openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /HierarchicalLowerInverseLemmas.lean
| import OAI.Computability.PerfectCompleteness.Machines.HierarchicalLowerMatrixInput | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedRows | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HierarchicalLowerInverse | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Fourier.MatrixLevelBridge | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter | |
| attribute [local instance] linearMapFintype | |
| variable {branch rows : Nat → Nat} {n t : Nat} | |
| (slots : RecursiveSpaces.Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (lowerLevel : Nat) | |
| (background : HierarchicalMatrixTable.Background (rows := rows) slots upper) | |
| local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _ | |
| abbrev Answer (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) := | |
| HierarchicalDecoderTables.Answer slots upper lowerLevel d | |
| local instance answerFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Fintype (Answer slots upper lowerLevel d) := | |
| HierarchicalAllDecoderTables.answerFintype slots upper lowerLevel d | |
| local instance upperDualFintype : Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) := | |
| HierarchicalAllDecoderTables.upperDualFintype slots upper | |
| def matrixTable {r : Nat} | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r → | |
| Option (Answer slots upper lowerLevel d)) : | |
| HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d → | |
| Option (Answer slots upper lowerLevel d) := | |
| fun X => table (HierarchicalLowerMatrixInput.input slots upper lowerLevel | |
| background d upperMatrix A X) | |
| variable (hbranch : ∀ k < n, 0 < branch k) | |
| theorem matrixTable_tested {r s : Nat} | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r → | |
| Option (Answer slots upper lowerLevel d)) | |
| (hvalid : DecoderTableAdmissibility.Admissible | |
| (HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch r d s) table) | |
| (X : HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d) | |
| (z : Answer slots upper lowerLevel d) | |
| (hz : matrixTable slots upper lowerLevel background d upperMatrix A table X = some z) : | |
| (LowerAffineSliceRank.testedGram | |
| (FunctionSpaceLowerRank.matrixForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) | |
| (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z)) X).rank ≤ 1 := by | |
| have h := hvalid | |
| (HierarchicalLowerMatrixInput.input slots upper lowerLevel background d upperMatrix A X) | |
| z hz | |
| have ht := h.2.1 | |
| simpa only [HierarchicalLowerMatrixInput.input_testedMatrix] using ht | |
| theorem matrixTable_high_rank {r s : Nat} | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) r) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel r → | |
| Option (Answer slots upper lowerLevel d)) | |
| (hvalid : DecoderTableAdmissibility.Admissible | |
| (HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch r d s) table) | |
| (X : HierarchicalLowerMatrixInput.Matrix (rows := rows) slots upper lowerLevel d) | |
| (z : Answer slots upper lowerLevel d) | |
| (hz : matrixTable slots upper lowerLevel background d upperMatrix A table X = some z) : | |
| s < Module.finrank F2 (LinearMap.range | |
| (FunctionSpaceLowerRank.matrixForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) | |
| (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z))) := by | |
| rw [FunctionSpaceLowerRank.matrixForm_rank] | |
| exact (hvalid | |
| (HierarchicalLowerMatrixInput.input slots upper lowerLevel background d upperMatrix A X) | |
| z hz).2.2 | |
| theorem matrixTable_agreement_lt {r s m adviceRows : Nat} | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows → | |
| Option (Answer slots upper lowerLevel d)) | |
| (hvalid : DecoderTableAdmissibility.Admissible | |
| (HierarchicalAllDecoderTables.RowValid slots upper lowerLevel hbranch adviceRows d s) table) | |
| (ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1) | |
| (hconstants : levelCutoffConstant r ρ + node (r + 1) < η) | |
| (hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d))) | |
| (herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) : | |
| PartialTableInverse.nonzeroAgreement | |
