openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedUpperPairErrorLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedProjectionErrorLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.UpperCutTriangle | |
| import OAI.Computability.PerfectCompleteness.Decoding.UpperScalarCutCalls | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedError | |
| import OAI.Computability.PerfectCompleteness.Sampling.CommonProductVariationLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.HierarchicalBackgroundVariation | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedUpperPairError | |
| open FixedParameters FixedRows DescendantSpaces | |
| noncomputable section | |
| variable {δ : ℚ} {hδ : 0 < δ} (params : Parameters δ hδ) | |
| {root upper lower : Nat} | |
| theorem lower_errors_lt (pref : Path (branch params) root (upper + 1)) | |
| (segment : Path (branch params) (upper + 1) (lower + 1)) | |
| (direction : BucketSampler.Direction (rows params.plan (upper + 1))) | |
| (hroot : root ≤ params.plan.depth) : | |
| let L : ℝ := ChildBlockCardinality.bound (branch params) lower | |
| (sourceLength params.plan hδ) | |
| (UpperScalarCutCalls.count (rows params.plan) (repeats params.plan) | |
| root (upper + 1) (lower + 1)) (rows params.plan) | |
| Real.sqrt ((1 + (projectionProbability params lower : ℝ) ^ 2 * (L - 1)) ^ | |
| branch params lower - 1) / 2 + | |
| Real.sqrt (L ^ 2 / branch params lower) / 2 < 2 * params.accuracy := by | |
| have hcalls := UpperScalarCutCalls.count_le_fixed_calls params.plan pref segment direction hroot | |
| have hsparse := FixedProjectionError.sparse_error_lt params hcalls lower | |
| have hhidden := FixedStoppedError.hidden_error_lt params hcalls lower | |
| dsimp only | |
| linarith | |
| theorem three_errors_lt (pref : Path (branch params) root (upper + 1)) | |
| (segment : Path (branch params) (upper + 1) (lower + 1)) | |
| (direction : BucketSampler.Direction (rows params.plan (upper + 1))) | |
| (hroot : root ≤ params.plan.depth) : | |
| let L : ℝ := ChildBlockCardinality.bound (branch params) lower | |
| (sourceLength params.plan hδ) | |
| (UpperScalarCutCalls.count (rows params.plan) (repeats params.plan) | |
| root (upper + 1) (lower + 1)) (rows params.plan) | |
| Real.sqrt ((1 + (projectionProbability params lower : ℝ) ^ 2 * (L - 1)) ^ | |
| branch params lower - 1) / 2 + | |
| Real.sqrt (L ^ 2 / branch params lower) / 2 + | |
| Real.sqrt ((ChildBlockCardinality.bound (branch params) upper | |
| (sourceLength params.plan hδ) | |
| (OriginalCutCalls.count (rows params.plan) (repeats params.plan) root (upper + 1) + 1) | |
| (rows params.plan) : ℝ) ^ 2 / branch params upper) / 2 < | |
| 3 * params.accuracy := by | |
| have hlower := lower_errors_lt params pref segment direction hroot | |
| have hupper := FixedStoppedError.collision_error_lt params hroot pref.height_le | |
| dsimp only at hlower ⊢ | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedUpperPairError | |
| end | |
| section | |
| namespace PerfectCompleteness.FixedUpperCutTriangle | |
| open FixedParameters FixedRows RecursiveSpaces DescendantSpaces | |
| open TreeSourceSpaces HierarchicalArrays OriginalWholeCutTape WholeArrayInteriorExterior | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| variable {δ : ℚ} {hδ : 0 < δ} (params : Parameters δ hδ) | |
| {root upper lower : Nat} | |
| theorem rows_positive (height : Nat) : 0 < rows params.plan height := | |
| lt_of_lt_of_le Nat.zero_lt_one (params.plan.rows_pos (params.plan.depth - height)) | |
| variable (pref : Path (branch params) root (upper + 1)) | |
| (hcut : lower + 1 ≤ upper + 1) | |
| (slots : Slots (branch params) root → Fin (sourceLength params.plan hδ) → MixedSupport.Slot) | |
| local instance backgroundFintype : | |
| Fintype (HierarchicalAgreementMean.Background (rows := rows params.plan) slots (upperNode pref)) := | |
| Fintype.ofFinite _ | |
| local instance rowSpaceFintype : Fintype (NodeEmbedding.RowSpace slots (upperNode pref)) := | |
| Fintype.ofFinite _ | |
| def upperReferenceLaw : | |
| FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows params.plan) slots (upperNode pref)) := | |
| HierarchicalAgreementMean.referenceLaw slots (upperNode pref) | |
| (OriginalOwnBucketSplit.backgroundLaw (rows params.plan) (repeats params.plan) pref slots) | |
| (by simpa only [upperNode_height] using rows_positive params (upper + 1)) | |
