openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /AllSamplingAccuracy.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingSemantics | |
| import OAI.Combinatorics.ContingencyTables.Transport.ListedScheduledSampler | |
| namespace OAI | |
| /-! # Accuracy of the single computed sampler for arbitrary margins -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal | |
| open ResidualMixture FirstSuccess FairBits SmallContextCoordinates | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {m n : 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 | |
| local notation "q" => largeTailCount r (d^20)*largeTailCount c (d^20) | |
| /-- The mathematical draw whose code is the output of the single public program. -/ | |
| def scheduledTableDraw (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) | |
| (bits : Fin (finiteSamplerBits r c k) → Bool) : Table r c := | |
| if h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true then | |
| listedOuterDraw r c (computedCellEquiv r c) | |
| ((hasLargeRectangle_spec r c k).mp h).1 ((hasLargeRectangle_spec r c k).mp h).2 | |
| b (outerAccuracy d b k) (smallBinaryLength_parameters r c).1 | |
| (tableState r c (GreedyFeasibleTable.table r c htotal)) | |
| (outerSteps d b k) (outerTrials d k) (GreedyFeasibleTable.table r c htotal) bits | |
| else | |
| unitCodeOuterDraw r c (computedCellEquiv r c) (allSmallFibre r c (allSmall_of_no_large r c k h)) | |
| (tableState r c (GreedyFeasibleTable.table r c htotal)) (outerSteps d b k) (outerTrials d k) | |
| (GreedyFeasibleTable.table r c htotal) (prefixWord (unitCodeBits_le r c k) bits) | |
| theorem samplingMatrixAll_code (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) | |
| (bits : Fin (finiteSamplerBits r c k) → Bool) : | |
| samplingMatrixAll (((List.ofFn r,List.ofFn c),k),List.ofFn bits)= | |
| matrixCode (scheduledTableDraw r c htotal k bits).val := by | |
| unfold samplingMatrixAll scheduledTableDraw | |
| by_cases h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true | |
| · rw [ite_eq_left h,dite_eq_left h] | |
| exact samplingMatrix_code r c ((hasLargeRectangle_spec r c k).mp h).1 | |
| ((hasLargeRectangle_spec r c k).mp h).2 htotal k bits | |
| · rw [ite_eq_right h,dite_eq_right h] | |
| exact unitSamplingMatrix_code r c (allSmall_of_no_large r c k h) htotal k bits | |
| theorem publicSamplingAll_spec (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) | |
| (bits : Fin (finiteSamplerBits r c k) → Bool) : | |
| publicSamplingAll (Algorithms.encodeSamplingInput r c k,List.ofFn bits)= | |
| Algorithms.encodeMatrix (scheduledTableDraw r c htotal k bits).val := by | |
| rw [publicSamplingAll,parsedSampling_spec,samplingMatrixAll_code r c htotal] | |
| exact GreedyFeasibleTable.printMatrix_ofFn _ | |
| private lemma law_uniform_of_subsingleton {Y : Type*} [Fintype Y] [Subsingleton Y] | |
| [Nonempty Y] (p : RationalLaw Y) : p=uniformLaw (α := Y) := by | |
| apply RationalLaw.ext | |
| funext y | |
| let : Unique Y := ⟨⟨y⟩,fun z => Subsingleton.elim z y⟩ | |
| have ht : p.mass (default : Y)=1 := by simpa using p.total | |
| have hy : p.mass y=1 := (congrArg p.mass (Subsingleton.elim y default)).trans ht | |
| simp only [uniformLaw,uniformMass,Fintype.card_unique,Nat.cast_one,div_one,hy] | |
| private lemma variation_uniform_of_subsingleton {Y : Type*} [Fintype Y] [Subsingleton Y] | |
| [Nonempty Y] (p : RationalLaw Y) : variation p (uniformLaw (α := Y))=0 := by | |
| rw [law_uniform_of_subsingleton p,variation_self] | |
| /-- Only the two cited published inputs on the completion grid are used. | |
| The small and zero-dimensional branches need neither input. -/ | |
| theorem scheduledTableDraw_accuracy | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8))) | |
| (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) (hk : 1≤k) : | |
| let _ : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩ | |
| (variation (mapLaw (uniformLaw (α := Fin (finiteSamplerBits r c k) → Bool)) | |
| (scheduledTableDraw r c htotal k)) (uniformLaw (α := Table r c)):ℝ)≤1/(2:ℝ)^k := by | |
| let : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩ | |
| let : Nonempty (PaperStates r c) := ⟨tableState r c (GreedyFeasibleTable.table r c htotal)⟩ | |
| dsimp only | |
| unfold scheduledTableDraw | |
| split_ifs with h | |
| · exact listedScheduledDraw_accuracy r c (computedCellEquiv r c) | |
| ((hasLargeRectangle_spec r c k).mp h).1 ((hasLargeRectangle_spec r c k).mp h).2 | |
| hPL hCheeger htotal (tableState r c (GreedyFeasibleTable.table r c htotal)) | |
| (GreedyFeasibleTable.table r c htotal) k hk | |
| · have hs := allSmall_of_no_large r c k h | |
| rcases isEmpty_or_nonempty (Fin m) with hm|hm | |
| · let := hm | |
| let : Subsingleton (Table r c) := ⟨fun X Y => Subtype.ext (by funext i; exact isEmptyElim i)⟩ | |
| rw [variation_uniform_of_subsingleton,Rat.cast_zero] | |
| positivity | |
| · rcases isEmpty_or_nonempty (Fin n) with hn|hn | |
| · let := hn | |
| let : Subsingleton (Table r c) := ⟨fun X Y => Subtype.ext (by funext i j; exact isEmptyElim j)⟩ | |
| rw [variation_uniform_of_subsingleton,Rat.cast_zero] | |
| positivity | |
| · let := hm | |
| let := hn | |
| rw [padded_draw_law,unitCodeOuterDraw_law r c (computedCellEquiv r c) htotal] | |
| exact unitOuterDraw_accuracy r c hs htotal k hk (allSmallFibre r c hs) | |
| (tableState r c (GreedyFeasibleTable.table r c htotal)) (GreedyFeasibleTable.table r c htotal) | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |