openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FinitePreliminaryCompletion.lean
| import OAI.Computability.PerfectCompleteness.Construction.CanonicalLocalCompletion | |
| import OAI.Computability.PerfectCompleteness.Construction.FlexibleLocalCompletionLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedQueryCodec | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FinitePreliminaryCompletion | |
| open MixedSupport CanonicalKeys CanonicalEdges CanonicalLabelEncoding | |
| open UniqueGamesTheorem.Foundations.Games | |
| open CompletionSoundness CompletionSoundness.LegalProjectionGame | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n t v m : Nat} {rows repeats : Nat → Nat} | |
| abbrev alphabet (branch : Nat → Nat) (n t : Nat) (δ : ℚ) : Nat := | |
| CanonicalLocalCompletion.alphabet (TreeCanonical.locationCount branch n t) δ | |
| def finiteInput (δ : ℚ) (data : MetadataFreeTape.Input branch n t rows repeats) : | |
| LocalCompletionFamily.Input (alphabet branch n t δ) := | |
| CanonicalPartialTable.lookup | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (CanonicalLocalCompletion.partitionWidth_le_alphabet | |
| (TreeCanonical.locationCount branch n t) δ) | |
| (FixedQueryCodec.partialInput data) | |
| def finiteResult (δ : ℚ) (data : MetadataFreeTape.Input branch n t rows repeats) : | |
| LocalCompletionFamily.Result (alphabet branch n t δ) := | |
| LocalCompletionFamily.lookup | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (finiteInput δ data) | |
| theorem finiteResult_correct (δ : ℚ) | |
| (data : MetadataFreeTape.Input branch n t rows repeats) : | |
| LocalCompletionFamily.Correct (finiteInput δ data) (finiteResult δ data) := | |
| LocalCompletionFamily.lookup_correct | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (finiteInput δ data) | |
| (FixedQueryCodec.partialInput_admissible | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (CanonicalLocalCompletion.partitionWidth_le_alphabet | |
| (TreeCanonical.locationCount branch n t) δ) data) | |
| abbrev blockFamily (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (branch : Nat → Nat) (n t : Nat) (rows repeats : Nat → Nat) := | |
| HierarchicalGame.toBlockFamily | |
| (PreliminarySampler.family clauses branch n t rows repeats) | |
| variable (clauses : Fin m → SourceClause.NormalizedClause v) | |
| local notation "F" => blockFamily clauses branch n t rows repeats | |
| def rawInput (δ : ℚ) (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| LocalCompletionFamily.Input (alphabet branch n t δ) := | |
| finiteInput δ (MetadataFreeTape.encodeRaw clauses rows repeats e) | |
| theorem rawInput_mask (δ : ℚ) | |
| (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| (rawInput clauses δ e).1 = paddedMask (CanonicalGame.leftKey F e) := by | |
| unfold rawInput finiteInput | |
| rw [FixedQueryCodec.partialInput_encodeRaw, CanonicalPartialTable.lookup_mask_actual, | |
| FixedQueryCodec.leftTable_key] | |
| rfl | |
| theorem rawInput_image_legal (δ : ℚ) | |
| (e : PreliminarySampler.Raw clauses branch n t rows repeats) | |
| (P : CanonicalGame.LeftLabel F (CanonicalGame.leftAt F e)) : | |
| (rawInput clauses δ e).2 | |
| (CanonicalLocalCompletion.legalL F δ (CanonicalGame.leftAt F e) P) = | |
| CanonicalLocalCompletion.legalR F δ (CanonicalGame.rightAt F e) | |
| (CanonicalGame.edge F e P) := by | |
| let slots := MetadataFreeSampler.sourceSlots clauses e.1.1 | |
| let arrays := WholeArraySampler.evaluate rows repeats | |
| (GeometricPath.leafPath e.1.2) slots e.2.1 | |
| let S := TreeCanonical.numberedSlots slots | |
| let f := TreeCanonical.numberedFunction slots (HierarchicalArrays.fullJoint arrays) | |
| let θ := FixedQueryCodec.rawTestIndex clauses e | |
| let Pencoded : Label S (FixedQueryCodec.leftEncode branch n rows ∘ f) := | |
| CanonicalCoarseningEncoding.transportFine S f | |
| (FixedQueryCodec.leftEncode branch n rows) | |
| (FixedQueryCodec.leftEncode_injective branch n rows) P | |
| have hfinite := CanonicalPartialTable.image_actual_legal | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (CanonicalLocalCompletion.partitionWidth_le_alphabet | |
| (TreeCanonical.locationCount branch n t) δ) | |
| S (FixedQueryCodec.leftTable arrays) (FixedQueryCodec.outputMap θ) Pencoded | |
| have hcoarse := CanonicalCoarseningEncoding.labelEmbedding_coarsen .right | |
| (CanonicalLocalCompletion.partitionWidth_le_alphabet | |
| (TreeCanonical.locationCount branch n t) δ) | |
| S f (FixedQueryCodec.project θ) (FixedQueryCodec.leftEncode branch n rows) | |
| (FixedQueryCodec.rightEncode θ) (FixedQueryCodec.outputMap θ) | |
| (FixedQueryCodec.leftEncode_injective branch n rows) | |
| (FixedQueryCodec.rightEncode_injective θ) (FixedQueryCodec.outputMap_leftEncode θ) P | |
| unfold rawInput finiteInput | |
| rw [FixedQueryCodec.partialInput_encodeRaw] | |
| exact hfinite.trans hcoarse.symm | |
| local instance leftVertex_fintype : Fintype (CanonicalGame.LeftVertex F) := | |
| Fintype.ofFinite _ | |
| local instance rightVertex_fintype : Fintype (CanonicalGame.RightVertex F) := | |
| Fintype.ofFinite _ | |
