openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedProjectionErrorLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.ChildFiniteCallCollisionTransfer | |
| import OAI.Computability.PerfectCompleteness.Foundations.ProjectedPrefixComparison | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedComparisonErrorsLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedProjectionError | |
| open FixedParameters FixedRows | |
| noncomputable section | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : Parameters δ hδ) | |
| theorem sparse_error_lt {retained : Nat} (hcalls : retained ≤ calls p.plan) | |
| (height : Nat) : | |
| Real.sqrt ((1 + (projectionProbability p height : ℝ) ^ 2 * | |
| ((ChildBlockCardinality.bound (branch p) height (sourceLength p.plan hδ) | |
| retained (rows p.plan) : ℝ) - 1)) ^ branch p height - 1) / 2 < p.accuracy := by | |
| let L : ℝ := ChildBlockCardinality.bound (branch p) height (sourceLength p.plan hδ) | |
| retained (rows p.plan) | |
| let m : ℝ := p.cubeSize height | |
| have hL : 1 ≤ L := by | |
| dsimp only [L] | |
| exact_mod_cast Nat.one_le_iff_ne_zero.mpr | |
| (Nat.ne_of_gt (ChildBlockCardinality.shape_bound_pos (branch p) height | |
| (sourceLength p.plan hδ) retained (rows p.plan))) | |
| have hm4 : 0 ≤ m ^ 4 := by positivity | |
| have hbeta : (projectionProbability p height : ℝ) ^ 2 = 1 / m ^ 4 := by | |
| simp only [projectionProbability, Rat.cast_div, Rat.cast_one, Rat.cast_pow, | |
| Rat.cast_natCast] | |
| dsimp only [m] | |
| ring | |
| have hbase : 1 + (projectionProbability p height : ℝ) ^ 2 * (L - 1) ≤ | |
| 1 + L ^ 2 / m ^ 4 := by | |
| rw [hbeta] | |
| have hsq : L - 1 ≤ L ^ 2 := by nlinarith | |
| have hdiv := div_le_div_of_nonneg_right hsq hm4 | |
| calc | |
| 1 + (1 / m ^ 4) * (L - 1) = 1 + (L - 1) / m ^ 4 := by ring | |
| _ ≤ 1 + L ^ 2 / m ^ 4 := by linarith | |
| have hnonneg : 0 ≤ 1 + (projectionProbability p height : ℝ) ^ 2 * (L - 1) := by | |
| have : 0 ≤ L - 1 := by linarith | |
| positivity | |
| have h := (FixedComparisonErrors.comparison_errors p hcalls height).2.1 | |
| change Real.sqrt ((1 + (projectionProbability p height : ℝ) ^ 2 * (L - 1)) ^ | |
| (p.cubeSize height ^ 3) - 1) / 2 < p.accuracy | |
| apply lt_of_le_of_lt _ h | |
| apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2) | |
| apply Real.sqrt_le_sqrt | |
| exact sub_le_sub_right (pow_le_pow_left₀ hnonneg hbase _) 1 | |
| end | |
| end PerfectCompleteness.FixedProjectionError | |
| end | |
| section | |
| namespace PerfectCompleteness.FixedChildCollisionTransfer | |
| noncomputable section | |
| open scoped Classical | |
| open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ) | |
| theorem posterior_error_lt {retained : Nat} | |
| (hcalls : retained ≤ calls parameters.plan) (height : Nat) : | |
| 2 * ((projectionProbability parameters height : ℝ) * | |
| (ChildBlockCardinality.bound (branch parameters) height | |
| (sourceLength parameters.plan hδ) retained (rows parameters.plan) : ℝ) / | |
| (1 - (projectionProbability parameters height : ℝ))) < parameters.accuracy := by | |
| have h := (FixedComparisonErrors.comparison_errors parameters hcalls height).2.2 | |
| simpa only [projectionProbability, Rat.cast_div, Rat.cast_one, Rat.cast_pow, | |
| Rat.cast_natCast] using h | |
| theorem lower_pair_calls_le {n height : Nat} | |
| (path : Path (branch parameters) n (height + 1)) (hn : n ≤ parameters.plan.depth) : | |
| Fintype.card (Option | |
| (WholeCutCalls.Index (rows parameters.plan) (repeats parameters.plan) path)) ≤ | |
| calls parameters.plan := by | |
| have h := FixedCallBudget.modified_calls_le (rows parameters.plan) | |
| (repeats parameters.plan) path hn | |
| rw [Fintype.card_option] | |
| unfold calls | |
| omega | |
| variable {height : Nat} {C : Type*} [Fintype C] | |
| {Z : Fin (branch parameters height) → Type*} [∀ i, Fintype (Z i)] | |
| theorem marked_restricted_lt_uniform_full | |
| (hcalls : Fintype.card C ≤ calls parameters.plan) | |
| (slots : Slots (branch parameters) (height + 1) → | |
| Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot) | |
| (projected : (i : Fin (branch parameters height)) → Z i → | |
| Slots (branch parameters) height → Fin (sourceLength parameters.plan hδ) → | |
| MixedSupport.Slot) | |
| (projection : ∀ i z s k, MixedSupport.Projection | |
| (childSlots slots i s k) (projected i z s k)) | |
| (choiceLaw : (i : Fin (branch parameters height)) → FiniteDistribution (Z i)) | |
| (F₀ F₁ : CutChildGrouping.Raw (C := C) slots (rows parameters.plan) → | |
| Option (ChildBilinearCollisionTransfer.ParentForm slots)) : | |
| (SparseReplacement.markedLaw | |
| (fun i => FiniteDistribution.uniform | |
| (ChildCallNumbering.Raw (C := C) (rows parameters.plan) (childSlots slots i))) | |
| (ChildCallNumberingLaws.replacementLaws (C := C) (rows parameters.plan) | |
| (childSlots slots) projected projection choiceLaw) | |
| (projectionProbability parameters height : ℝ) | |
| (by exact_mod_cast (projectionProbability_bounds parameters height).1.le) | |
| (by exact_mod_cast (projectionProbability_bounds parameters height).2.le)).probability | |
| (fun z => BilinearCollisionTransfer.restrictedCollision | |
| (F₀ z.2) (F₁ z.2) (ChildUnmarkedSpace.space slots z.1)) < | |
| (CutChildGrouping.rawLaw (C := C) slots (rows parameters.plan)).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision (F₀ raw) (F₁ raw)) + | |
| 2 * parameters.accuracy := by | |
| have hβ : 0 ≤ (projectionProbability parameters height : ℝ) := by | |
| exact_mod_cast (projectionProbability_bounds parameters height).1.le | |
| have hβ' : (projectionProbability parameters height : ℝ) < 1 := by | |
| exact_mod_cast (projectionProbability_bounds parameters height).2 | |
| have htransfer := ChildFiniteCallCollisionTransfer.marked_restricted_le_uniform_full | |
| (C := C) (rows parameters.plan) slots projected projection choiceLaw | |
| (projectionProbability parameters height : ℝ) hβ hβ' F₀ F₁ | |
| have hposterior := posterior_error_lt parameters hcalls height | |
| have hvariation := FixedProjectionError.sparse_error_lt parameters hcalls height | |
| change _ < (FiniteProduct.law (fun i => FiniteDistribution.uniform | |
| (ChildCallNumbering.Raw (C := C) (rows parameters.plan) (childSlots slots i)))).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision (F₀ raw) (F₁ raw)) + | |
| 2 * parameters.accuracy | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedChildCollisionTransfer | |
| end | |
| end OAI | |