| variable {Z : UpperCutTriangle.Prefix (branch params) upper lower → Fin (branch params lower) → Type*} | |
| [∀ lowerPref child, Fintype (Z lowerPref child)] | |
| (projected : (lowerPref : UpperCutTriangle.Prefix (branch params) upper lower) → | |
| (child : Fin (branch params lower)) → Z lowerPref child → | |
| Slots (branch params) lower → Fin (sourceLength params.plan hδ) → MixedSupport.Slot) | |
| (projection : ∀ lowerPref child z s a, MixedSupport.Projection | |
| (childSlots (cutSlots (pref.append (GeometricCutSplit.prefixPath hcut lowerPref)) slots) child s a) | |
| (projected lowerPref child z s a)) | |
| (choiceLaw : (lowerPref : UpperCutTriangle.Prefix (branch params) upper lower) → | |
| (child : Fin (branch params lower)) → FiniteDistribution (Z lowerPref child)) | |
| def modifiedPairLaw : | |
| FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows params.plan) slots (upperNode pref)) := | |
| UpperCutTriangle.modifiedPairLaw (rows params.plan) (repeats params.plan) pref hcut slots | |
| (fun k _ => branch_pos params k) (rows_positive params (upper + 1)) | |
| projected projection choiceLaw (projectionProbability params lower : ℝ) | |
| (by exact_mod_cast (projectionProbability_bounds params lower).1.le) | |
| (by exact_mod_cast (projectionProbability_bounds params lower).2.le) | |
| def originalReferenceLaw : | |
| FiniteDistribution (HierarchicalAgreementMean.PairRecord (rows := rows params.plan) slots (upperNode pref)) := | |
| HierarchicalAgreementMean.referenceLaw slots (upperNode pref) | |
| ((modifiedPairLaw params pref hcut slots projected projection choiceLaw).pushforward Prod.fst) | |
| (by simpa only [upperNode_height] using rows_positive params (upper + 1)) | |
| theorem modified_reference_lt (hroot : root ≤ params.plan.depth) : | |
| (modifiedPairLaw params pref hcut slots projected projection choiceLaw).totalVariation | |
| (upperReferenceLaw params pref slots) < 3 * params.accuracy := by | |
| have hcomparison := UpperCutTriangle.modified_reference_variation | |
| (rows params.plan) (repeats params.plan) pref hcut slots | |
| (fun k _ => branch_pos params k) (rows_positive params (upper + 1)) | |
| projected projection choiceLaw (projectionProbability params lower : ℝ) | |
| (by exact_mod_cast (projectionProbability_bounds params lower).1.le) | |
| (by exact_mod_cast (projectionProbability_bounds params lower).2.le) | |
| let lowerPref : UpperCutTriangle.Prefix (branch params) upper lower := | |
| Classical.choice (GeometricCutSplit.prefix_nonempty hcut (fun k _ => branch_pos params k)) | |
| let direction : BucketSampler.Direction (rows params.plan (upper + 1)) := | |
| BucketUniform.coordinateDirection ⟨0, rows_positive params (upper + 1)⟩ | |
| have herror := FixedUpperPairError.three_errors_lt params pref | |
| (GeometricCutSplit.prefixPath hcut lowerPref) direction hroot | |
| simpa only [modifiedPairLaw, upperReferenceLaw] using hcomparison.trans_lt herror | |
| theorem original_reference_lt (hroot : root ≤ params.plan.depth) : | |
| (modifiedPairLaw params pref hcut slots projected projection choiceLaw).totalVariation | |
| (originalReferenceLaw params pref hcut slots projected projection choiceLaw) < | |
| 6 * params.accuracy := by | |
| have h := HierarchicalBackgroundVariation.original_background_variation slots (upperNode pref) | |
| (modifiedPairLaw params pref hcut slots projected projection choiceLaw) | |
| (OriginalOwnBucketSplit.backgroundLaw (rows params.plan) (repeats params.plan) pref slots) | |
| (by simpa only [upperNode_height] using rows_positive params (upper + 1)) | |
| change (modifiedPairLaw params pref hcut slots projected projection choiceLaw).totalVariation | |
| (originalReferenceLaw params pref hcut slots projected projection choiceLaw) ≤ | |
| 2 * (modifiedPairLaw params pref hcut slots projected projection choiceLaw).totalVariation | |
| (upperReferenceLaw params pref slots) at h | |
| have hthree := modified_reference_lt params pref hcut slots projected projection choiceLaw hroot | |
| linarith | |
| theorem original_reference_le_ten (hroot : root ≤ params.plan.depth) : | |
| (modifiedPairLaw params pref hcut slots projected projection choiceLaw).totalVariation | |
| (originalReferenceLaw params pref hcut slots projected projection choiceLaw) ≤ | |
| 10 * params.accuracy := by | |
| have h := original_reference_lt params pref hcut slots projected projection choiceLaw hroot | |
