File size: 4,885 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.PaddedResidualProgram
import OAI.Combinatorics.ContingencyTables.Sampling.SmallProfileValidator
import Mathlib.Data.List.NodupEquivFin
namespace OAI
/-! # Computing the large rectangle and its padded margins -/
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS.TreeTyped
def largeIndexTest (z : (ℕ×List ℕ)×ℕ) : Bool :=
decide (z.1.1≤z.1.2.getD z.2 0)
def largeIndexTestRealizer : Realizer largeIndexTest :=
composition (pair (composition first first)
(composition (pair (composition first second) second) (getDFast 0))) leNat
theorem polynomial_largeIndexTest : PolynomialTime largeIndexTestRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_first)
(polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_second) polynomial_second)
(polynomial_getDFast 0))) polynomial_leNat
def largeIndices (z : ℕ×List ℕ) : List ℕ :=
(List.range z.2.length).filter (fun i => largeIndexTest (z,i))
def largeIndicesRealizer : Realizer largeIndices :=
composition (pair identity (composition (composition second listLength) range))
(filterWith largeIndexTestRealizer)
theorem polynomial_largeIndices : PolynomialTime largeIndicesRealizer := by
have hr : PolynomialTime (composition (composition
(second (A := ℕ) (B := List ℕ)) listLength) range) := by
apply polynomial_boundedRange (polynomial_composition polynomial_second polynomial_listLength)
Polynomial.X
intro z
simpa only [Function.comp_apply,Polynomial.eval_X] using
(list_length_le_weight z.2).trans (by rw [weight_prod]; omega)
exact polynomial_composition (polynomial_pair polynomial_identity hr)
(polynomial_filterWith polynomial_largeIndexTest)
def selectList {A : Type} (a : A) (z : List ℕ×List A) : List A :=
z.1.map (fun i => z.2.getD i a)
def selectListRealizer {A : Type} [Codec A] (a : A) : Realizer (selectList a) :=
composition (pair second first) (mapWith (getDFast a))
theorem polynomial_selectList {A : Type} [Codec A] (a : A) :
PolynomialTime (selectListRealizer a) :=
polynomial_composition (polynomial_pair polynomial_second polynomial_first)
(polynomial_mapWith (polynomial_getDFast a))
def largePaddedMargins (row : Bool) (z : PaddingData) : List ℕ :=
selectList 0 (largeIndices (z.1.2,marginList row z.1),paddedMargins row z)
def largePaddedMarginsRealizer (row : Bool) : Realizer (largePaddedMargins row) :=
composition (pair
(composition (pair (composition first second)
(composition first (marginListRealizer row))) largeIndicesRealizer)
(paddedMarginsRealizer row)) (selectListRealizer 0)
theorem polynomial_largePaddedMargins (row : Bool) :
PolynomialTime (largePaddedMarginsRealizer row) :=
polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_second)
(polynomial_composition polynomial_first (polynomial_marginList row)))
polynomial_largeIndices) (polynomial_paddedMargins row)) (polynomial_selectList 0)
def profilePaddedMargins (row : Bool) (z : ProfileCheckData×ℕ) : List ℕ :=
largePaddedMargins row
(((((z.1.1.1.1,if row then z.1.1.1.2.1 else complementValues (z.1.2,z.1.1.1.2.2)),
z.1.1.2),z.1.2)),z.2)
def profilePaddedMarginsRealizer (row : Bool) : Realizer (profilePaddedMargins row) :=
composition (pair (composition first (profileLineData row)) second)
(largePaddedMarginsRealizer row)
theorem polynomial_profilePaddedMargins (row : Bool) :
PolynomialTime (profilePaddedMarginsRealizer row) :=
polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first (polynomial_profileLineData row)) polynomial_second)
(polynomial_largePaddedMargins row)
lemma mem_largeIndices (U : ℕ) (r : List ℕ) (i : ℕ) :
i∈largeIndices (U,r) ↔ i<r.length ∧ U≤r.getD i 0 := by
simp only [largeIndices,List.mem_filter,List.mem_range,largeIndexTest,decide_eq_true_eq]
lemma largeIndices_nodup (U : ℕ) (r : List ℕ) : (largeIndices (U,r)).Nodup :=
(List.nodup_range (n := r.length)).filter _
lemma largePaddedMargins_eq (row : Bool) (z : PaddingData) :
largePaddedMargins row z=
(largeIndices (z.1.2,marginList row z.1)).map (fun i => paddedMarginAt row (z,i)) := by
change (largeIndices (z.1.2,marginList row z.1)).map
(fun i => (paddedMargins row z).getD i 0)=_
apply List.map_congr_left
intro i hi
have hi' := (mem_largeIndices _ _ _).mp hi
rw [List.getD_eq_getElem (l := paddedMargins row z) (n := i) (d := 0)
(by simpa only [paddedMargins,List.length_map,List.length_range] using hi'.1)]
simp only [paddedMargins,List.getElem_map,List.getElem_range]
end ContingencyTables.ProfilePrograms
end OAI
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