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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import OAI.Combinatorics.ContingencyTables.Dense.DenseTabulationScalarBounds
import OAI.Combinatorics.ContingencyTables.Sampling.ProfileMatrixEntryRuntime
namespace OAI
/-! # The finite boxes controlling one cached transition entry -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms Polynomial
def completionBoxMeasure (z : CanonicalParameters) : ℕ :=
rawBoxMeasure (canonicalCountInput z).1+
gridStateCount (completionEnumerationData z.1)+weight z
theorem completionWeightMeasure_polynomial : ∃ p : Polynomial ℕ,∀ z,
completionWeightMeasure z ≤ p.eval (completionBoxMeasure z) := by
obtain ⟨p,hp⟩ := polynomial_canonicalCountInput
obtain ⟨q,hq⟩ := rawTabulationMeasure_polynomial
refine ⟨q+p+2*X,?_⟩
intro z
let M := completionBoxMeasure z
have hz : weight z ≤ M := by unfold M completionBoxMeasure; omega
have hw : weight (canonicalCountInput z) ≤ p.eval M :=
(Nat.le_add_left _ _).trans ((hp z).trans (polynomial_eval_monotone p hz))
have hb : rawBoxMeasure (canonicalCountInput z).1 ≤ M := by
unfold M completionBoxMeasure; omega
have hr : rawTabulationMeasure (canonicalCountInput z).1 ≤ q.eval M :=
(hq (canonicalCountInput z).1).trans (polynomial_eval_monotone q hb)
have hc : gridStateCount (completionEnumerationData z.1)+weight z.1 ≤ M := by
have := weight_fst z
unfold M completionBoxMeasure
omega
change rawTabulationMeasure (canonicalCountInput z).1+weight (canonicalCountInput z)+
weight z+(gridStateCount (completionEnumerationData z.1)+weight z.1) ≤ _
change _ ≤ (q+p+2*X).eval M
simp only [eval_add,eval_mul,eval_ofNat,eval_X]
omega
end ContingencyTables.DensePrograms
namespace ContingencyTables.ProfilePrograms
open MatchingFPRAS MatchingFPRAS.TreeTyped DensePrograms Polynomial
def matrixEntryBox (z : MatrixEntryInput) : ℕ :=
completionBoxMeasure (residualParameters (entryCacheParameters z).1)+
(entryCacheParameters z).2+gridStateCount (z.1.2,2)+weight z
def entryResidualParameters (z : MatrixEntryInput) : CanonicalParameters :=
residualParameters (entryCacheParameters z).1
def entryResidualRealizer : Realizer entryResidualParameters :=
composition (f := fun z : MatrixEntryInput => (entryCacheParameters z).1)
(g := residualParameters)
(composition (f := entryCacheParameters) (g := Prod.fst) entryCacheParametersRealizer first)
residualParametersRealizer
lemma polynomial_entryResidual : PolynomialTime entryResidualRealizer :=
polynomial_composition (polynomial_composition polynomial_entryCacheParameters polynomial_first)
polynomial_residualParameters
/-- Separating binary preparation size from the four enumerated boxes. -/
theorem matrixEntryBox_bound : ∃ p : Polynomial ℕ,∀ z,
let r := entryResidualParameters z
let a := (canonicalCountInput r).1
matrixEntryBox z ≤ (4*a.2.1^8)^tabulationDimension a+
gridStateCount (completionEnumerationData a.1)+
gridStateCount (completionEnumerationData r.1)+gridStateCount (z.1.2,2)+
r.2.1+r.2.2+(entryCacheParameters z).2+p.eval (weight z) := by
obtain ⟨p,hp⟩ := polynomial_entryResidual
obtain ⟨q,hq⟩ := polynomial_canonicalCountInput
refine ⟨q.comp p+p+X,?_⟩
intro z
dsimp only
have hr : weight (entryResidualParameters z) ≤ p.eval (weight z) :=
(Nat.le_add_left _ _).trans (hp z)
have ha : weight (canonicalCountInput (entryResidualParameters z)) ≤
q.eval (p.eval (weight z)) :=
(Nat.le_add_left _ _).trans ((hq _).trans (polynomial_eval_monotone q hr))
have hf := (weight_fst (canonicalCountInput (entryResidualParameters z))).trans ha
change weight (canonicalCountInput (entryResidualParameters z)).1 ≤
q.eval (p.eval (weight z)) at hf
simp only [eval_add,eval_comp,eval_X]
change (4*((canonicalCountInput (entryResidualParameters z)).1).2.1^8)^
tabulationDimension (canonicalCountInput (entryResidualParameters z)).1+
gridStateCount (completionEnumerationData (canonicalCountInput (entryResidualParameters z)).1.1)+
((entryResidualParameters z).2.1+(entryResidualParameters z).2.2+
weight (canonicalCountInput (entryResidualParameters z)).1)+
gridStateCount (completionEnumerationData (entryResidualParameters z).1)+
weight (entryResidualParameters z)+(entryCacheParameters z).2+
gridStateCount (z.1.2,2)+weight z ≤ _
omega
theorem matrixEntryMeasure_polynomial : ∃ p : Polynomial ℕ,∀ z,
matrixEntryMeasure z ≤ p.eval (matrixEntryBox z) := by
obtain ⟨p,hp⟩ := polynomial_entryCacheParameters
obtain ⟨q,hq⟩ := completionWeightMeasure_polynomial
refine ⟨q+p+X,?_⟩
intro z
let M := matrixEntryBox z
have hz : weight z ≤ M := by unfold M matrixEntryBox; omega
have hw : weight (entryCacheParameters z) ≤ p.eval M :=
(Nat.le_add_left _ _).trans ((hp z).trans (polynomial_eval_monotone p hz))
have hc : completionBoxMeasure (residualParameters (entryCacheParameters z).1) ≤ M := by
unfold M matrixEntryBox; omega
have hq' : completionWeightMeasure (residualParameters (entryCacheParameters z).1) ≤
q.eval M := (hq _).trans (polynomial_eval_monotone q hc)
have hs : (entryCacheParameters z).2+gridStateCount (z.1.2,2)+weight z ≤ M := by
unfold M matrixEntryBox; omega
change (entryCacheParameters z).2+
completionWeightMeasure (residualParameters (entryCacheParameters z).1)+
weight (entryCacheParameters z)+gridStateCount (z.1.2,2)+weight z ≤ _
change _ ≤ (q+p+X).eval M
simp only [eval_add,eval_X]
omega
end ContingencyTables.ProfilePrograms
end OAI
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