openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /CompletionEnergyComparison.lean
| import OAI.Combinatorics.ContingencyTables.Transport.CompletionChainFlow | |
| namespace OAI | |
| /-! # From literal acceptance counts to weighted graph energy -/ | |
| namespace ContingencyTables.CompletionTranslation | |
| open scoped BigOperators Classical | |
| open CommonBasesFPRAS | |
| variable {X I J : Type*} [Fintype X] [Nonempty X] [Fintype I] [Fintype J] | |
| /-- This is the actual chain energy comparison. The half-acceptance | |
| premise is supplied by the proved reference-star small-entry estimate. -/ | |
| theorem completionChain_energy_lower | |
| (R : X → Option I → ℕ) (P : X → Option J → ℕ) | |
| (htotal : ∀ x,(∑ i,R x i)=∑ j,P x j) | |
| (Adj : X → X → Prop) (hsym : ∀ x y,Adj x y ↔ Adj y x) | |
| (β : ℝ) (hβ : 0 ≤ β) (D : ℕ) | |
| (hdegree : ∀ x,(Finset.univ.filter (Adj x)).card ≤ D) | |
| (hproposal : (D:ℝ)*β ≤ 1/2) | |
| (haccept : ∀ x y,Adj x y → (blockCount R P x:ℝ)/2 ≤ acceptedCount R P x y) | |
| (H : X → ℝ) : | |
| β/(2*MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)))* | |
| ((∑ x,∑ y,if Adj x y then min (blockCount R P x:ℝ) (blockCount R P y:ℝ)* | |
| (H x-H y)^2 else 0)/2) ≤ | |
| (completionChain R P htotal Adj hsym β hβ D hdegree hproposal).energy H := by | |
| let Z := MetropolisChain.normalizer (fun z => (blockCount R P z:ℝ)) | |
| have hZ : 0 < Z := MetropolisChain.normalizer_pos _ | |
| (fun z => by exact_mod_cast blockCount_pos R P htotal z) | |
| have hterm (x y : X) : | |
| β/2*(if Adj x y then min (blockCount R P x:ℝ) (blockCount R P y:ℝ)*(H x-H y)^2 else 0) ≤ | |
| if Adj x y then β*(acceptedCount R P x y:ℝ)*(H x-H y)^2 else 0 := by | |
| by_cases hxy : Adj x y | |
| · rw [ite_eq_left hxy,ite_eq_left hxy] | |
| have hm : min (blockCount R P x:ℝ) (blockCount R P y:ℝ)/2 ≤ acceptedCount R P x y := | |
| (div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num)).trans (haccept x y hxy) | |
| have hs := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm hβ) (sq_nonneg (H x-H y)) | |
| convert hs using 1 | |
| all_goals ring | |
| · rw [ite_eq_right hxy,ite_eq_right hxy,mul_zero] | |
| have hsum := Finset.sum_le_sum (fun x (_ : x ∈ (Finset.univ : Finset X)) => | |
| Finset.sum_le_sum (fun y (_ : y ∈ (Finset.univ : Finset X)) => hterm x y)) | |
| simp_rw [← Finset.mul_sum] at hsum | |
| rw [completionChain_energy] | |
| have hb := div_le_div_of_nonneg_right hsum (show 0 ≤ 2*Z by positivity) | |
| convert hb using 1 | |
| all_goals | |
| dsimp only [Z] | |
| ring | |
| end ContingencyTables.CompletionTranslation | |
| end OAI | |