| 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 |
|
|
