| import OAI.Combinatorics.ContingencyTables.Sampling.OriginalTableEnumeration |
| import OAI.Combinatorics.ContingencyTables.Sampling.ProfileBoxEnumeration |
| import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperCorrectionBudget |
| import OAI.Combinatorics.ContingencyTables.Sampling.RareBranchRuntime |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace ContingencyTables.SmallGraphProfiles |
| open ProfilePrograms DensePrograms CompletionCounts SmallContextCoordinates |
| open PhysicalCompletionFibres FirstPaperPhysicalMarginal |
| open scoped BigOperators Classical |
| noncomputable section |
|
|
| variable {m n q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) |
| local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n) |
| local notation "b" => smallBinaryLength r c |
|
|
| lemma originalEnumeration_count : |
| gridStateCount (originalEnumerationData (List.ofFn r,List.ofFn c))=((∑ i,r i)+1)^(m*n) := by |
| rw [gridStateCount_eq] |
| simp only [originalEnumerationData,originalLayoutMargins,completionEnumerationData, |
| zeroGrid,List.tail_cons,cellPairs,List.length_map,List.length_range,List.length_ofFn, |
| List.sum_cons,List.sum_ofFn,zero_add] |
|
|
| lemma originalEnumeration_count_bound : |
| gridStateCount (originalEnumerationData (List.ofFn r,List.ofFn c))≤2^(d*b) := by |
| rw [originalEnumeration_count] |
| have hN : (∑ i,r i)+1≤2^b := |
| (show (∑ i,r i)+1≤ stateBound r c+1 by unfold stateBound; omega).trans |
| (smallBinaryLength_parameters r c).2.2 |
| have hdim : m*n≤d := by unfold dimensionAllowance; simp only [Fintype.card_fin]; nlinarith |
| calc |
| ((∑ i,r i)+1)^(m*n) ≤ (2^b)^(m*n) := Nat.pow_le_pow_left hN _ |
| _ = 2^(b*(m*n)) := (pow_mul _ _ _).symm |
| _ ≤ 2^(d*b) := Nat.pow_le_pow_right (by decide) (by nlinarith) |
|
|
| lemma profileEnumeration_count (e : Fin q ≃ Cells (paperSmall r c)) (x : PaperStates r c) : |
| gridStateCount (profileEnumerationData (stateCodeData r c e x))=(d^20+1)^(q+q) := by |
| rw [gridStateCount_eq] |
| simp only [profileEnumerationData,stateCodeData,codeCheckData,cellCatalog,List.length_map, |
| List.length_append,List.length_ofFn] |
|
|
| lemma profileEnumeration_count_bound (e : Fin q ≃ Cells (paperSmall r c)) (x : PaperStates r c) : |
| gridStateCount (profileEnumerationData (stateCodeData r c e x))≤2^(2*(d*b)) := by |
| rw [profileEnumeration_count] |
| have hU : d^20+1≤2^b := |
| (show d^20+1≤ stateBound r c+1 by unfold stateBound; omega).trans |
| (smallBinaryLength_parameters r c).2.2 |
| have hq : q≤d := by |
| have he : q=Fintype.card (Cells (paperSmall r c)) := by |
| simpa only [Fintype.card_fin] using Fintype.card_congr e |
| rw [he] |
| exact (RowMajorEnumeration.small_card_le_dimension (paperSmall r c)).trans dimension_le_allowance |
| calc |
| (d^20+1)^(q+q) ≤ (2^b)^(q+q) := Nat.pow_le_pow_left hU _ |
| _ = 2^(b*(q+q)) := (pow_mul _ _ _).symm |
| _ ≤ 2^(2*(d*b)) := Nat.pow_le_pow_right (by decide) (by nlinarith) |
|
|
| /-- Any fixed polynomial of the product of these enumeration sizes is |
| absorbed by the actual correction probability. -/ |
| lemma exact_boxes_absorbed (p : Polynomial ℕ) (A : ℕ) : |
| p.eval (A*2^(3*(d*b)))≤2^(correctionPrecision r c)* |
| (p.eval A*2^((3*p.natDegree)^2)) := by |
| apply RareBranchRuntime.polynomial_tabulation_bound |
| exact RareBranchRuntime.square_le_precision d b (by unfold dimensionAllowance; omega) |
|
|
| end |
| end ContingencyTables.SmallGraphProfiles |
|
|
| end OAI |
|
|
