| import OAI.Combinatorics.ContingencyTables.Dense.DenseExponentListSemantics |
|
|
| namespace OAI |
|
|
| /-! # A polynomial exact acceptance test for a free integer matrix -/ |
|
|
| namespace ContingencyTables.DensePrograms |
| open MatchingFPRAS.TreeTyped ProfilePrograms |
|
|
| def completionNonnegative (z : CompletionCell) : Bool := |
| decide ((completionParts z).2≤(completionParts z).1) |
|
|
| def completionNonnegativeRealizer : Realizer completionNonnegative := |
| composition (composition completionPartsRealizer (pair second first)) leNat |
|
|
| theorem polynomial_completionNonnegative : PolynomialTime completionNonnegativeRealizer := |
| polynomial_composition (polynomial_composition polynomial_completionParts |
| (polynomial_pair polynomial_second polynomial_first)) polynomial_leNat |
|
|
| def completionValid (z : CompletionData) : Bool := |
| (cellPairs z.1).all (fun ij => completionNonnegative (z,ij)) |
|
|
| def completionValidRealizer : Realizer completionValid := |
| (composition (composition (pair identity (composition first cellPairsRealizer)) |
| (mapWith completionNonnegativeRealizer)) all).congr (by |
| intro z |
| simp only [Function.comp_apply,List.all_map,Function.comp_def,id_eq,completionValid]) |
|
|
| theorem polynomial_completionValid : PolynomialTime completionValidRealizer := |
| polynomial_congr _ (polynomial_composition (polynomial_composition |
| (polynomial_pair polynomial_identity (polynomial_composition polynomial_first polynomial_cellPairs)) |
| (polynomial_mapWith polynomial_completionNonnegative)) polynomial_all) |
|
|
| theorem completionNonnegative_code {m n : ℕ} (R : Option (Fin m) → ℕ) |
| (P : Option (Fin n) → ℕ) (z : Fin m → Fin n → ℕ) |
| (i : Option (Fin m)) (j : Option (Fin n)) : |
| completionNonnegative (((marginCode R,marginCode P),matrixCode z), |
| (optionIndex i,optionIndex j))=true ↔ |
| 0≤DenseAffine.integerTable R P (fun ij => z ij.1 ij.2) i j := by |
| have h := completionParts_signed R P z i j |
| simp only [completionNonnegative,decide_eq_true_eq] |
| dsimp only at h |
| omega |
|
|
| theorem completionValid_code {m n : ℕ} (R : Option (Fin m) → ℕ) |
| (P : Option (Fin n) → ℕ) (z : Fin m → Fin n → ℕ) : |
| completionValid ((marginCode R,marginCode P),matrixCode z)=true ↔ |
| ∀ i j,0≤DenseAffine.integerTable R P (fun ij => z ij.1 ij.2) i j := by |
| simp only [completionValid,List.all_eq_true] |
| constructor |
| · intro h i j |
| apply (completionNonnegative_code R P z i j).mp |
| apply h |
| apply (mem_cellPairs _ _ _ _).mpr |
| simpa only [marginCode,List.length_cons,List.length_ofFn] using |
| And.intro (optionIndex_lt i) (optionIndex_lt j) |
| · intro h ij hij |
| obtain ⟨i,j⟩ := ij |
| have hh := (mem_cellPairs _ _ i j).mp hij |
| simp only [marginCode,List.length_cons,List.length_ofFn] at hh |
| obtain ⟨a,ha⟩ := exists_optionIndex i hh.1 |
| obtain ⟨b,hb⟩ := exists_optionIndex j hh.2 |
| subst i |
| subst j |
| exact (completionNonnegative_code R P z a b).mpr (h a b) |
|
|
| end ContingencyTables.DensePrograms |
|
|
| end OAI |
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|
