File size: 2,955 Bytes
97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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)
/-- This identifies every output, including the all-rejections fallback,
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
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