type, Finset.sum_comm]
	calc
	_ = ∑ s : SelectedCommonData (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected j,
	∑ u : {i : Fin n // i ∉ selected} → Q₁ × Q₂,
	partialRevealWeight G.questions j s.2 u *
	selectedOutsideLikelihood G strategy selected s.1 u *
	(if event (selectedQuestionTuple selected s.1.1 u) then 1 else 0) := by
	apply Finset.sum_congr rfl
	intro s _
	exact selected_completion_event_row G strategy selected positive j s event
	_ = _ := selected_partial_event_total G strategy selected j event

	theorem selectedFullCompletionMixture_eq_conditionedQuestions (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}) :
	selectedFullCompletionMixture G strategy selected positive j =
	(G.repetition n).questions.condition (G.selectedWins strategy selected) positive := by
	classical
	apply FiniteDistribution.eq_of_weight_eq
	intro q
	rw [FiniteDistribution.weight_eq_probability_singleton,
	FiniteDistribution.weight_eq_probability_singleton]
	exact selectedFullCompletionMixture_probability G strategy selected positive j _

	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