File size: 6,392 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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import OAI.Combinatorics.ContingencyTables.Dense.DenseCompletionProgram
import OAI.Combinatorics.ContingencyTables.Dense.DenseExponentProgram
namespace OAI
/-! # Computing all dense exponent components and their maximum -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS.TreeTyped ContingencyTables.ProfilePrograms Polynomial
open scoped BigOperators
def listMaximum (z : List ℕ×ℕ) : ℕ := z.1.foldl (fun a b => max b a) z.2
def listMaximumRealizer : Realizer listMaximum := fold maximum
lemma polynomial_maximum : PolynomialTime maximum := by
unfold maximum
exact polynomial_congr _ (polynomial_choose polynomial_less polynomial_second polynomial_first)
lemma foldMaximum_size (xs : List ℕ) (a S : ℕ) (ha : a.size≤S)
(hx : ∀ b∈xs,b.size≤S) :
(xs.foldl (fun a b => max b a) a).size≤S := by
induction xs generalizing a with
| nil => exact ha
| cons b bs ih =>
apply ih
· rcases le_total b a with h|h
· simpa only [max_eq_right h] using ha
· simpa only [max_eq_left h] using hx b (by simp)
· intro c hc
exact hx c (by simp [hc])
theorem polynomial_listMaximum : PolynomialTime listMaximumRealizer := by
apply polynomial_fold polynomial_maximum (4*X+1)
intro xs a pre post he
let S := weight (xs,a)
have ha : a.size≤S := (size_le_weight a).trans (weight_snd (xs,a))
have hb : ∀ b∈pre,b.size≤S := by
intro b hb
exact (size_le_weight b).trans ((weight_mem (as := xs) (by simp [he,hb])).trans
(weight_fst (xs,a)))
have hs := foldMaximum_size pre a S ha hb
have hw := weight_nat_le (pre.foldl (fun a b => max b a) a)
simp only [eval_add,eval_mul,eval_ofNat,eval_X,eval_one]
change _≤4*S+1
omega
lemma listMaximum_le (xs : List ℕ) (a M : ℕ) :
listMaximum (xs,a)≤M ↔ a≤M ∧ ∀ b∈xs,b≤M := by
induction xs generalizing a with
| nil => simp [listMaximum]
| cons b bs ih =>
simp only [listMaximum,List.foldl_cons] at ih ⊢
rw [ih]
simp only [max_le_iff,List.mem_cons]
constructor
· rintro ⟨⟨hb,ha⟩,hbs⟩
refine ⟨ha,?_⟩
intro c hc
rcases hc with rfl|hc
· exact hb
· exact hbs c hc
· rintro ⟨ha,hxs⟩
exact ⟨⟨hxs b (Or.inl rfl),ha⟩,fun c hc => hxs c (Or.inr hc)⟩
lemma le_listMaximum_default (xs : List ℕ) (a : ℕ) : a≤listMaximum (xs,a) := by
exact ((listMaximum_le xs a _).mp le_rfl).1
lemma le_listMaximum_member (xs : List ℕ) (a b : ℕ) (hb : b∈xs) :
b≤listMaximum (xs,a) := ((listMaximum_le xs a _).mp le_rfl).2 b hb
abbrev ExponentData := CompletionData×ℕ
def exponentCellData (z : ExponentData×(ℕ×ℕ)) : ComponentData :=
((z.1.2,min (z.1.1.1.1.getD z.2.1 0) (z.1.1.1.2.getD z.2.2 0)),
completionParts (z.1.1,z.2))
def exponentCellDataRealizer : Realizer exponentCellData := by
let R : Realizer (fun z : ExponentData×(ℕ×ℕ) => z.1.1.1.1) :=
composition (composition (composition first first) first) first
let C : Realizer (fun z : ExponentData×(ℕ×ℕ) => z.1.1.1.2) :=
composition (composition (composition first first) first) second
let I : Realizer (fun z : ExponentData×(ℕ×ℕ) => z.2.1) := composition second first
let J : Realizer (fun z : ExponentData×(ℕ×ℕ) => z.2.2) := composition second second
exact pair (pair (composition first second)
(composition (pair (composition (pair R I) (getDFast 0))
(composition (pair C J) (getDFast 0))) minimum))
(composition (pair (composition first first) second) completionPartsRealizer)
theorem polynomial_exponentCellData : PolynomialTime exponentCellDataRealizer := by
unfold exponentCellDataRealizer
apply polynomial_pair
· apply polynomial_pair (polynomial_composition polynomial_first polynomial_second)
apply polynomial_composition _ polynomial_minimum
apply polynomial_pair
· apply polynomial_composition _ (polynomial_getDFast 0)
exact polynomial_pair (polynomial_composition
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_first) polynomial_first)
(polynomial_composition polynomial_second polynomial_first)
· apply polynomial_composition _ (polynomial_getDFast 0)
exact polynomial_pair (polynomial_composition
(polynomial_composition (polynomial_composition polynomial_first polynomial_first)
polynomial_first) polynomial_second)
(polynomial_composition polynomial_second polynomial_second)
· exact polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_first) polynomial_second)
polynomial_completionParts
def exponentCell (z : ExponentData×(ℕ×ℕ)) : ℕ := componentValue (exponentCellData z)
def exponentCellRealizer : Realizer exponentCell :=
composition exponentCellDataRealizer componentRealizer
theorem polynomial_exponentCell : PolynomialTime exponentCellRealizer :=
polynomial_composition polynomial_exponentCellData polynomial_component
def exponentCells (z : ExponentData) : List (ℕ×ℕ) := cellPairs z.1.1
def exponentCellsRealizer : Realizer exponentCells :=
composition (composition first first) cellPairsRealizer
theorem polynomial_exponentCells : PolynomialTime exponentCellsRealizer :=
polynomial_composition (polynomial_composition polynomial_first polynomial_first) polynomial_cellPairs
def exponentValues (z : ExponentData) : List ℕ :=
(exponentCells z).map (fun ij => exponentCell (z,ij))
def exponentValuesRealizer : Realizer exponentValues :=
composition (pair identity exponentCellsRealizer) (mapWith exponentCellRealizer)
theorem polynomial_exponentValues : PolynomialTime exponentValuesRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_exponentCells)
(polynomial_mapWith polynomial_exponentCell)
def fullExponent (z : ExponentData) : ℕ := listMaximum (exponentValues z,0)
def fullExponentRealizer : Realizer fullExponent :=
composition (pair exponentValuesRealizer (constant 0)) listMaximumRealizer
/-- The exponent evaluation traverses only the supplied matrix dimensions. -/
theorem polynomial_fullExponent : PolynomialTime fullExponentRealizer :=
polynomial_composition (polynomial_pair polynomial_exponentValues (polynomial_constant 0))
polynomial_listMaximum
end ContingencyTables.DensePrograms
end OAI
|