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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 | import OAI.Combinatorics.ContingencyTables.Machines.CellMinimumProgram
namespace OAI
/-! # Computing every attained cell endpoint
Both scans use the dimensions of the stored margin lists. Each entry is
computed by the binary feasibility search; neither scan enumerates tables.
-/
namespace ContingencyTables.EndpointMatrixProgram
open MatchingFPRAS.TreeTyped Polynomial ResidualGraphProgram
abbrev Input := NetworkVertexCode.Input
abbrev RowInput := Input × ℕ
abbrev CellInput := RowInput × ℕ
def context (z : CellInput) : CellLowerTestProgram.Context := (z.1.1,(z.1.2,z.2))
def contextRealizer : Realizer context := pair (composition first first)
(pair (composition first second) second)
lemma polynomial_context : PolynomialTime contextRealizer :=
polynomial_pair (polynomial_composition polynomial_first polynomial_first)
(polynomial_pair (polynomial_composition polynomial_first polynomial_second)
polynomial_second)
def lowerEntry (z : CellInput) : ℕ := CellMinimumProgram.minimum (context z)
def lowerEntryRealizer : Realizer lowerEntry :=
composition (f := context) (g := CellMinimumProgram.minimum)
contextRealizer CellMinimumProgram.minimumRealizer
lemma polynomial_lowerEntry : PolynomialTime lowerEntryRealizer :=
polynomial_composition polynomial_context CellMinimumProgram.polynomial_minimum
def upperEntry (z : CellInput) : ℕ := CellMaximumProgram.maximum (context z)
def upperEntryRealizer : Realizer upperEntry :=
composition (f := context) (g := CellMaximumProgram.maximum)
contextRealizer CellMaximumProgram.maximumRealizer
lemma polynomial_upperEntry : PolynomialTime upperEntryRealizer :=
polynomial_composition polynomial_context CellMaximumProgram.polynomial_maximum
def columns (z : RowInput) : List ℕ := List.range z.1.1.2.length
def columnsRealizer : Realizer columns :=
composition (composition (composition (composition first first) second) listLength) range
lemma polynomial_columns : PolynomialTime columnsRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_composition
(polynomial_composition polynomial_first polynomial_first) polynomial_second)
polynomial_listLength) X
intro z
simpa only [Function.comp_apply,eval_X] using
(list_length_le_weight z.1.1.2).trans
((weight_snd z.1.1).trans ((weight_fst z.1).trans (weight_fst z)))
def lowerRow (z : RowInput) : List ℕ := (columns z).map (fun j => lowerEntry (z,j))
def lowerRowRealizer : Realizer lowerRow :=
composition (pair identity columnsRealizer) (mapWith lowerEntryRealizer)
lemma polynomial_lowerRow : PolynomialTime lowerRowRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_columns)
(polynomial_mapWith polynomial_lowerEntry)
def upperRow (z : RowInput) : List ℕ := (columns z).map (fun j => upperEntry (z,j))
def upperRowRealizer : Realizer upperRow :=
composition (pair identity columnsRealizer) (mapWith upperEntryRealizer)
lemma polynomial_upperRow : PolynomialTime upperRowRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_columns)
(polynomial_mapWith polynomial_upperEntry)
def rows (z : Input) : List ℕ := List.range z.1.1.length
def rowsRealizer : Realizer rows :=
composition (composition (composition first first) listLength) range
lemma polynomial_rows : PolynomialTime rowsRealizer := 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))
def lowerMatrix (z : Input) : NatMatrix := (rows z).map (fun i => lowerRow (z,i))
def lowerMatrixRealizer : Realizer lowerMatrix :=
composition (pair identity rowsRealizer) (mapWith lowerRowRealizer)
lemma polynomial_lowerMatrix : PolynomialTime lowerMatrixRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_rows)
(polynomial_mapWith polynomial_lowerRow)
def upperMatrix (z : Input) : NatMatrix := (rows z).map (fun i => upperRow (z,i))
def upperMatrixRealizer : Realizer upperMatrix :=
composition (pair identity rowsRealizer) (mapWith upperRowRealizer)
lemma polynomial_upperMatrix : PolynomialTime upperMatrixRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_rows)
(polynomial_mapWith polynomial_upperRow)
def endpoints (z : Input) : NatMatrix × NatMatrix := (lowerMatrix z,upperMatrix z)
def endpointsRealizer : Realizer endpoints := pair lowerMatrixRealizer upperMatrixRealizer
lemma polynomial_endpoints : PolynomialTime endpointsRealizer :=
polynomial_pair polynomial_lowerMatrix polynomial_upperMatrix
lemma lowerMatrix_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) :
lowerMatrix (NetworkVertexCode.data r c b) = natCode (fun (i : Fin m) (j : Fin n) =>
CellMinimumProgram.minimum (NetworkVertexCode.data r c b,(i.val,j.val))) := by
simp only [lowerMatrix,rows,NetworkVertexCode.data,List.length_ofFn,
DirectedBFSCode.range_eq_map,←List.ofFn_eq_map,List.map_ofFn,natCode]
apply congrArg List.ofFn
funext i
dsimp only [Function.comp_apply]
simp only [lowerRow,columns,List.length_ofFn,DirectedBFSCode.range_eq_map,
←List.ofFn_eq_map,List.map_ofFn]
rfl
lemma upperMatrix_code {m n : ℕ} (r : Fin m→ℕ) (c : Fin n→ℕ) (b : Fin m→Fin n→ℕ) :
upperMatrix (NetworkVertexCode.data r c b) = natCode (fun (i : Fin m) (j : Fin n) =>
CellMaximumProgram.maximum (NetworkVertexCode.data r c b,(i.val,j.val))) := by
simp only [upperMatrix,rows,NetworkVertexCode.data,List.length_ofFn,
DirectedBFSCode.range_eq_map,←List.ofFn_eq_map,List.map_ofFn,natCode]
apply congrArg List.ofFn
funext i
dsimp only [Function.comp_apply]
simp only [upperRow,columns,List.length_ofFn,DirectedBFSCode.range_eq_map,
←List.ofFn_eq_map,List.map_ofFn]
rfl
end ContingencyTables.EndpointMatrixProgram
end OAI
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