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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Combinatorics.ContingencyTables.Transport.BoundedTableComplement
import OAI.Combinatorics.ContingencyTables.Machines.NetworkFeasibilityProgram
namespace OAI
/-! # Computing complementary bounded-table margins in binary -/
namespace ContingencyTables.BoundedComplementProgram
open MatchingFPRAS.TreeTyped Polynomial NetworkVertexCode
abbrev EntryInput := Input×ℕ
def rowValue (z : EntryInput) : ℕ :=
(z.1.2.getD z.2 []).sum-z.1.1.1.getD z.2 0
def rowValueRealizer : Realizer rowValue :=
composition (pair
(composition (composition (pair (composition first second) second) (getDFast [])) listSum)
(composition (pair (composition (composition first first) first) second) (getDFast 0))) subtract
lemma polynomial_rowValue : PolynomialTime rowValueRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_second) polynomial_second)
(polynomial_getDFast [])) polynomial_listSum)
(polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_first) polynomial_second) (polynomial_getDFast 0))) polynomial_subtract
def columnValues (z : EntryInput) : List ℕ := z.1.2.map (fun row=>row.getD z.2 0)
def columnValuesRealizer : Realizer columnValues :=
composition (pair second (composition first second))
(mapWith (composition (pair second first) (getDFast 0)))
lemma polynomial_columnValues : PolynomialTime columnValuesRealizer :=
polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition polynomial_first polynomial_second))
(polynomial_mapWith (polynomial_composition (polynomial_pair polynomial_second polynomial_first)
(polynomial_getDFast 0)))
def columnValue (z : EntryInput) : ℕ :=
(columnValues z).sum-z.1.1.2.getD z.2 0
def columnValueRealizer : Realizer columnValue :=
composition (pair (composition columnValuesRealizer listSum)
(composition (pair (composition (composition first first) second) second) (getDFast 0))) subtract
lemma polynomial_columnValue : PolynomialTime columnValueRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition polynomial_columnValues polynomial_listSum)
(polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_second) polynomial_second) (polynomial_getDFast 0))) polynomial_subtract
def rowIndices (z : Input) : List ℕ := List.range z.1.1.length
def columnIndices (z : Input) : List ℕ := List.range z.1.2.length
def rowIndicesRealizer : Realizer rowIndices :=
composition (composition (composition first first) listLength) range
def columnIndicesRealizer : Realizer columnIndices :=
composition (composition (composition first second) listLength) range
lemma polynomial_rowIndices : PolynomialTime rowIndicesRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_listLength) X
intro z
simpa only [Function.comp_apply,eval_X] using
(list_length_le_weight z.1.1).trans ((weight_fst z.1).trans (weight_fst z))
lemma polynomial_columnIndices : PolynomialTime columnIndicesRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition polynomial_first polynomial_second)
polynomial_listLength) X
intro z
simpa only [Function.comp_apply,eval_X] using
(list_length_le_weight z.1.2).trans ((weight_snd z.1).trans (weight_fst z))
def complement (z : Input) : Input :=
(((rowIndices z).map (fun i=>rowValue (z,i)),
(columnIndices z).map (fun j=>columnValue (z,j))),z.2)
def complementRealizer : Realizer complement :=
pair (pair
(composition (pair identity rowIndicesRealizer) (mapWith rowValueRealizer))
(composition (pair identity columnIndicesRealizer) (mapWith columnValueRealizer))) second
lemma polynomial_complement : PolynomialTime complementRealizer :=
polynomial_pair (polynomial_pair
(polynomial_composition (polynomial_pair polynomial_identity polynomial_rowIndices)
(polynomial_mapWith polynomial_rowValue))
(polynomial_composition (polynomial_pair polynomial_identity polynomial_columnIndices)
(polynomial_mapWith polynomial_columnValue))) polynomial_second
lemma rowValue_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) (i : Fin m) :
rowValue (data r c b,i.val)=BoundedTableComplement.rows b r i := by
simp only [rowValue,data,ResidualGraphProgram.natCode,List.getD_eq_getElem?_getD,
List.getElem?_ofFn,i.isLt,dite_true,Option.getD_some,Fin.eta,List.sum_ofFn,
BoundedTableComplement.rows,BoundedTableComplement.rowTotal]
lemma columnValue_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) (j : Fin n) :
columnValue (data r c b,j.val)=BoundedTableComplement.columns b c j := by
simp only [columnValue,columnValues,data,ResidualGraphProgram.natCode,List.map_ofFn,
Function.comp_def,List.getD_eq_getElem?_getD,List.getElem?_ofFn,j.isLt,
dite_true,Option.getD_some,Fin.eta,List.sum_ofFn,
BoundedTableComplement.columns,BoundedTableComplement.columnTotal]
lemma complement_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) :
complement (data r c b)=data (BoundedTableComplement.rows b r)
(BoundedTableComplement.columns b c) b := by
unfold complement
apply Prod.ext
· apply Prod.ext
· simp only [rowIndices,data,List.length_ofFn,DirectedBFSCode.range_eq_map,←List.ofFn_eq_map,
List.map_ofFn]
apply congrArg List.ofFn
funext i
exact rowValue_code r c b i
· simp only [columnIndices,data,List.length_ofFn,DirectedBFSCode.range_eq_map,←List.ofFn_eq_map,
List.map_ofFn]
apply congrArg List.ofFn
funext j
exact columnValue_code r c b j
· rfl
end ContingencyTables.BoundedComplementProgram
end OAI
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