| (matrixTable slots upper lowerLevel background d upperMatrix A table) < η := by | |
| let f := matrixTable slots upper lowerLevel background d upperMatrix A table | |
| let form := fun z : Answer slots upper lowerLevel d => | |
| FunctionSpaceLowerRank.matrixForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) | |
| (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) z) | |
| have tested : ∀ X z, f X = some z → (LowerAffineSliceRank.testedGram (form z) X).rank ≤ 1 := | |
| fun X z hz => matrixTable_tested slots upper lowerLevel background hbranch | |
| d upperMatrix A table hvalid X z hz | |
| have high : ∀ X z, f X = some z → s < Module.finrank F2 (LinearMap.range (form z)) := | |
| fun X z hz => matrixTable_high_rank slots upper lowerLevel background hbranch | |
| d upperMatrix A table hvalid X z hz | |
| have h := LowerAffineSliceRank.high_rank_agreement_lt f form tested r s ρ η | |
| hm hρ hρ1 hconstants hrows herror | |
| have herase : MatrixErasure.erase f (fun X => ∃ z, f X = some z ∧ | |
| Module.finrank F2 (LinearMap.range (form z)) ≤ s) = f := by | |
| funext X | |
| apply MatrixErasure.erase_eq_of_not | |
| rintro ⟨z, hz, hrank⟩ | |
| exact (not_le_of_gt (high X z hz)) hrank | |
| rw [herase] at h | |
| exact h | |
| variable (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch n t)) | |
| (useful : (bg : HierarchicalMatrixTable.Background (rows := rows) slots upper) → | |
| HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop) | |
| include hbranch σ useful | |
| theorem supported_matrixTable_agreement_lt {r s m adviceRows : Nat} | |
| (decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows → | |
| Option (Answer slots upper lowerLevel d)) | |
| (hsupport : (HierarchicalAllDecoderTables.tableLaw slots upper lowerLevel hbranch σ useful | |
| adviceRows decoderDensity d s).weight table ≠ 0) | |
| (ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1) | |
| (hconstants : levelCutoffConstant r ρ + node (r + 1) < η) | |
| (hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d))) | |
| (herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) : | |
| PartialTableInverse.nonzeroAgreement | |
| (matrixTable slots upper lowerLevel background d upperMatrix A table) < η := by | |
| exact matrixTable_agreement_lt slots upper lowerLevel background hbranch | |
| d upperMatrix A table | |
| (HierarchicalAllDecoderTables.supported_table_valid slots upper lowerLevel hbranch σ useful | |
| adviceRows decoderDensity hdecoderDensity d s table hsupport) | |
| ρ η hm hρ hρ1 hconstants hrows herror | |
| theorem mean_matrixTable_agreement_le {r s m adviceRows : Nat} | |
| (decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1) | |
| (hconstants : levelCutoffConstant r ρ + node (r + 1) < η) | |
| (hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d))) | |
| (herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) : | |
| (HierarchicalAllDecoderTables.tableLaw slots upper lowerLevel hbranch σ useful | |
| adviceRows decoderDensity d s).expectation | |
| (fun table => PartialTableInverse.nonzeroAgreement | |
| (matrixTable slots upper lowerLevel background d upperMatrix A table)) ≤ η := by | |
| apply DecoderTableAdmissibility.expectation_le_of_support | |
| intro table hsupport | |
| exact (supported_matrixTable_agreement_lt slots upper lowerLevel background hbranch | |
| σ useful decoderDensity hdecoderDensity d upperMatrix A table hsupport | |
| ρ η hm hρ hρ1 hconstants hrows herror).le | |
| theorem supported_upper_matrixTable_agreement_lt {r s m adviceRows : Nat} | |
| (decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := rows) slots upper) | |
| (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) slots upper lowerLevel adviceRows → | |
| HierarchicalAllDecoderTables.UpperAnswer slots upper) | |
| (hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful | |
| adviceRows decoderDensity).weight table ≠ 0) | |
| (ρ η : ℝ) (hm : 0 < m) (hρ : 0 < ρ) (hρ1 : ρ < 1) | |
| (hconstants : levelCutoffConstant r ρ + node (r + 1) < η) | |
| (hrows : 2 * m + r ≤ rows (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d))) | |
| (herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) + | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) : | |
| PartialTableInverse.nonzeroAgreement | |
| (matrixTable slots upper lowerLevel background d upperMatrix A | |
