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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactEntryBoxBound
import OAI.Combinatorics.ContingencyTables.Sampling.ExactResidualInput
namespace OAI
/-! # A single exponential scale for the executed rare-branch measure -/
namespace ContingencyTables.SmallGraphProfiles
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms DensePrograms CompletionCounts Polynomial
open scoped BigOperators Classical
noncomputable section
lemma originalEnumeration_scale {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ) :
gridStateCount (originalEnumerationData (List.ofFn r,List.ofFn c)) ≤ exactBoxScale r c := by
apply (originalEnumeration_count_bound r c).trans
have hd : 1 ≤ dimensionAllowance (I := Fin m) (J := Fin n) := by unfold dimensionAllowance; omega
have hp := Nat.le_self_pow (by decide : 3 ≠ 0) (dimensionAllowance (I := Fin m) (J := Fin n))
apply Nat.pow_le_pow_right (by decide)
nlinarith
lemma initialProfile_scale {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
(htotal : (∑ i,r i)=∑ j,c j) :
gridStateCount (profileEnumerationData (initialCheck (List.ofFn r,List.ofFn c))) ≤ exactBoxScale r c := by
rw [initialCheck_code r c htotal]
apply (profileEnumeration_count_bound r c _ _).trans
have hp := Nat.le_self_pow (by decide : 3 ≠ 0) (dimensionAllowance (I := Fin m) (J := Fin n))
apply Nat.pow_le_pow_right (by decide)
nlinarith
theorem exact_rare_measure_bound : ∃ p a : Polynomial ℕ,
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j →
exactRareMeasure (List.ofFn r,List.ofFn c) ≤
p.eval (a.eval (Algorithms.samplingSize n r 0)*exactBoxScale r c) := by
obtain ⟨a,ha⟩ := exact_count_input_measure_polynomial
obtain ⟨b,hb⟩ := tabulation_preparation_polynomial
obtain ⟨c,hc⟩ := exact_entry_box_bound
obtain ⟨d,hd⟩ := exact_parameter_polynomial
obtain ⟨e,he⟩ := work_exactResidualInput
obtain ⟨q,hq⟩ := matrixEntryMeasure_polynomial
let base : Polynomial ℕ := a+b.comp a+c+d+e.comp d+1
let Q : Polynomial ℕ := q+X
refine ⟨8*X+2*vectorMeasurePolynomial.comp Q,base,?_⟩
intro m n r cols htotal
let z := exactSamplingParameters (List.ofFn r,List.ofFn cols)
let E := exactBoxScale r cols
let S := Algorithms.samplingSize n r 0
let W := base.eval S*E
let M := Q.eval W
have hE : 1 ≤ E := exactBoxScale_pos r cols
have hbase : 1 ≤ base.eval S := by simp only [base,eval_add,eval_one]; omega
have hEW : E ≤ W := by simpa only [Nat.one_mul] using Nat.mul_le_mul_right E hbase
have hbaseW : base.eval S ≤ W := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (base.eval S) hE
have hWM : W ≤ M := by unfold M Q; simp only [eval_add,eval_X]; omega
have haW : a.eval S ≤ W := (by simp only [base,eval_add,eval_one]; omega : a.eval S ≤ base.eval S).trans hbaseW
have hbW : (b.comp a).eval S ≤ W :=
(by simp only [base,eval_add,eval_one]; omega : (b.comp a).eval S ≤ base.eval S).trans hbaseW
have hcW : c.eval S*E ≤ W := Nat.mul_le_mul_right E (by simp only [base,eval_add,eval_one]; omega)
have hdW : d.eval S ≤ W := (by simp only [base,eval_add,eval_one]; omega : d.eval S ≤ base.eval S).trans hbaseW
have heW : (e.comp d).eval S ≤ W :=
(by simp only [base,eval_add,eval_one]; omega : (e.comp d).eval S ≤ base.eval S).trans hbaseW
have hcount : scheduledCountInputMeasure z ≤ W := (ha m n r cols htotal).trans haW
