| import OAI.Combinatorics.ContingencyTables.Dense.DenseCandidateCounts |
| import OAI.Combinatorics.ContingencyTables.Transport.IndexedTableLaw |
| import OAI.Combinatorics.ContingencyTables.Machines.VectorRetryProgram |
|
|
| namespace OAI |
|
|
| /-! # Complete indexed integer law of a bounded dense draw -/ |
|
|
| namespace ContingencyTables.DensePrograms |
| open MatchingFPRAS MatchingFPRAS.TreeTyped GridBoundary |
| open scoped BigOperators Classical |
| noncomputable section |
|
|
| theorem candidateCounts_indexed {m n d : ℕ} (R : Option (Fin m) → ℕ) |
| (P : Option (Fin n) → ℕ) (hd : 14≤d) (hq : m*n≤d) |
| (hR : ∀ i,d^12≤R i) (hP : ∀ j,d^12≤P j) (htotal : ∑ i,R i=∑ j,P j) |
| (h : ℕ) (cap : List Bool) |
| (hcap : ∀ j : Fin (m*n),FairBits.offsetBitLength |
| (integerWidth R P d (finProdFinEquiv.symm j)) h d≤cap.length) : |
| candidateCounts ((tabulateDenseWalk ((marginCode R,marginCode P),(d,h)),cap), |
| ((marginCode R,marginCode P),(d,h)))= |
| List.ofFn (fun k => (IndexedTableLaw.law R P (le_refl (∑ i,R i)) |
| ((denseTrialCounts R P d hd hR hP finProdFinEquiv h |
| (canonicalOrigin (m*n) d (by omega))).push |
| (trialSuccess R P d finProdFinEquiv htotal))).count (some k)) := by |
| unfold candidateCounts |
| rw [completionCandidates_code R P htotal,List.map_ofFn] |
| apply congrArg List.ofFn |
| funext k |
| simp only [Function.comp_apply] |
| rw [candidateCount_actual R P hd hq hR hP h cap hcap, |
| IndexedTableLaw.count_some R P htotal] |
| unfold IndexedTableLaw.tableAt |
| simp_rw [denseTrial_success_count R P d hd hR hP finProdFinEquiv htotal] |
|
|
| / |
| recurrence, after the accepted-trial vector has been tabulated once. -/ |
| theorem denseRetryVector_actual {m n d : ℕ} (R : Option (Fin m) → ℕ) |
| (P : Option (Fin n) → ℕ) (hd : 14≤d) (hq : m*n≤d) |
| (hR : ∀ i,d^12≤R i) (hP : ∀ j,d^12≤P j) (htotal : ∑ i,R i=∑ j,P j) |
| (h : ℕ) (cap xs : List Bool) |
| (hcap : ∀ j : Fin (m*n),FairBits.offsetBitLength |
| (integerWidth R P d (finProdFinEquiv.symm j)) h d≤cap.length) |
| (fallback : Table R P) : |
| VectorRetryPrograms.counts ((xs,2^denseTrialBits R P d finProdFinEquiv h), |
| (candidateCounts ((tabulateDenseWalk ((marginCode R,marginCode P),(d,h)),cap), |
| ((marginCode R,marginCode P),(d,h))), |
| (IndexedTableLaw.index R P (le_refl (∑ i,R i)) fallback).val))= |
| List.ofFn (((IndexedTableLaw.law R P (le_refl (∑ i,R i)) |
| ((denseTrialCounts R P d hd hR hP finProdFinEquiv h |
| (canonicalOrigin (m*n) d (by omega))).push |
| (trialSuccess R P d finProdFinEquiv htotal))).retry |
| (IndexedTableLaw.index R P (le_refl (∑ i,R i)) fallback) xs.length).count) := by |
| rw [candidateCounts_indexed R P hd hq hR hP htotal h cap hcap] |
| exact VectorRetryPrograms.counts_actual _ xs _ |
|
|
| end |
| end ContingencyTables.DensePrograms |
|
|
| end OAI |
|
|
