openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactEntryBoxBound.lean
| import OAI.Combinatorics.ContingencyTables.Transport.ResidualBoxScale | |
| import OAI.Combinatorics.ContingencyTables.Transport.TabulationPreparationBounds | |
| namespace OAI | |
| /-! # The actual transition-cache boxes in the exact sampler -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms DensePrograms CompletionCounts Polynomial | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| attribute [local irreducible] unaryMarkers parameterProposalWidth scheduledCacheParameters | |
| residualParameters canonicalCountInput | |
| private lemma proposalBox_eq (z : SamplingParameters) (u v : List ℕ×List ℕ) : | |
| gridStateCount ((scheduledMatrixParameters z,(u,v)).1.2,2)=2^(parameterProposalWidth z) := by | |
| have hm : (scheduledMatrixParameters z,(u,v)).1.2=unaryMarkers (parameterProposalWidth z) := by | |
| unfold scheduledMatrixParameters | |
| rfl | |
| rw [hm,gridStateCount_eq,unaryMarkers_length] | |
| theorem exact_entry_box_bound : ∃ p : Polynomial ℕ, | |
| ∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j → | |
| ∀ u∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn c)), | |
| ∀ v∈profileBoxViews (initialCheck (List.ofFn r,List.ofFn c)), | |
| matrixEntryBox (scheduledMatrixParameters (exactSamplingParameters (List.ofFn r,List.ofFn c)),(u,v)) ≤ | |
| p.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 ⟨p,hp⟩ := matrixEntryBox_bound | |
| let A := b.comp a | |
| refine ⟨p.comp A+A+4,?_⟩ | |
| intro m n r c htotal u hu v hv | |
| let z := exactSamplingParameters (List.ofFn r,List.ofFn c) | |
| let e : MatrixEntryInput := (scheduledMatrixParameters z,(u,v)) | |
| let V := candidateData (initialCheck (List.ofFn r,List.ofFn c),u) | |
| let dim := dimensionAllowance (I := Fin m) (J := Fin n) | |
| let bits := smallBinaryLength r c | |
| let S := Algorithms.samplingSize n r 0 | |
| let E := exactBoxScale r c | |
| have hd : 1 ≤ dim := by unfold dim dimensionAllowance; omega | |
| have hbits : 1 ≤ bits := hd.trans (smallBinaryLength_parameters r c).2.1 | |
| have h23 : dim^2 ≤ dim^3 := Nat.pow_le_pow_right hd (by decide) | |
| have he : 1 ≤ E := exactBoxScale_pos r c | |
| have hdim : parameterDimension z=dim := by | |
| change marginDimension (List.ofFn r,List.ofFn c)=dim | |
| exact marginDimension_ofFn r c | |
| have hVrow : V.1.2.1=List.ofFn r := rfl | |
| have hVcol : V.1.2.2=List.ofFn c := rfl | |
| have hres : entryResidualParameters e=residualParameters ((V,dim^12),(dim,parameterAccuracy z)) := by | |
| unfold entryResidualParameters | |
| dsimp only [e,entryCacheParameters,sourceData,scheduledMatrixParameters,scheduledCacheParameters,matrixData] | |
| unfold scheduledCacheParameters | |
| dsimp only | |
| rw [show z.1=(List.ofFn r,List.ofFn c) from rfl,marginPadding_ofFn,hdim] | |
| have hnorm := normalized_profile_completion_count r c V hVrow hVcol (parameterAccuracy z) | |
| have hdense := normalized_profile_dense_grid r c V hVrow hVcol (parameterAccuracy z) | |
| have horig : gridStateCount (completionEnumerationData (entryResidualParameters e).1) ≤ E := by | |
| rw [hres] | |
| unfold residualParameters | |
| have hh := arbitrary_profile_completion_count r c V hVrow hVcol | |
| apply hh.trans | |
| apply Nat.pow_le_pow_right (by decide) | |
| have hmul : dim^2*bits ≤ dim^3*bits := Nat.mul_le_mul_right bits h23 | |
| change 4*(dim^2*bits) ≤ 40*(dim^3*bits) | |
| omega | |
| have hprop : gridStateCount (e.1.2,2) ≤ E := by | |
| rw [proposalBox_eq z u v,parameterProposalWidth,hdim] | |
| apply Nat.pow_le_pow_right (by decide) | |
| have hc : Nat.clog 2 (32*dim^2) ≤ 32*dim^2 := | |
| Nat.clog_le_of_le_pow (Nat.le_of_lt Nat.lt_two_pow_self) | |
| have hmul : dim^3 ≤ dim^3*bits := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (dim^3) hbits | |
| exact hc.trans (by change 32*dim^2 ≤ 40*(dim^3*bits); nlinarith) | |
| have hprepare := hb z | |
| have hnum : parameterDimension z+parameterAccuracy z+parameterCompletionWidth z ≤ A.eval S := | |
| hprepare.2.1.trans (by simp only [A,eval_comp]; exact polynomial_eval_monotone b (ha m n r c htotal)) | |
| have hw : weight e ≤ A.eval S := (hprepare.2.2.2.2 u hu v hv).trans (by | |
| simp only [A,eval_comp]; exact polynomial_eval_monotone b (ha m n r c htotal)) | |
| have hpoly : p.eval (weight e) ≤ p.eval (A.eval S) := polynomial_eval_monotone p hw | |
| have hbound := hp e | |
| dsimp only at hbound | |
| have hn : gridStateCount (completionEnumerationData (canonicalCountInput (entryResidualParameters e)).1.1) ≤ E := by | |
| rw [hres] | |
| exact hnorm | |
| have hg : (4*(canonicalCountInput (entryResidualParameters e)).1.2.1^8)^ | |
| tabulationDimension (canonicalCountInput (entryResidualParameters e)).1 ≤ E := by | |
| rw [hres] | |
| exact hdense | |
| change matrixEntryBox e ≤ _ | |
| have hrs : (entryResidualParameters e).2=(parameterDimension z,parameterAccuracy z) := by | |
| rw [hres] | |
| unfold residualParameters | |
| rw [hdim] | |
| have hcache : (entryCacheParameters e).2=parameterCompletionWidth z := by | |
| unfold e entryCacheParameters scheduledMatrixParameters | |
| unfold scheduledCacheParameters | |
| rfl | |
| have hnum' : (entryResidualParameters e).2.1+(entryResidualParameters e).2.2+ | |
| (entryCacheParameters e).2 ≤ A.eval S := by | |
| rw [hrs,hcache] | |
| exact hnum | |
| have hfirst : matrixEntryBox e ≤ 4*E+A.eval S+p.eval (A.eval S) := by | |
| omega | |
| apply hfirst.trans | |
| have hA : A.eval S ≤ A.eval S*E := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (A.eval S) he | |
| have hpA : p.eval (A.eval S) ≤ p.eval (A.eval S)*E := by | |
| simpa only [Nat.mul_one] using Nat.mul_le_mul_left (p.eval (A.eval S)) he | |
| simp only [eval_add,eval_comp,eval_ofNat] | |
| nlinarith | |
| end | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |