openAI-math-dataset / lean /OAI /Computability /VertexCover /Repetition /SelectedCommonMarginal.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Repetition.SelectedJointGame | |
| import OAI.Computability.VertexCover.Repetition.QuestionMarginal | |
| import OAI.Computability.VertexCover.Repetition.DistributionMaps | |
| import OAI.Computability.VertexCover.Repetition.SelectedCommon | |
| namespace OAI | |
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| namespace UniqueGames.Foundations.Repetition | |
| open scoped BigOperators | |
| open Games Information | |
| noncomputable section | |
| theorem pushforward_snd_weight_eq_secondMarginal {A B : Type*} | |
| [Fintype A] [Fintype B] (μ : FiniteDistribution (A × B)) : | |
| (μ.pushforward Prod.snd).weight = secondMarginal μ.weight := by | |
| classical | |
| funext b | |
| simp [FiniteDistribution.pushforward, secondMarginal, Fintype.sum_prod_type] | |
| variable {I T X Y : Type*} [Fintype I] [DecidableEq I] [Fintype T] | |
| [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y] | |
| def revealDataSplitEquiv (j : I) : | |
| (T × (I → X ⊕ Y)) ≃ ((T × ({i : I // i ≠ j} → X ⊕ Y)) × (X ⊕ Y)) where | |
| toFun z := ((z.1, fun i => z.2 i.1), z.2 j) | |
| invFun z := (z.1.1, mergeAt j z.2 z.1.2) | |
| left_inv z := by | |
| apply Prod.ext | |
| · rfl | |
| · funext i | |
| by_cases h : i = j <;> simp [mergeAt, h] | |
| right_inv z := by | |
| apply Prod.ext | |
| · apply Prod.ext | |
| · rfl | |
| · funext i | |
| exact mergeAt_other j z.2 z.1.2 i | |
| · exact mergeAt_self j z.2 z.1.2 | |
| theorem partialRevealMarginal_secondMarginal | |
| (μ : FiniteDistribution (X × Y)) (j : I) | |
| (likelihood : T → (I → X × Y) → ℝ) : | |
| secondMarginal (partialRevealMarginal μ j likelihood) = | |
| secondMarginal (fullRevealMarginal μ j likelihood) := by | |
| classical | |
| funext q | |
| calc | |
| _ = ∑ z : (T × ({i : I // i ≠ j} → X ⊕ Y)) × (X ⊕ Y), | |
| maskedJoint (partialRevealMarginal μ j likelihood) (z,q) := by | |
| simp [secondMarginal, Fintype.sum_prod_type, maskedJoint] | |
| _ = _ := by | |
| apply Fintype.sum_equiv (revealDataSplitEquiv (T := T) j).symm | |
| intro z | |
| exact (fullRevealMarginal_merge μ j likelihood z.1.1 z.1.2 z.2 q).symm | |
| variable {Q₁ Q₂ A₁ A₂ : Type*} | |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] | |
| [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat} | |
| def selectedObservationDataEquiv (selected : Finset (Fin n)) : | |
| (SelectedInput Q₁ Q₂ selected × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) ≃ | |
| (((selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) × | |
| ({i : Fin n // i ∉ selected} → Q₁ ⊕ Q₂)) where | |
| toFun z := ((z.1.1,z.2),z.1.2) | |
| invFun z := ((z.1.1,z.2),z.1.2) | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| theorem selectedFullReveal_secondMarginal (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}) : | |
| secondMarginal (fullRevealMarginal G.questions j | |
| (selectedOutsideLikelihood G strategy selected)) = | |
| secondMarginal (selectedRawMarginal G strategy selected j) := by | |
| classical | |
| funext q | |
| apply Fintype.sum_equiv | |
| (selectedObservationDataEquiv (Q₁ := Q₁) (Q₂ := Q₂) | |
| (A₁ := A₁) (A₂ := A₂) selected).symm | |
| intro z | |
| exact (selectedRawMarginal_fullReveal G strategy selected j (z,q)).symm | |
| theorem selectedRawMarginal_secondMarginal (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}) : | |
| secondMarginal (selectedRawMarginal G strategy selected j) = | |
| (selectedQuestionMarginal G strategy selected positive j.1).weight := by | |
| classical | |
| funext q | |
| calc | |
| _ = (selectedJointLaw G strategy selected positive).probability | |
| (fun z => decide (z.2 j = q)) := by | |
| simp only [secondMarginal, selectedRawMarginal, FiniteDistribution.probability, | |
| selectedJointLaw, toGameLaw, Fintype.sum_prod_type, decide_eq_true_eq] | |
| _ = ((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).probability | |
| (fun questions => decide ((questions.1 j.1,questions.2 j.1) = q)) := by | |
| have h := selectedJointLaw_question_probability G strategy selected positive | |
| (fun questions => decide ((questions.1 j.1,questions.2 j.1) = q)) | |
| simpa [selectedQuestionTuple, mergeCoordinates, j.property] using h | |
| _ = _ := by | |
| rw [FiniteDistribution.weight_eq_probability_singleton, selectedQuestionMarginal, | |
| FiniteDistribution.probability_pushforward] | |
| theorem selectedCommonLaw_secondMarginal (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}) : | |
| secondMarginal (selectedCommonLaw G strategy selected positive j).weight = | |
| (selectedQuestionMarginal G strategy selected positive j.1).weight := by | |
| change secondMarginal (partialRevealMarginal G.questions j | |
| (selectedOutsideLikelihood G strategy selected)) = _ | |
| rw [partialRevealMarginal_secondMarginal, selectedFullReveal_secondMarginal] | |
| exact selectedRawMarginal_secondMarginal G strategy selected positive j | |
| theorem selectedCommonLaw_questions_pushforward (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}) : | |
| (selectedCommonLaw G strategy selected positive j).pushforward Prod.snd = | |
| selectedQuestionMarginal G strategy selected positive j.1 := by | |
| classical | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro q | |
| exact (congrFun (pushforward_snd_weight_eq_secondMarginal | |
| (selectedCommonLaw G strategy selected positive j)) q).trans | |
| (congrFun (selectedCommonLaw_secondMarginal G strategy selected positive j) q) | |
| end | |
| end UniqueGames.Foundations.Repetition | |
| end | |
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| end | |
| end OAI | |