File size: 3,591 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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseExponentArithmetic
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperPreparedProposal
namespace OAI
/-! # Binary execution of one dense exponent component -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS.TreeTyped ContingencyTables.ProfilePrograms
abbrev ComponentData := (ℕ×ℕ)×(ℕ×ℕ)
def eighthPower : Realizer (fun n : ℕ => n^8) :=
(composition fourthPower squareNat).congr (by
intro n
simp only [Function.comp_apply]
ring)
lemma polynomial_eighthPower : PolynomialTime eighthPower :=
polynomial_congr _ (polynomial_composition polynomial_fourthPower polynomial_squareNat)
def componentValue (z : ComponentData) : ℕ :=
GridBoundary.componentExponent z.1.1 z.1.2 z.2.1 z.2.2
def componentRealizer : Realizer componentValue := by
let D : Realizer (fun z : ComponentData => z.1.1) := composition first first
let S : Realizer (fun z : ComponentData => z.1.2) := composition first second
let P : Realizer (fun z : ComponentData => z.2.1) := composition second first
let N : Realizer (fun z : ComponentData => z.2.2) := composition second second
let D4 := composition D fourthPower
let D8 := composition D eighthPower
let pos := composition (pair D4
(composition (pair (composition (pair D8 P) multiply)
(composition (pair D S) multiply)) add)) multiply
let neg := composition (pair (composition (pair D4 D8) multiply) N) multiply
let den := composition (pair S D8) multiply
exact (composition (pair (composition (pair neg pos) subtract) den) quotient).congr
(by intro z; rfl)
/-- Cost is polynomial in the binary data, even when the represented
margin values and the quotient are large. -/
theorem polynomial_component : PolynomialTime componentRealizer := by
have hD : PolynomialTime (composition
(first (A := ℕ×ℕ) (B := ℕ×ℕ)) first) :=
polynomial_composition polynomial_first polynomial_first
have hS : PolynomialTime (composition
(first (A := ℕ×ℕ) (B := ℕ×ℕ)) second) :=
polynomial_composition polynomial_first polynomial_second
have hP : PolynomialTime (composition
(second (A := ℕ×ℕ) (B := ℕ×ℕ)) first) :=
polynomial_composition polynomial_second polynomial_first
have hN : PolynomialTime (composition
(second (A := ℕ×ℕ) (B := ℕ×ℕ)) second) :=
polynomial_composition polynomial_second polynomial_second
have h4 := polynomial_composition hD polynomial_fourthPower
have h8 := polynomial_composition hD polynomial_eighthPower
have hp := polynomial_composition (polynomial_pair h4
(polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair h8 hP) polynomial_multiply)
(polynomial_composition (polynomial_pair hD hS) polynomial_multiply)) polynomial_add))
polynomial_multiply
have hn := polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair h4 h8) polynomial_multiply) hN) polynomial_multiply
have hd := polynomial_composition (polynomial_pair hS h8) polynomial_multiply
unfold componentRealizer
exact polynomial_congr _ (polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair hn hp) polynomial_subtract) hd) polynomial_quotient)
theorem componentValue_eq_floor (d s positive negative : ℕ) :
componentValue ((d,s),(positive,negative))=
⌊(d : ℚ)^4*(-((positive : ℚ)-negative)/s-(d : ℚ)/(d : ℚ)^8)⌋₊ :=
GridBoundary.componentExponent_eq d s positive negative
end ContingencyTables.DensePrograms
end OAI
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