File size: 4,007 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 | import OAI.Combinatorics.ContingencyTables.Dense.DenseMatrixWalk
import OAI.Combinatorics.ContingencyTables.Transport.FiniteGridWords
namespace OAI
/-! # Computing every state of the dense transition table -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms GridBoundary
open scoped BigOperators
abbrev EnumeratedMatrixInput := CountMatrixContext×List Bool
def enumerationData (z : EnumeratedMatrixInput) : GridEnumerationData :=
(z.2,4*z.1.2.2^8)
def enumerationDataRealizer : Realizer enumerationData := by
let d : Realizer (fun z : EnumeratedMatrixInput => z.1.2.2) :=
composition (composition first second) second
let K := composition (pair (constant 4) (composition d eighthPower)) multiply
exact pair second K
lemma polynomial_enumerationData : PolynomialTime enumerationDataRealizer := by
have hd : PolynomialTime (composition (composition
(first (A := CountMatrixContext) (B := List Bool)) second) second) :=
polynomial_composition (polynomial_composition polynomial_first polynomial_second)
polynomial_second
unfold enumerationDataRealizer
exact polynomial_pair polynomial_second (polynomial_composition
(polynomial_pair (polynomial_constant 4) (polynomial_composition hd polynomial_eighthPower))
polynomial_multiply)
def enumeratedMatrixInput (z : EnumeratedMatrixInput) : CountMatrixData :=
(z.1,gridWords (enumerationData z))
def enumeratedMatrixInputRealizer : Realizer enumeratedMatrixInput :=
pair first (composition enumerationDataRealizer gridWordsRealizer)
def enumerationMeasure (z : EnumeratedMatrixInput) : ℕ :=
gridStateCount (enumerationData z)+weight z
lemma work_enumeratedMatrixInput : WorkBound enumeratedMatrixInputRealizer enumerationMeasure := by
have hp : WorkBound enumerationDataRealizer enumerationMeasure :=
work_of_polynomial polynomial_enumerationData Polynomial.X (fun z => by
simp only [enumerationMeasure,Polynomial.eval_X]; omega)
obtain ⟨p,hp'⟩ := hp
have hg : WorkBound (composition enumerationDataRealizer gridWordsRealizer)
enumerationMeasure := by
apply work_composition ⟨p,hp'⟩ work_gridWords (Polynomial.X+p)
intro z
have hw := hp' z
simp only [enumerationMeasure,Polynomial.eval_add,Polynomial.eval_X]
change gridStateCount (enumerationData z)+weight (enumerationData z) ≤ _
change _≤p.eval (enumerationMeasure z) at hw
unfold enumerationMeasure at hw
omega
unfold enumeratedMatrixInputRealizer
exact work_pair (work_of_polynomial polynomial_first Polynomial.X (fun z => by
simp only [enumerationMeasure,Polynomial.eval_X]; omega)) hg Polynomial.X
(fun z => by simp only [enumerationMeasure,Polynomial.eval_X]; omega)
def enumeratedMatrix (z : EnumeratedMatrixInput) : List (List ℕ) :=
countMatrix (enumeratedMatrixInput z)
def enumeratedMatrixRealizer : Realizer enumeratedMatrix :=
composition enumeratedMatrixInputRealizer countMatrixRealizer
/-- All states are generated from the dimension marker and grid size.
The work is polynomial in their count and the supplied binary data. -/
theorem work_enumeratedMatrix : WorkBound enumeratedMatrixRealizer enumerationMeasure :=
work_then_polynomial work_enumeratedMatrixInput polynomial_countMatrix
theorem enumeratedMatrix_actual {m n d : ℕ} (R : Option (Fin m) → ℕ)
(P : Option (Fin n) → ℕ) (hq : m*n≤d)
(xs dims : List Bool) (hxs : xs.length=8*d^5) (hdims : dims.length=m*n) :
enumeratedMatrix ((((xs,2^denseStepBits d),((marginCode R,marginCode P),d)),dims))=
List.ofFn (fun j => List.ofFn (fun i =>
denseTransitionCount R P d finProdFinEquiv
(gridEnumeration (m*n) (4*d^8) i) (gridEnumeration (m*n) (4*d^8) j))) := by
unfold enumeratedMatrix enumeratedMatrixInput enumerationData
rw [gridWords_eq_length dims (4*d^8) (m*n) hdims]
exact countMatrix_actual R P hq (gridEnumeration (m*n) (4*d^8)) xs hxs
end ContingencyTables.DensePrograms
end OAI
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