openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactRareMeasureBound.lean
| 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 | |