openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /HierarchicalProjectedRawAverage.lean
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalProjectedRawMeeting | |
| import OAI.Computability.PerfectCompleteness.Foundations.TupleIndexEmitterLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalProjectedRawAverage | |
| noncomputable section | |
| open scoped BigOperators Classical TensorProduct | |
| open TreeSourceSpaces HierarchicalArrays UpperParameterScalars | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| private theorem event_mass_le_expectation {X : Type*} [Fintype X] | |
| (μ : FiniteDistribution X) (event : X → Bool) (value : X → ℝ) (factor : ℝ) | |
| (hnonneg : ∀ x, 0 ≤ value x) | |
| (hbound : ∀ x, event x = true → factor ≤ value x) : | |
| factor * μ.probability event ≤ μ.expectation value := by | |
| unfold FiniteDistribution.probability FiniteDistribution.expectation | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_le_sum | |
| intro x _ | |
| by_cases hx : event x = true | |
| · rw [ite_eq_left hx] | |
| calc | |
| factor * μ.weight x = μ.weight x * factor := mul_comm _ _ | |
| _ ≤ μ.weight x * value x := | |
| mul_le_mul_of_nonneg_left (hbound x hx) (μ.nonnegative x) | |
| · rw [ite_eq_right hx, mul_zero] | |
| exact mul_nonneg (μ.nonnegative x) (hnonneg x) | |
| variable {δ : ℚ} (plan : FixedRows.Plan δ) {branch : Nat → Nat} {t : Nat} | |
| (slots projected : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot) | |
| (projection : ∀ s j, MixedSupport.Projection (slots s j) (projected s j)) | |
| (upper lower : Nodes branch plan.depth) (lowerLevel : Nat) | |
| {Ω : Type*} [Fintype Ω] | |
| (original : FiniteDistribution Ω) (originalArrays : Ω → Arrays slots (FixedRows.rows plan)) | |
| (lowerEvent : Ω → Bool) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t)) | |
| (A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) plan.order) | |
| (a : Block (FixedRows.rows plan) lower) (cut : OwnInputReference.Cut upper lower) | |
| abbrev Raw := HierarchicalProjectedRawPrediction.Raw projected (FixedRows.rows plan) upper lower A | |
| (FixedRows.repeats plan) cut | |
| abbrev Visible := HierarchicalProjectedRawPrediction.Visible projected (FixedRows.rows plan) upper lower A | |
| (FixedRows.repeats plan) cut | |
| def law (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : | |
| FiniteDistribution (Raw plan projected upper lower A cut) := | |
| OwnInputReference.rawLaw projected (FixedRows.rows plan) upper lower (LinearMap.ker A) | |
| (FixedRows.repeats plan) cut externalLaw | |
| abbrev J := HierarchicalProjectedRawPrediction.rawJ slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut | |
| abbrev prediction := HierarchicalProjectedRawPrediction.rawPrediction slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut a | |
| def goodRecord (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) | |
| (v : Visible plan projected upper lower A cut) : Bool := | |
| PositiveFiberMass.goodRecord (law plan projected upper lower A cut externalLaw) | |
| (fun sample : Raw plan projected upper lower A cut => sample.2) | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut) | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) | |
| (rhoPred (InitialParameters.useful δ) plan.density) v | |
| def goodEvent (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) | |
| (sample : Raw plan projected upper lower A cut) : Bool := | |
| J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && | |
| goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut | |
| externalLaw sample.2 | |
| def factor : ℝ := | |
| (1 / (2 : ℝ) ^ plan.order) * | |
| (tau (InitialParameters.useful δ) plan.density plan.order | |
| (FixedRows.rows plan (Nodes.height upper)) ^ 2 / | |
| (2 * (2 : ℝ) ^ FixedRows.rows plan (Nodes.height upper))) | |
| theorem factor_nonneg : 0 ≤ factor plan upper := by | |
| unfold factor | |
| positivity | |
| abbrev meetingProbability := | |
| HierarchicalProjectedRawMeeting.meetingProbability plan slots projected projection upper lower lowerLevel | |
| original originalArrays lowerEvent σ A a cut | |
| def expectedMeeting (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : ℝ := | |
| (law plan projected upper lower A cut externalLaw).expectation | |
| (fun sample => meetingProbability plan slots projected projection upper lower lowerLevel | |
| original originalArrays lowerEvent σ A a cut sample.2) | |
| theorem meeting_lower_of_goodRecord (hδ : 0 < δ) | |
| (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) | |
| (v : Visible plan projected upper lower A cut) | |
| (hgood : goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ | |
| A a cut externalLaw v = true) : | |
| factor plan upper ≤ | |
| meetingProbability plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut v := by | |
| have hpositive := (PositiveFiberMass.goodRecord_iff | |
| (law plan projected upper lower A cut externalLaw) | |
| (fun sample : Raw plan projected upper lower A cut => sample.2) | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut) | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) | |
| (rhoPred (InitialParameters.useful δ) plan.density) v).mp hgood | |
| let originalGiven : Raw plan projected upper lower A cut → Bool := fun sample => | |
| J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && | |
| @decide (sample.2 = v) (Classical.propDecidable _) | |
| have hgiven : originalGiven = | |
| HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut v := by | |
| funext sample | |