| (HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d s table)) < η := by | |
| exact matrixTable_agreement_lt slots upper lowerLevel background hbranch | |
| d upperMatrix A | |
| (HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d s table) | |
| (HierarchicalAllDecoderTables.decoded_supported_table_valid slots upper lowerLevel hbranch | |
| σ useful adviceRows decoderDensity hdecoderDensity table hsupport d s) | |
| ρ η hm hρ hρ1 hconstants hrows herror | |
| end | |
| end PerfectCompleteness.HierarchicalLowerInverse | |
| end | |
| section | |
| namespace PerfectCompleteness.HierarchicalFixedLowerInverse | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| variable {δ : ℚ} (plan : FixedRows.Plan δ) (hδ : 0 < δ) | |
| {branch : Nat → Nat} {t : Nat} | |
| (slots : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch plan.depth) (lowerLevel : Nat) | |
| (background : HierarchicalMatrixTable.Background | |
| (rows := FixedRows.rows plan) slots upper) | |
| local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _ | |
| local instance upperDualFintype : Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) := | |
| HierarchicalAllDecoderTables.upperDualFintype slots upper | |
| variable (hbranch : ∀ k < plan.depth, 0 < branch k) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t)) | |
| (useful : (bg : HierarchicalMatrixTable.Background | |
| (rows := FixedRows.rows plan) slots upper) → | |
| HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop) | |
| include hbranch σ useful | |
| theorem supported_upper_matrixTable_agreement_lt {adviceRows : Nat} | |
| (decoderDensity : ℝ) (hdecoderDensity : 0 < decoderDensity) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix | |
| (rows := FixedRows.rows plan) slots upper) | |
| (A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input | |
| (rows := FixedRows.rows plan) slots upper lowerLevel adviceRows → | |
| HierarchicalAllDecoderTables.UpperAnswer slots upper) | |
| (hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful | |
| adviceRows decoderDensity).weight table ≠ 0) : | |
| PartialTableInverse.nonzeroAgreement | |
| (HierarchicalLowerInverse.matrixTable slots upper lowerLevel background d upperMatrix A | |
| (HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d | |
| (FixedRows.cutoff plan hδ) table)) < | |
| UpperParameterScalars.gamma (InitialParameters.useful δ) plan.density | |
| (InitialParameters.inverse δ) plan.order | |
| (FixedRows.rows plan (Nodes.height upper)) ^ 2 / 8 := by | |
| let lower := DescendingRows.lowerChoice (InitialParameters.useful_pos hδ) | |
| plan.density_pos (InitialParameters.inverse_pos hδ) plan.order | |
| (FixedRows.rows plan (Nodes.height upper)) | |
| have hheight : Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d) = | |
| lowerLevel := | |
| (RelativeDescendantProducts.relativeNode_height upper d.val.val).trans d.property | |
| have hproper : lowerLevel < Nodes.height upper := by | |
| rw [← hheight] | |
| exact (RelativeDescendantProducts.descendantIndex_below upper d.val).height_lt | |
| have hminimum : lower.minimumRows ≤ FixedRows.rows plan lowerLevel := | |
| FixedRows.lower_row_requirement plan hδ hproper (Nodes.height_le upper) | |
| have hrows : 2 * lower.crossSize + lower.order ≤ | |
| FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)) := by | |
| rw [hheight] | |
| exact lower.row_bound _ hminimum | |
| have hcutoff : lower.cutoff ≤ FixedRows.cutoff plan hδ := by | |
| simpa only [lower, FixedRows.rows, FixedRows.cutoff] using | |
| DescendingRows.tree_cutoff_le_common (InitialParameters.useful_pos hδ) | |
| plan.density_pos (InitialParameters.inverse_pos hδ) plan.order plan.depth | |
| plan.rootFirstRows (Nat.zero_lt_of_lt hproper) (Nodes.height_le upper) | |
| exact HierarchicalLowerInverse.supported_upper_matrixTable_agreement_lt | |
| (r := lower.order) (s := FixedRows.cutoff plan hδ) (m := lower.crossSize) | |
| slots upper lowerLevel background hbranch σ useful decoderDensity hdecoderDensity | |
| d upperMatrix A table hsupport lower.density _ lower.cross_pos lower.density_pos | |
| lower.density_lt_one lower.spectral hrows (lower.error_bound _ hcutoff).le | |
| end | |
| end PerfectCompleteness.HierarchicalFixedLowerInverse | |
| end | |
| end OAI | |