openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /ContextChildVariance.lean
| import OAI.Combinatorics.ContingencyTables.Transport.ContextAdjacentTransport | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteExposureVariance | |
| namespace OAI | |
| /-! # Between-child variance for actual physical exposure contexts -/ | |
| namespace ContingencyTables.ExcessContextProfiles | |
| open scoped BigOperators Classical | |
| open SmallContextCoordinates SmallContextEnumeration SmallContextFibres | |
| open PhysicalCompletionFibres LargePairEnumeration BalancedFamily BalancedBox | |
| open IntegerWeightedTransport FirstPaperProfiles FirstPaperContextTransport | |
| open FirstPaperPhysicalMarginal PaddedCompletions CompletionCounts | |
| universe u | |
| lemma graphEnergy_mono {C : Type*} [DecidableEq C] | |
| (V W : Finset (C → ℕ)) (f H : (C → ℕ) → ℝ) | |
| (hVW : V ⊆ W) (hf : ∀ x, 0 ≤ f x) : | |
| integerGraphEnergy V f H ≤ integerGraphEnergy W f H := by | |
| unfold integerGraphEnergy | |
| apply div_le_div_of_nonneg_right _ (by norm_num) | |
| apply Finset.sum_le_sum_of_subset_of_nonneg | |
| · intro e he | |
| obtain ⟨he,hadj⟩ := Finset.mem_filter.mp he | |
| obtain ⟨hx,hy⟩ := Finset.mem_product.mp he | |
| exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨hVW hx,hVW hy⟩,hadj⟩ | |
| · intro e _ _ | |
| exact mul_nonneg (le_min (hf _) (hf _)) (sq_nonneg _) | |
| noncomputable def contextVertices (U n : ℕ) : | |
| Finset (Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ) := | |
| Finset.univ.biUnion (fun l : Fin (U+1) => leafVertices _ (rootDisplay U l.val)) | |
| lemma leafVertices_subset_context (U n l : ℕ) (hl : l ≤ U) : | |
| leafVertices (widths (fun _ : Fin n => U)) (rootDisplay U l) ⊆ contextVertices U n := by | |
| intro x hx | |
| exact Finset.mem_biUnion.mpr ⟨⟨l,Nat.lt_succ_of_le hl⟩,Finset.mem_univ _,hx⟩ | |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] | |
| /-- The physical conditional masses and means satisfy the paper's | |
| between-child estimate, allowing arbitrary zero-mass children. -/ | |
| theorem physical_child_variance (small : I → J → Bool) (B : I → J → ℕ) | |
| (r : I → ℕ) (c : J → ℕ) (L : ℕ) (s : I×J) (hs : small s.1 s.2=true) | |
| (σ : I×J → ℕ) (U : ℕ) (hU : 2 ≤ U) | |
| (hB : ∀ i j, small i j=true → B i j=U) | |
| (hlarge : ∀ i j, small i j=false → 2 ≤ B i j) | |
| (hσ : ∀ a : Earlier small s, σ a.val.val ≤ U) {n : ℕ} (e : Fin n ≃ Later small s) | |
| (hcap : ∀ T : ResidualTables small r c | |
| (fun i => r i+rowLargeCount small i*L) (fun j => c j+columnLargeCount small j*L), | |
| ∀ i j, T.val.val i j ≤ B i j) | |
| (H : (Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ) → ℝ) | |
| (hZ : 0 < ∑ j : Fin (U+1), eliminate (widths (fun _ : Fin n => U)) | |
| (contextHard small B r c L _ (contextEquiv small s hs e U) (prefixRaw small s σ U)) | |
| (childProfile U j.val)) : | |
| let Ms := widths (fun _ : Fin n => U) | |
| let f := contextHard small B r c L Ms (contextEquiv small s hs e U) (prefixRaw small s σ U) | |
| let z := fun j => eliminate Ms f (childProfile U j) | |
| let μ := fun j => eliminate Ms (fun p => f p*H p) (childProfile U j)/z j | |
| let E := integerGraphEnergy ((contextVertices U n).filter | |
| (fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum)) f H | |
| (∑ j : Fin (U+1), z j.val*(μ j.val- | |
| (∑ k : Fin (U+1),z k.val*μ k.val)/(∑ k : Fin (U+1),z k.val))^2) ≤ | |
| ((U:ℝ)+1)^4*(2+2*n*((U:ℝ)+1)^2)*E := by | |
| dsimp only | |
| let Ms := widths (fun _ : Fin n => U) | |
| let f := contextHard small B r c L Ms (contextEquiv small s hs e U) (prefixRaw small s σ U) | |
| let z := fun j => eliminate Ms f (childProfile U j) | |
| let μ := fun j => eliminate Ms (fun p => f p*H p) (childProfile U j)/z j | |
| let V := (contextVertices U n).filter | |
| (fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum) | |
| have hf : ∀ x, 0 ≤ f x := contextHard_nonnegative small B r c L Ms _ _ | |
| have hz (j : ℕ) : 0 ≤ z j := eliminate_nonnegative Ms f hf _ | |
| have hE : 0 ≤ integerGraphEnergy V f H := graphEnergy_nonnegative V f H hf | |
| apply child_variance_bound z μ U (2+2*n*((U:ℝ)+1)^2) (integerGraphEnergy V f H) | |
| (fun j _ => hz j) hZ (by positivity) hE | |
| · intro a j b haj hjb hbU | |
| exact context_child_minimum small B r c L Ms (contextEquiv small s hs e U) | |
| (prefixRaw small s σ U) (fun _ => U) U (prefixRaw_inBox small s σ U hσ) hlarge | |
| (fun w hw => by rw [widths_constant_mem U w hw]; exact hU) a j b haj hjb hbU | |
| · intro j hj | |
| by_cases hl : 0 < z j | |
| · by_cases hr : 0 < z (j+1) | |
| · have htransport := adjacent_context_transport small B r c L s hs σ U (j+1) hU | |
| (by omega) (by omega) hB hlarge hσ e hcap H (by simpa only [Nat.add_sub_cancel] using hl) hr | |
| dsimp only at htransport | |
| simp only [Nat.add_sub_cancel] at htransport | |
| have hsub : (leafVertices Ms (rootDisplay U (j+1))).filter | |
| (fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum) ⊆ V := by | |
| intro x hx | |
| obtain ⟨hx,hbox⟩ := Finset.mem_filter.mp hx | |
| exact Finset.mem_filter.mpr ⟨leafVertices_subset_context U n (j+1) (by omega) hx,hbox⟩ | |
| exact htransport.trans (mul_le_mul_of_nonneg_left (graphEnergy_mono _ V f H hsub hf) (by positivity)) | |
| · have hr0 : z (j+1)=0 := le_antisymm (le_of_not_gt hr) (hz (j+1)) | |
| rw [hr0,min_eq_right (hz j),zero_mul] | |
| positivity | |
| · have hl0 : z j=0 := le_antisymm (le_of_not_gt hl) (hz j) | |
| rw [hl0,min_eq_left (hz (j+1)),zero_mul] | |
| positivity | |
| end ContingencyTables.ExcessContextProfiles | |
| end OAI | |