| local instance leftLabel_fintype (x : CanonicalGame.LeftVertex F) : | |
| Fintype (CanonicalGame.LeftLabel F x) := Fintype.ofFinite _ | |
| local instance rightLabel_fintype (y : CanonicalGame.RightVertex F) : | |
| Fintype (CanonicalGame.RightLabel F y) := Fintype.ofFinite _ | |
| local instance rightAlphabet_nonempty (δ : ℚ) : Nonempty (Fin (alphabet branch n t δ)) := | |
| ⟨⟨0, CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ⟩⟩ | |
| local instance leftAlphabet_nonempty (δ : ℚ) : Nonempty (Fin (2 * alphabet branch n t δ)) := | |
| ⟨⟨0, Nat.mul_pos (by decide) | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ)⟩⟩ | |
| theorem compatible (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) : | |
| FlexibleLocalCompletion.Compatible (CanonicalGame.game F μ) | |
| (CanonicalLocalCompletion.legalL F δ) (CanonicalLocalCompletion.legalR F δ) | |
| (rawInput clauses δ) where | |
| mask e a := by | |
| rw [rawInput_mask] | |
| exact paddedMask_eq_true_iff (CanonicalGame.leftKey F e) | |
| (CanonicalLocalCompletion.partitionWidth_le_leftAlphabet | |
| (TreeCanonical.locationCount branch n t) δ) a | |
| image e P := rawInput_image_legal clauses δ e P | |
| def family (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) : | |
| (CanonicalGame.game F μ).CompletionFamily | |
| (CanonicalLocalCompletion.legalL F δ) (CanonicalLocalCompletion.legalR F δ) := | |
| FlexibleLocalCompletion.family (CanonicalGame.game F μ) | |
| (CanonicalLocalCompletion.legalL F δ) (CanonicalLocalCompletion.legalR F δ) | |
| (rawInput clauses δ) (compatible clauses μ δ) | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (by | |
| intro e b | |
| rw [Fintype.card_eq_nat_card] | |
| have h := CanonicalLocalCompletion.small F μ e b | |
| rw [Fintype.card_eq_nat_card] at h | |
| exact h) | |
| @[simp] theorem family_size | |
| (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| (family clauses μ δ).size e = LocalCompletionFamily.activeCount | |
| (finiteResult δ (MetadataFreeTape.encodeRaw clauses rows repeats e)) := rfl | |
| @[simp] theorem family_map | |
| (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| (family clauses μ δ).map e = LocalCompletionFamily.table | |
| (finiteResult δ (MetadataFreeTape.encodeRaw clauses rows repeats e)) := rfl | |
| theorem family_size_le | |
| (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) (e : PreliminarySampler.Raw clauses branch n t rows repeats) : | |
| (family clauses μ δ).size e ≤ 2 * alphabet branch n t δ := | |
| LocalCompletionFamily.activeCount_le _ | |
| theorem family_value_le | |
| (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) (hδ : 0 < δ) : | |
| (family clauses μ δ).game.value ≤ (CanonicalGame.game F μ).value + (δ : ℝ) / 3 := by | |
| apply (family clauses μ δ).value_le_of_bound ((δ : ℝ) / 3) | |
| intro e | |
| have hδreal : (0 : ℝ) < (δ : ℝ) := by exact_mod_cast hδ | |
| have h := CompletionParameters.residual_le_third | |
| (CanonicalLocalCompletion.alphabet_positive (TreeCanonical.locationCount branch n t) δ) | |
| (CanonicalLocalCompletion.leftLabel_card_le F δ (CanonicalGame.leftAt F e)) hδreal | |
| (CompletionAlphabet.size_budget | |
| (CanonicalKeyEncoding.partitionWidth (TreeCanonical.locationCount branch n t)) δ hδ) | |
| simp only [Fintype.card_fin] | |
| rw [Fintype.card_eq_nat_card] | |
| rw [Fintype.card_eq_nat_card] at h | |
| simpa only [CanonicalGame.game] using h | |
| theorem family_value_of_satisfying | |
| (μ : FiniteDistribution (PreliminarySampler.Raw clauses branch n t rows repeats)) | |
| (δ : ℚ) (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) : | |
| (family clauses μ δ).game.value = 1 := by | |
| apply (family clauses μ δ).value_eq_one | |
| exact HierarchicalGame.value_eq_one | |
| (PreliminarySampler.family clauses branch n t rows repeats) μ assignment hsat | |
| def actualFamily [NeZero m] (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) := | |
| family clauses (PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows) δ | |
| def actualValue [NeZero m] (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) (δ : ℚ) : ℝ := | |
| (actualFamily (t := t) clauses rows repeats hn hbranch hrows δ).game.value | |
| theorem actualValue_le [NeZero m] (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (δ : ℚ) (hδ : 0 < δ) : | |
| actualValue (t := t) clauses rows repeats hn hbranch hrows δ ≤ | |
| (PreliminarySampler.game clauses branch n t rows repeats hn hbranch hrows).value + | |
| (δ : ℝ) / 3 := by | |
| exact family_value_le clauses | |
| (PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows) δ hδ | |
| theorem actualValue_of_satisfying [NeZero m] (rows repeats : Nat → Nat) (hn : 0 < n) | |
| (hbranch : ∀ k < n, 0 < branch k) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (δ : ℚ) (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) : | |
| actualValue (t := t) clauses rows repeats hn hbranch hrows δ = 1 := | |
| family_value_of_satisfying clauses | |
| (PreliminarySampler.law (t := t) clauses rows repeats hn hbranch hrows) δ assignment hsat | |
| end | |
| end PerfectCompleteness.FinitePreliminaryCompletion | |
| end | |
| end OAI | |