openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /ContextGraphEnergy.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalGraphVertices | |
| namespace OAI | |
| /-! # Contextual transport energy embeds in the actual physical graph -/ | |
| namespace ContingencyTables.SmallGraphProfiles | |
| open scoped BigOperators Classical | |
| open SmallContextCoordinates SmallContextEnumeration SmallContextFibres | |
| open PhysicalCompletionFibres LargePairEnumeration BalancedFamily BalancedBox | |
| open IntegerWeightedTransport FirstPaperProfiles ExcessContextProfiles | |
| open FirstPaperContextTransport FirstPaperPhysicalMarginal | |
| universe u | |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] | |
| def PhysicalPrefix (small : I → J → Bool) (s : I×J) (σ : I×J → ℕ) | |
| (z : SmallProfile small) : Prop := | |
| ∀ a : Earlier small s, z (a.val,false)=σ a.val.val | |
| omit [Fintype I] [Fintype J] in | |
| lemma assemble_physicalPrefix (small : I → J → Bool) (s : I×J) (hs : small s.1 s.2=true) | |
| (σ : I×J → ℕ) (U : ℕ) {n : ℕ} (e : Fin n ≃ Later small s) | |
| (x : Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ) : | |
| PhysicalPrefix small s σ (assemble small s hs σ U e x) := by | |
| intro a | |
| exact assemble_prefix small s hs σ U e x a false | |
| noncomputable def physicalContextEdges (small : I → J → Bool) (B : I → J → ℕ) | |
| (s : I×J) (σ : I×J → ℕ) : Finset (SmallProfile small×SmallProfile small) := | |
| ((vertices small B) ×ˢ (vertices small B)).filter | |
| (fun p => IntegerExchangeAdjacent p.1 p.2 ∧ PhysicalPrefix small s σ p.1 ∧ PhysicalPrefix small s σ p.2) | |
| noncomputable def physicalContextEnergy (small : I → J → Bool) (B : I → J → ℕ) | |
| (s : I×J) (σ : I×J → ℕ) (f H : SmallProfile small → ℝ) : ℝ := | |
| (∑ p ∈ physicalContextEdges small B s σ, edgeTerm f H p)/2 | |
| omit [LinearOrder I] [LinearOrder J] in | |
| lemma hardMarginal_nonnegative (small : I → J → Bool) (B : I → J → ℕ) | |
| (r : I → ℕ) (c : J → ℕ) (L : ℕ) (z : SmallProfile small) : | |
| 0 ≤ hardMarginal small B r c L z := by | |
| unfold hardMarginal fullHardFamily NestedPenaltyApplication.hardWeight | |
| apply Finset.sum_nonneg | |
| intro a _ | |
| split_ifs <;> norm_num | |
| /-- The actual conditional variance energy uses physical allowed states | |
| and ordinary exchange edges in its fixed row-major prefix only. -/ | |
| theorem context_graphEnergy_le (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 : 1 ≤ U) (hB : ∀ i j, small i j=true → B i j=U) | |
| (hσ : ∀ a : Earlier small s, σ a.val.val ≤ U) {n : ℕ} (e : Fin n ≃ Later small s) | |
| (H : SmallProfile small → ℝ) : | |
| 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) | |
| integerGraphEnergy ((contextVertices U n).filter | |
| (fun x => InBox (bounds (fun _ : Bool => U) Ms) x ∧ (∑ i, x i)=U+Ms.sum)) | |
| f (fun x => H (assemble small s hs σ U e x)) ≤ | |
| physicalContextEnergy small B s σ (hardMarginal small B r c L) H := by | |
| dsimp only | |
| let V := (contextVertices U n).filter | |
| (fun x => InBox (bounds (fun _ : Bool => U) (widths (fun _ : Fin n => U))) x ∧ | |
| (∑ i, x i)=U+(widths (fun _ : Fin n => U)).sum) | |
| let E := (V ×ˢ V).filter (fun p => IntegerExchangeAdjacent p.1 p.2) | |
| let j : ((Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ)× | |
| (Coordinates Bool (widths (fun _ : Fin n => U)) → ℕ)) → | |
| (SmallProfile small×SmallProfile small) := | |
| fun p => (assemble small s hs σ U e p.1,assemble small s hs σ U e p.2) | |
| have hj : Function.Injective j := by | |
| intro a b he | |
| apply Prod.ext | |
| · exact assemble_injective small s hs σ U e (congrArg Prod.fst he) | |
| · exact assemble_injective small s hs σ U e (congrArg Prod.snd he) | |
| have hmem (x) (hx : x ∈ V) : assemble small s hs σ U e x ∈ vertices small B := by | |
| obtain ⟨hx,hbox,_⟩ := Finset.mem_filter.mp hx | |
| exact context_vertex_mem small B s hs σ U hU hB hσ e x hx hbox | |
| have hsub : E.image j ⊆ physicalContextEdges small B s σ := by | |
| intro p hp | |
| obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hp | |
| obtain ⟨hq,hadj⟩ := Finset.mem_filter.mp hq | |
| obtain ⟨hx,hy⟩ := Finset.mem_product.mp hq | |
| apply Finset.mem_filter.mpr | |
| exact ⟨Finset.mem_product.mpr ⟨hmem _ hx,hmem _ hy⟩, | |
| assemble_exchange_adjacent small s hs σ U e _ _ hadj, | |
| assemble_physicalPrefix small s hs σ U e _,assemble_physicalPrefix small s hs σ U e _⟩ | |
| unfold integerGraphEnergy physicalContextEnergy | |
| apply div_le_div_of_nonneg_right _ (by norm_num) | |
| change (∑ p ∈ E, edgeTerm (hardMarginal small B r c L) H (j p)) ≤ _ | |
| calc | |
| _ = ∑ p ∈ E.image j, edgeTerm (hardMarginal small B r c L) H p := | |
| (Finset.sum_image (fun a _ b _ he => hj he)).symm | |
| _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg hsub (fun p _ _ => | |
| mul_nonneg (le_min (hardMarginal_nonnegative small B r c L _) | |
| (hardMarginal_nonnegative small B r c L _)) (sq_nonneg _)) | |
| end ContingencyTables.SmallGraphProfiles | |
| end OAI | |