| import OAI.NumberTheory.Ostmann.Characters.BinaryExposure |
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| namespace OAI |
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| noncomputable section |
| namespace Ostmann.Characters.BinaryExposure |
| attribute [local instance] Classical.propDecidable |
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| def Budget : ℕ → Type |
| | 0 => PUnit |
| | k+1 => NNReal × (Budget k × Budget k) |
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| def Budget.value : {k:ℕ} → Budget k → NNReal |
| | 0,_ => 1 |
| | _k+1,b => b.1 * b.2.1.value * b.2.2.value |
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| variable {G H:Type*} [Group G] [Fintype G] |
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| def Controlled (c:ℕ→H→G→G→Prop) (left right:ℕ→H→G→G→H) : |
| (k:ℕ) → H → G → Budget k → Prop |
| | 0,_,_,_ => True |
| | k+1,h,t,b => |
| (Nat.card {x:G // c k h t x}:ℝ) / Nat.card G ≤ b.1 ∧ |
| ∀ x:G, Controlled c left right k (left k h t x) x b.2.1 ∧ |
| Controlled c left right k (right k h t x) (x⁻¹*t) b.2.2 |
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| theorem probability_le_tree (c:ℕ→H→G→G→Prop) |
| (left right:ℕ→H→G→G→H) (k:ℕ) (h:H) (t:G) (b:Budget k) |
| (hc:Controlled c left right k h t b) : |
| probability c left right k h t ≤ (b.value:ℝ) := by |
| induction k generalizing h t with |
| | zero => exact le_rfl |
| | succ k ih => |
| change avg (fun x:G=>(if c k h t x then 1 else 0)* |
| probability c left right k (left k h t x) x * |
| probability c left right k (right k h t x) (x⁻¹*t)) ≤ _ |
| apply le_trans (avg_mono _ (fun x:G=>(if c k h t x then 1 else 0)* |
| ((b.2.1.value:ℝ)*(b.2.2.value:ℝ))) ?_) |
| · rw [avg_mul_const,avg_indicator] |
| change _ ≤ (b.1:ℝ)*(b.2.1.value:ℝ)*(b.2.2.value:ℝ) |
| rw [mul_assoc] |
| exact mul_le_mul_of_nonneg_right hc.1 (mul_nonneg b.2.1.value.property b.2.2.value.property) |
| · intro x |
| by_cases hx:c k h t x |
| · simp only [hx,ite_true,one_mul] |
| exact mul_le_mul (ih _ _ b.2.1 (hc.2 x).1) |
| (ih _ _ b.2.2 (hc.2 x).2) |
| (probability_nonneg c left right k _ _) b.2.1.value.property |
| · simp only [hx,ite_false,zero_mul,le_refl] |
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| theorem count_le_tree (c:ℕ→H→G→G→Prop) |
| (left right:ℕ→H→G→G→H) (k:ℕ) (h:H) (t:G) (b:Budget k) |
| (hc:Controlled c left right k h t b) : |
| (Nat.card {z:BinaryHaar.Splits G k // admissible c left right k h t z}:ℝ)/ |
| Nat.card (BinaryHaar.Splits G k) ≤ (b.value:ℝ) := by |
| rw [← probability_eq_count] |
| exact probability_le_tree c left right k h t b hc |
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| end Ostmann.Characters.BinaryExposure |
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| end |
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| end OAI |
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