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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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Combinatorics.ContingencyTables.Dense.DenseCompletionTest
namespace OAI
/-! # Binary construction of the completed integer matrix -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS.TreeTyped Polynomial
def completedEntry (z : CompletionCell) : ℕ := (completionParts z).1-(completionParts z).2
def completedEntryRealizer : Realizer completedEntry :=
composition (composition completionPartsRealizer (pair first second)) subtract
theorem polynomial_completedEntry : PolynomialTime completedEntryRealizer :=
polynomial_composition (polynomial_composition polynomial_completionParts
(polynomial_pair polynomial_first polynomial_second)) polynomial_subtract
def completionRows (z : CompletionData) : List ℕ := List.range z.1.1.length
def completionColumns (z : CompletionData) : List ℕ := List.range z.1.2.length
def completionRowsRealizer : Realizer completionRows :=
composition (composition (composition first first) listLength) range
def completionColumnsRealizer : Realizer completionColumns :=
composition (composition (composition first second) listLength) range
lemma polynomial_completionRows : PolynomialTime completionRowsRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_listLength) X
intro z
simp only [Function.comp_apply,eval_X]
exact (list_length_le_weight z.1.1).trans ((weight_fst z.1).trans (weight_fst z))
lemma polynomial_completionColumns : PolynomialTime completionColumnsRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition polynomial_first polynomial_second) polynomial_listLength) X
intro z
simp only [Function.comp_apply,eval_X]
exact (list_length_le_weight z.1.2).trans ((weight_snd z.1).trans (weight_fst z))
def completedRowEntry (z : (CompletionData×ℕ)×ℕ) : ℕ := completedEntry (z.1.1,(z.1.2,z.2))
def completedRowEntryRealizer : Realizer completedRowEntry :=
composition (pair (composition first first) (pair (composition first second) second)) completedEntryRealizer
lemma polynomial_completedRowEntry : PolynomialTime completedRowEntryRealizer :=
polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_first)
(polynomial_pair (polynomial_composition polynomial_first polynomial_second) polynomial_second))
polynomial_completedEntry
def completedRow (z : CompletionData×ℕ) : List ℕ :=
(completionColumns z.1).map (fun j => completedRowEntry (z,j))
def completedRowRealizer : Realizer completedRow :=
composition (pair identity (composition first completionColumnsRealizer)) (mapWith completedRowEntryRealizer)
lemma polynomial_completedRow : PolynomialTime completedRowRealizer :=
polynomial_composition (polynomial_pair polynomial_identity
(polynomial_composition polynomial_first polynomial_completionColumns))
(polynomial_mapWith polynomial_completedRowEntry)
def completedMatrix (z : CompletionData) : MatrixCode :=
(completionRows z).map (fun i => completedRow (z,i))
def completedMatrixRealizer : Realizer completedMatrix :=
composition (pair identity completionRowsRealizer) (mapWith completedRowRealizer)
theorem polynomial_completedMatrix : PolynomialTime completedMatrixRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_completionRows)
(polynomial_mapWith polynomial_completedRow)
lemma completedEntry_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)) :
completedEntry (((marginCode R,marginCode P),matrixCode z),(optionIndex i,optionIndex j))=
DenseAffine.naturalTable R P (fun ij => z ij.1 ij.2) i j := by
have h := completionParts_signed R P z i j
unfold completedEntry DenseAffine.naturalTable
dsimp only at h
rw [←h]
omega
lemma optionIndex_finSucc {m : ℕ} (i : Fin (m+1)) : optionIndex (finSuccEquiv m i)=i.val := by
refine Fin.cases ?_ (fun j => ?_) i
· simp only [finSuccEquiv_zero,optionIndex,Fin.val_zero]
· simp only [finSuccEquiv_succ,optionIndex,Fin.val_succ]
private lemma completedRow_code {m n : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(z : Fin m → Fin n → ℕ) (i : Fin (m+1)) :
completedRow (((marginCode R,marginCode P),matrixCode z),i.val)=
List.ofFn (fun j : Fin (n+1) => DenseAffine.naturalTable R P
(fun ij => z ij.1 ij.2) (finSuccEquiv m i) (finSuccEquiv n j)) := by
have hc : completionColumns ((marginCode R,marginCode P),matrixCode z)=List.range (n+1) := by
simp only [completionColumns,marginCode,List.length_cons,List.length_ofFn]
unfold completedRow
rw [hc]
apply List.ext_getElem
· simp only [List.length_map,List.length_range,List.length_ofFn]
· intro j hj _
have hjn : j < n+1 := by simpa only [List.length_map,List.length_range] using hj
simp only [List.getElem_map,List.getElem_range,List.getElem_ofFn,completedRowEntry]
simpa only [optionIndex_finSucc] using
completedEntry_code R P z (finSuccEquiv m i) (finSuccEquiv n ⟨j,hjn⟩)
theorem completedMatrix_code {m n : ℕ} (R : Option (Fin m) → ℕ) (P : Option (Fin n) → ℕ)
(z : Fin m → Fin n → ℕ) :
completedMatrix ((marginCode R,marginCode P),matrixCode z)=
matrixCode (fun i : Fin (m+1) => fun j : Fin (n+1) =>
DenseAffine.naturalTable R P (fun ij => z ij.1 ij.2) (finSuccEquiv m i) (finSuccEquiv n j)) := by
have hr : completionRows ((marginCode R,marginCode P),matrixCode z)=List.range (m+1) := by
simp only [completionRows,marginCode,List.length_cons,List.length_ofFn]
unfold completedMatrix
rw [hr]
unfold matrixCode
apply List.ext_getElem
· simp only [List.length_map,List.length_range,List.length_ofFn]
· intro i hi _
have him : i < m+1 := by simpa only [List.length_map,List.length_range] using hi
simp only [List.getElem_map,List.getElem_range,List.getElem_ofFn]
exact completedRow_code R P z ⟨i,him⟩
end ContingencyTables.DensePrograms
end OAI
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