| import Mathlib |
| import OAI.Computability.VertexCover.Repetition.SelectedInformation |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Repetition |
| open Games |
| noncomputable section |
| variable {Q₁ Q₂ A₁ A₂ : Type*} |
| [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] |
| [DecidableEq Q₁] [DecidableEq Q₂] {n : Nat} |
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| def selectedLabelAccepts (G : Game Q₁ Q₂ A₁ A₂) (selected : Finset (Fin n)) |
| (fixed : selected → Q₁ × Q₂) |
| (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) : Bool := by |
| classical |
| exact decide (∀ i : selected, G.accepts (fixed i).1 (fixed i).2 (label.1 i) (label.2 i) = true) |
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| def selectedLocalAnswers {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A)) |
| (selected : Finset (Fin n)) (fixed : selected → Q) |
| (remaining : {i : Fin n // i ∉ selected} → Q) : selected → A := |
| fun i => strategy (mergeCoordinates selected fixed remaining) i.1 |
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| def selectedLocalTest {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A)) |
| (selected : Finset (Fin n)) (fixed : selected → Q) (label : selected → A) |
| (remaining : {i : Fin n // i ∉ selected} → Q) : ℝ := by |
| classical |
| exact if selectedLocalAnswers strategy selected fixed remaining = label then 1 else 0 |
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| theorem selectedLocalTest_nonnegative {Q A : Type*} (strategy : (Fin n → Q) → (Fin n → A)) |
| (selected : Finset (Fin n)) (fixed : selected → Q) (label : selected → A) |
| (remaining : {i : Fin n // i ∉ selected} → Q) : |
| 0 ≤ selectedLocalTest strategy selected fixed label remaining := by |
| classical |
| unfold selectedLocalTest |
| split <;> norm_num |
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|
| omit [Fintype Q₁] [Fintype Q₂] [DecidableEq Q₁] [DecidableEq Q₂] in |
| theorem selectedQuestionTuple_left (selected : Finset (Fin n)) |
| (fixed : selected → Q₁ × Q₂) (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) : |
| (selectedQuestionTuple selected fixed remaining).1 = |
| mergeCoordinates selected (fun i => (fixed i).1) (fun i => (remaining i).1) := by |
| funext i |
| simp only [selectedQuestionTuple, mergeCoordinates] |
| split <;> rfl |
|
|
| omit [Fintype Q₁] [Fintype Q₂] [DecidableEq Q₁] [DecidableEq Q₂] in |
| theorem selectedQuestionTuple_right (selected : Finset (Fin n)) |
| (fixed : selected → Q₁ × Q₂) (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) : |
| (selectedQuestionTuple selected fixed remaining).2 = |
| mergeCoordinates selected (fun i => (fixed i).2) (fun i => (remaining i).2) := by |
| funext i |
| simp only [selectedQuestionTuple, mergeCoordinates] |
| split <;> rfl |
|
|
| omit [DecidableEq Q₁] [DecidableEq Q₂] in |
| theorem selectedWins_of_answerLabel (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (fixed : selected → Q₁ × Q₂) |
| (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) |
| (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) |
| (hlabel : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) = label) : |
| G.selectedWins strategy selected (selectedQuestionTuple selected fixed remaining) = |
| selectedLabelAccepts G selected fixed label := by |
| classical |
| have hleft (i : selected) := congrArg (fun l => l.1 i) hlabel |
| have hright (i : selected) := congrArg (fun l => l.2 i) hlabel |
| apply Bool.eq_iff_iff.mpr |
| simp only [Game.selectedWins, selectedLabelAccepts, decide_eq_true_eq] |
| constructor |
| · intro h i |
| have hi := h i.1 i.property |
| simpa [Game.coordinateWin, selectedQuestionTuple, mergeCoordinates, i.property, |
| ← hleft i, ← hright i, selectedAnswerLabel] using hi |
| · intro h i hi |
| have hv := h ⟨i,hi⟩ |
| simpa [Game.coordinateWin, selectedQuestionTuple, mergeCoordinates, hi, |
| ← hleft ⟨i,hi⟩, ← hright ⟨i,hi⟩, selectedAnswerLabel] using hv |
|
|
| omit [DecidableEq Q₁] [DecidableEq Q₂] in |
| theorem selectedLikelihood_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)) (fixed : selected → Q₁ × Q₂) |
| (label : SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) |
| (remaining : {i : Fin n // i ∉ selected} → Q₁ × Q₂) : |
| selectedLikelihood G strategy selected fixed label remaining = |
| (if selectedLabelAccepts G selected fixed label then 1 else 0) * |
| selectedLocalTest strategy.1 selected (fun i => (fixed i).1) label.1 (fun i => (remaining i).1) * |
| selectedLocalTest strategy.2 selected (fun i => (fixed i).2) label.2 (fun i => (remaining i).2) := by |
| classical |
| have he : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) = |
| (selectedLocalAnswers strategy.1 selected (fun i => (fixed i).1) (fun i => (remaining i).1), |
| selectedLocalAnswers strategy.2 selected (fun i => (fixed i).2) (fun i => (remaining i).2)) := by |
| simp only [selectedAnswerLabel, |
| selectedQuestionTuple_left, selectedQuestionTuple_right] |
| rfl |
| by_cases hl : selectedLocalAnswers strategy.1 selected (fun i => (fixed i).1) |
| (fun i => (remaining i).1) = label.1 |
| · by_cases hr : selectedLocalAnswers strategy.2 selected (fun i => (fixed i).2) |
| (fun i => (remaining i).2) = label.2 |
| · have hlabel : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) = label := by |
| rw [he, hl, hr] |
| simp only [selectedLikelihood, ite_eq_left hlabel, |
| selectedWins_of_answerLabel G strategy selected fixed label remaining hlabel, |
| selectedLocalTest, ite_eq_left hl, ite_eq_left hr, mul_one] |
| · have hn : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) ≠ label := by |
| intro h |
| rw [he] at h |
| exact hr (congrArg Prod.snd h) |
| simp [selectedLikelihood, hn, selectedLocalTest, hl, hr] |
| · have hn : selectedAnswerLabel strategy selected (selectedQuestionTuple selected fixed remaining) ≠ label := by |
| intro h |
| rw [he] at h |
| exact hl (congrArg Prod.fst h) |
| simp [selectedLikelihood, hn, selectedLocalTest, hl] |
|
|
| end |
| end UniqueGames.Foundations.Repetition |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
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| end |
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| end |
| end |
| end |
| end |
| end |
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|
| end OAI |
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