File size: 2,995 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 70 | 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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