have hparam : exactParameterMeasure (List.ofFn r,List.ofFn cols) ≤ W := (hd m n r cols htotal).trans hdW
have hres : weight (exactResidualInput (List.ofFn r,List.ofFn cols)) ≤ W := by
have ht : weight (exactResidualInput (List.ofFn r,List.ofFn cols)) ≤
e.eval (d.eval S) :=
(Nat.le_add_left _ _).trans ((he _).trans (polynomial_eval_monotone e (hd m n r cols htotal)))
exact ht.trans (by simpa only [eval_comp] using heW)
have hz : weight (List.ofFn r,List.ofFn cols) ≤ W := by
have hz' : weight z ≤ scheduledCountInputMeasure z := by
simpa only [eval_X] using weight_le_scheduledCountInputMeasure z
exact ((weight_fst z).trans hz').trans hcount
have hprep := hb z
have hprep_bound : b.eval (scheduledCountInputMeasure z) ≤ W :=
(polynomial_eval_monotone b (ha m n r cols htotal)).trans (by simpa only [eval_comp] using hbW)
have houter : weight (scheduledOuterInput z) ≤ M := (hprep.1.trans hprep_bound).trans hWM
have hN : gridStateCount (profileEnumerationData (initialCheck (List.ofFn r,List.ofFn cols))) ≤ M :=
(initialProfile_scale r cols htotal).trans (hEW.trans hWM)
have hO : gridStateCount (originalEnumerationData (List.ofFn r,List.ofFn cols)) ≤ W :=
(originalEnumeration_scale r cols).trans hEW
have hv : ∀ v∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn cols)),weight v ≤ M := by
intro v hv
exact ((hprep.2.2.1 v hv).trans hprep_bound).trans hWM
have ho : ∀ o∈originalCandidates (List.ofFn r,List.ofFn cols),weight o ≤ M ∧ weight (o.getD []) ≤ M := by
intro o ho
exact ⟨((hprep.2.2.2.1 o ho).1.trans hprep_bound).trans hWM,
((hprep.2.2.2.1 o ho).2.trans hprep_bound).trans hWM⟩
have hentry : ∀ u∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn cols)),
∀ v∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn cols)),
matrixEntryMeasure (scheduledMatrixParameters z,(u,v)) ≤ M := by
intro u hu v hv
have hx := (hc m n r cols htotal u hu v hv).trans hcW
have ht := (hq (scheduledMatrixParameters z,(u,v))).trans (polynomial_eval_monotone q hx)
apply ht.trans
unfold M Q
simp only [eval_add,eval_X]
omega
have hsteps : parameterSteps z ≤ M := by
have hs : parameterSteps z ≤ scheduledCountInputMeasure z := by unfold scheduledCountInputMeasure; omega
exact hs.trans (hcount.trans hWM)
have hvector : outerVectorMeasure (scheduledOuterInput z) ≤ vectorMeasurePolynomial.eval M :=
outerVectorMeasure_bound _ M houter hsteps (hO.trans hWM) hN hv ho hentry
have hunit := (unit_outerVectorMeasure_le (scheduledOuterInput z)).trans hvector
have hen : originalEnumerationMeasure (List.ofFn r,List.ofFn cols) ≤ 2*W := by
unfold originalEnumerationMeasure
omega
change _ ≤ (8*X+2*vectorMeasurePolynomial.comp Q).eval W
simp only [eval_add,eval_mul,eval_ofNat,eval_X,eval_comp]
change exactParameterMeasure (List.ofFn r,List.ofFn cols)+
(scheduledCountInputMeasure z+outerVectorMeasure (scheduledOuterInput z)+
UnitCountPrograms.outerVectorMeasure (scheduledOuterInput z)+
originalEnumerationMeasure (List.ofFn r,List.ofFn cols)+
weight (exactResidualInput (List.ofFn r,List.ofFn cols)))+
originalEnumerationMeasure (List.ofFn r,List.ofFn cols)+weight (List.ofFn r,List.ofFn cols) ≤ _
change _ ≤ 8*W+2*vectorMeasurePolynomial.eval M
omega
end
end ContingencyTables.SmallGraphProfiles
end OAI
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