| import OAI.Combinatorics.BalancedRyser.Selection.SelectionKernel |
| import OAI.Combinatorics.BalancedRyser.Selection.SelectionPushforward2 |
|
|
| namespace OAI |
|
|
| namespace Balanced.DeletionLaw |
|
|
| theorem sum_kernelOfSlotWeights {q n : ℕ} |
| (w : Offset q n → Offset q n → ((Fin 2 × Fin n) → Fin q) → ℝ) |
| (m₀ m₁ : Offset q n) : |
| ∑ s, kernelOfSlotWeights w m₀ m₁ s = ∑ t, w m₀ m₁ t := by |
| exact Fintype.sum_equiv (Equiv.curry (Fin 2) (Fin n) (Fin q)).symm |
| (fun s => kernelOfSlotWeights w m₀ m₁ s) (w m₀ m₁) (fun _ => rfl) |
|
|
| theorem kernelOfSlotWeights_subprobability {q n : ℕ} |
| (w : Offset q n → Offset q n → ((Fin 2 × Fin n) → Fin q) → ℝ) |
| (hw : ∀ m₀ m₁ t, 0 ≤ w m₀ m₁ t) |
| (hmass : ∀ m₀ m₁, ∑ t, w m₀ m₁ t ≤ 1) : |
| IsSubprobabilityKernel (kernelOfSlotWeights w) := by |
| constructor |
| · intro m₀ m₁ s |
| exact hw m₀ m₁ _ |
| · intro m₀ m₁ |
| rw [sum_kernelOfSlotWeights] |
| exact hmass m₀ m₁ |
|
|
| theorem kernelOfSlotWeights_cylinders {q n : ℕ} |
| (w : Offset q n → Offset q n → ((Fin 2 × Fin n) → Fin q) → ℝ) (K : ℝ) |
| (hbound : ∀ m₀ m₁ J t, |
| FinitePushforward.cylinderMass (w m₀ m₁) J t ≤ (K / q) ^ J.card) : |
| CylinderBound (kernelOfSlotWeights w) K := by |
| intro m₀ m₁ J t |
| calc |
| (∑ s, if ∀ j ∈ J, s j.1 j.2 = t j then kernelOfSlotWeights w m₀ m₁ s else 0) = |
| FinitePushforward.cylinderMass (w m₀ m₁) J t := by |
| exact Fintype.sum_equiv (Equiv.curry (Fin 2) (Fin n) (Fin q)).symm |
| _ _ (fun _ => rfl) |
| _ ≤ (K / q) ^ J.card := hbound m₀ m₁ J t |
|
|
| end Balanced.DeletionLaw |
|
|
| end OAI |
|
|
