| import OAI.Combinatorics.ContingencyTables.Dense.DenseDrawProgram |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseTrialSemantics |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanDraw |
|
|
| namespace OAI |
|
|
| /-! # The bounded binary draw is exactly the analyzed Boolean sampler -/ |
|
|
| namespace ContingencyTables.DensePrograms |
| open GridBoundary FairBits |
| open scoped BigOperators Classical |
|
|
| noncomputable section |
|
|
| lemma trialWidth_code {m n d : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ) |
| (h : ℕ) (a : Grid (I := Fin (m*n)) (4*d^8)) : |
| trialWidth ((((marginCode R,marginCode P),gridNatCode a),d),h)= |
| denseTrialBits R P d finProdFinEquiv h := by |
| unfold trialWidth |
| rw [flatToGridData_code,offsetLengths_code,List.sum_ofFn] |
| rfl |
|
|
| lemma denseTrialBits_pos {m n d : ℕ} (hd : 0 < d) |
| (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ) (h : ℕ) : |
| 0 < denseTrialBits R P d finProdFinEquiv h := by |
| have hw := denseStepBits_pos hd |
| have ht : 0 < (d^50*(h+1)^2)*denseStepBits d := by positivity |
| unfold denseTrialBits |
| omega |
|
|
| def drawInput {m n d : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ) |
| (h : ℕ) (a : Grid (I := Fin (m*n)) (4*d^8)) (fallback : Table R P) |
| (bits : List Bool) : DenseDrawData := |
| ((((((marginCode R,marginCode P),gridNatCode a),d),h),tableOutput fallback),bits) |
|
|
| / |
| with the actual finite Boolean sampler whose total-variation bound is proved. -/ |
| theorem denseDrawCode_eq {m n d : ℕ} |
| (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ) |
| (hd : 14 ≤ d) (hR : ∀ i,d^12 ≤ R i) (hP : ∀ j,d^12 ≤ P j) |
| (htotal : ∑ i,R i=∑ j,P j) (h : ℕ) |
| (a : Grid (I := Fin (m*n)) (4*d^8)) (fallback : Table R P) |
| (bits : Fin (denseDrawBits R P d finProdFinEquiv h) → Bool) : |
| denseDrawCode (drawInput R P h a fallback (List.ofFn bits))= |
| tableOutput (denseBooleanDraw R P d hd hR hP finProdFinEquiv htotal h a fallback bits) := by |
| have hd0 : 0 < d := by omega |
| unfold denseDrawCode drawChunks drawInput |
| dsimp only [Prod.fst,Prod.snd] |
| rw [trialWidth_code] |
| have hc := wordChunks_ofFn (n := d^4*(h+1)) (denseTrialBits_pos hd0 R P h) bits |
| let params : DenseDrawParameters := ((((marginCode R,marginCode P),gridNatCode a),d),h) |
| have he := congrArg (fun ws => FirstSuccessProgram.retry denseTrialCode |
| ((params,tableOutput fallback),ws)) hc |
| refine he.trans ?_ |
| rw [FirstSuccessProgram.retry_ofFn] |
| have ht : (fun word => denseTrialCode |
| (((((marginCode R,marginCode P),gridNatCode a),d),h),List.ofFn word))= |
| fun word => (trialSuccess R P d finProdFinEquiv htotal |
| (denseBooleanTrial R P d hd hR hP finProdFinEquiv h a word)).map tableOutput := by |
| funext word |
| exact denseTrialCode_eq R P hd hR hP htotal h a word |
| rw [ht,FirstSuccessProgram.wordRetry_map] |
| rfl |
|
|
| end |
| end ContingencyTables.DensePrograms |
|
|
| end OAI |
|
|