| have hpositive := params.accuracy_pos | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedUpperCutTriangle | |
| end | |
| section | |
| namespace PerfectCompleteness.FixedUpperCutFamily | |
| open FixedParameters FixedRows RecursiveSpaces DescendantSpaces | |
| open TreeSourceSpaces HierarchicalArrays OriginalWholeCutTape WholeArrayInteriorExterior | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance] linearMapFintype | |
| variable {δ : ℚ} {hδ : 0 < δ} (params : Parameters δ hδ) | |
| {root upper lower : Nat} {O : Type*} [Fintype O] | |
| (pref : O → Path (branch params) root (upper + 1)) | |
| (hcut : lower + 1 ≤ upper + 1) | |
| (slots : O → Slots (branch params) root → Fin (sourceLength params.plan hδ) → MixedSupport.Slot) | |
| local instance backgroundFintype (o : O) : | |
| Fintype (HierarchicalAgreementMean.Background (rows := rows params.plan) (slots o) (upperNode (pref o))) := | |
| Fintype.ofFinite _ | |
| local instance rowSpaceFintype (o : O) : | |
| Fintype (NodeEmbedding.RowSpace (slots o) (upperNode (pref o))) := Fintype.ofFinite _ | |
| abbrev Record := Σ o : O, | |
| HierarchicalAgreementMean.PairRecord (rows := rows params.plan) (slots o) (upperNode (pref o)) | |
| variable {Z : O → UpperCutTriangle.Prefix (branch params) upper lower → Fin (branch params lower) → Type*} | |
| [∀ o lowerPref child, Fintype (Z o lowerPref child)] | |
| (projected : (o : O) → (lowerPref : UpperCutTriangle.Prefix (branch params) upper lower) → | |
| (child : Fin (branch params lower)) → Z o lowerPref child → | |
| Slots (branch params) lower → Fin (sourceLength params.plan hδ) → MixedSupport.Slot) | |
| (projection : ∀ o lowerPref child z s a, MixedSupport.Projection | |
| (childSlots (cutSlots ((pref o).append (GeometricCutSplit.prefixPath hcut lowerPref)) (slots o)) child s a) | |
| (projected o lowerPref child z s a)) | |
| (choiceLaw : (o : O) → (lowerPref : UpperCutTriangle.Prefix (branch params) upper lower) → | |
| (child : Fin (branch params lower)) → FiniteDistribution (Z o lowerPref child)) | |
| def modifiedLaw (outerLaw : FiniteDistribution O) : | |
| FiniteDistribution (Record params pref slots) := | |
| CompletionSoundness.sigmaLaw outerLaw (fun o => | |
| FixedUpperCutTriangle.modifiedPairLaw params (pref o) hcut (slots o) | |
| (projected o) (projection o) (choiceLaw o)) | |
| def referenceLaw (outerLaw : FiniteDistribution O) : | |
| FiniteDistribution (Record params pref slots) := | |
| CompletionSoundness.sigmaLaw outerLaw (fun o => | |
| FixedUpperCutTriangle.originalReferenceLaw params (pref o) hcut (slots o) | |
| (projected o) (projection o) (choiceLaw o)) | |
| theorem totalVariation_le_six (outerLaw : FiniteDistribution O) | |
| (hroot : root ≤ params.plan.depth) : | |
| (modifiedLaw params pref hcut slots projected projection choiceLaw outerLaw).totalVariation | |
| (referenceLaw params pref hcut slots projected projection choiceLaw outerLaw) ≤ | |
| 6 * params.accuracy := by | |
| have h := ConditionalVariation.observed_sigma_le_const outerLaw | |
| (fun o => FixedUpperCutTriangle.modifiedPairLaw params (pref o) hcut (slots o) | |
| (projected o) (projection o) (choiceLaw o)) | |
| (fun o => FixedUpperCutTriangle.originalReferenceLaw params (pref o) hcut (slots o) | |
| (projected o) (projection o) (choiceLaw o)) | |
| (6 * params.accuracy) | |
| (fun o => (FixedUpperCutTriangle.original_reference_lt params (pref o) hcut (slots o) | |
| (projected o) (projection o) (choiceLaw o) hroot).le) | |
| (id : Record params pref slots → Record params pref slots) | |
| simpa only [modifiedLaw, referenceLaw, FiniteDistribution.pushforward_id] using h | |
| theorem observed_totalVariation_le_ten {Γ : Type*} [Fintype Γ] | |
| (outerLaw : FiniteDistribution O) (hroot : root ≤ params.plan.depth) | |
| (observe : Record params pref slots → Γ) : | |
| ((modifiedLaw params pref hcut slots projected projection choiceLaw outerLaw).pushforward observe).totalVariation | |
| ((referenceLaw params pref hcut slots projected projection choiceLaw outerLaw).pushforward observe) ≤ | |
| 10 * params.accuracy := by | |
| have h := (DensityVariation.variation_pushforward_le _ _ observe).trans | |
| (totalVariation_le_six params pref hcut slots projected projection choiceLaw outerLaw hroot) | |
| have hpositive := params.accuracy_pos | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedUpperCutFamily | |
| end | |
| end OAI | |