| unfold originalGiven HierarchicalProjectedRawPrediction.given | |
| apply congrArg (fun flag : Bool => | |
| J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && flag) | |
| exact (@decide_eq_decide _ _ _ _).mpr Iff.rfl | |
| have positive : 0 < (law plan projected upper lower A cut externalLaw).probability | |
| (HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut v) := by | |
| rw [← hgiven] | |
| exact hpositive.1 | |
| have conditional : rhoPred (InitialParameters.useful δ) plan.density ≤ | |
| ((law plan projected upper lower A cut externalLaw).condition | |
| (HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut v) positive).probability | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) := by | |
| have hprediction := hpositive.2 | |
| have hmatch : (fun sample : Raw plan projected upper lower A cut => | |
| originalGiven sample && | |
| prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) = | |
| (fun sample => | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && | |
| prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) && | |
| @decide (sample.2 = v) (Classical.propDecidable _)) := by | |
| funext sample | |
| dsimp only [originalGiven] | |
| cases J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample <;> | |
| cases prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample <;> | |
| cases @decide (sample.2 = v) (Classical.propDecidable _) <;> rfl | |
| change rhoPred (InitialParameters.useful δ) plan.density ≤ | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (fun sample => | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && | |
| prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) && | |
| @decide (sample.2 = v) (Classical.propDecidable _)) / | |
| (law plan projected upper lower A cut externalLaw).probability originalGiven at hprediction | |
| rw [← hmatch, hgiven] at hprediction | |
| exact hprediction.trans_eq | |
| (FiniteDistribution.probability_condition (law plan projected upper lower A cut externalLaw) | |
| (HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel | |
| original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A | |
| (FixedRows.repeats plan) cut v) | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) | |
| positive).symm | |
| exact HierarchicalProjectedRawMeeting.meeting_probability_lower plan slots projected projection upper lower lowerLevel | |
| original originalArrays lowerEvent σ A a cut v hδ externalLaw positive conditional | |
| theorem expectedMeeting_ge_good_mass (hδ : 0 < δ) | |
| (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : | |
| factor plan upper * | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (goodEvent plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw) ≤ | |
| expectedMeeting plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw := by | |
| apply event_mass_le_expectation | |
| · intro sample | |
| exact HierarchicalProjectedRawMeeting.meetingProbability_nonneg plan slots projected projection upper lower lowerLevel | |
| original originalArrays lowerEvent σ A a cut sample.2 | |
| · intro sample hs | |
| exact meeting_lower_of_goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ | |
| A a cut hδ externalLaw sample.2 (Bool.and_eq_true_iff.mp hs).2 | |
| theorem J_and_prediction : | |
| (fun sample => J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && | |
| prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) = | |
| prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut := by | |
| funext sample | |
| unfold J prediction HierarchicalProjectedRawPrediction.rawJ HierarchicalProjectedRawPrediction.rawPrediction | |
| HierarchicalPrediction.prediction | |
| exact Bool.and_self_left _ _ | |
| theorem prediction_difference_le_good_mass (hδ : 0 < δ) | |
| (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) - | |
| rhoPred (InitialParameters.useful δ) plan.density * | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut) ≤ | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (goodEvent plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw) := by | |
| have h := PositiveFiberMass.difference_le_good_mass | |
| (law plan projected upper lower A cut externalLaw) | |
| (fun sample : Raw plan projected upper lower A cut => sample.2) | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut) | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) | |
| (rhoPred (InitialParameters.useful δ) plan.density) | |
| (rhoPred_pos (InitialParameters.useful_pos hδ) plan.density_pos).le | |
| rw [J_and_prediction] at h | |
| exact h | |
| theorem expectedMeeting_ge_prediction_difference (hδ : 0 < δ) | |
| (externalLaw : FiniteDistribution | |
| (OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : | |
| factor plan upper * | |
| ((law plan projected upper lower A cut externalLaw).probability | |
| (prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) - | |
| rhoPred (InitialParameters.useful δ) plan.density * | |
| (law plan projected upper lower A cut externalLaw).probability | |
| (J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut)) ≤ | |
| expectedMeeting plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw := | |
| (mul_le_mul_of_nonneg_left | |
| (prediction_difference_le_good_mass plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ | |
| A a cut hδ externalLaw) (factor_nonneg plan upper)).trans | |
| (expectedMeeting_ge_good_mass plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ | |
| A a cut hδ externalLaw) | |
| end | |
| end PerfectCompleteness.HierarchicalProjectedRawAverage | |
| end | |
| end OAI | |