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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | 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
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