| import OAI.Dynamics.ConditionalShuffle.FiniteMinorization |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Classical |
| open Filter Topology |
| namespace Thorp.Conditional.Reset |
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| lemma kernel_add (D b s t : ℕ) (x y : Injection D b) : |
| kernel D b (s+t) x y = ∑ z, kernel D b s x z * kernel D b t z y := by |
| unfold kernel |
| rw [fairMass_test, fairMass_eq_mean, mean_history_add] |
| apply mean_congr; intro ω |
| rw [fairMass_eq_mean] |
| apply mean_congr; intro η |
| rw [Reachability.run_append, act_mul] |
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| lemma reset_minorization (ρ : ℝ) (hρ : 0 < ρ) (hρ₁ : ρ ≤ 1) : |
| ∀ᶠ d : ℕ in atTop, ∀ (b : ℕ), (b : ℝ) ≤ (1-ρ)*(2^(d+2) : ℝ) → |
| ∀ x y : Injection (d+2) b, |
| (1-2*(1/2 : ℝ)^(8*(d+2))) / Fintype.card (Injection (d+2) b) ≤ |
| kernel (d+2) b (4*resetS ρ*(d+2)) x y := by |
| filter_upwards [kernel_row_bound ρ hρ hρ₁, kernel_column_bound ρ hρ hρ₁] with d hr hc |
| intro b hb x y |
| let : Nonempty (Injection (d+2) b) := ⟨x⟩ |
| have hh := square_minorization (kernel (d+2) b (2*resetS ρ*(d+2))) |
| (kernel_nonneg _ _ _) (kernel_row_sum _ _ _) (kernel_column_sum _ _ _) |
| ((1/2 : ℝ)^(8*(d+2))) (hr b hb) (hc b hb) x y |
| have ht : 4*resetS ρ*(d+2) = 2*resetS ρ*(d+2)+2*resetS ρ*(d+2) := by ring |
| rw [ht, kernel_add] |
| exact hh |
|
|
| def resetRho : ℝ := (2 : ℝ)⁻¹^20 |
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| def resetBlocks : ℕ := 128 |
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| def resetSuccess (D : ℕ) : ℝ := 1-2*(1/2 : ℝ)^(8*D) |
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| lemma resetSuccess_pos (d : ℕ) : 0 < resetSuccess (d+2) := by |
| have hh : (1/2 : ℝ)^(8*(d+2)) ≤ (1/2 : ℝ)^2 := |
| pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega) |
| norm_num at hh |
| unfold resetSuccess; linarith |
|
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| lemma resetS_exact : resetS resetRho = 335544320 := by norm_num [resetS, resetRho] |
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| lemma reset_time_exact : resetBlocks*(4*resetS resetRho+1) = 171798691968 := by |
| rw [resetS_exact]; rfl |
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| theorem reset_split : |
| ∀ᶠ d : ℕ in atTop, ∀ (b : ℕ), (b : ℝ) ≤ (1-resetRho)*(2^(d+2) : ℝ) → |
| ∀ x : Injection (d+2) b, |
| (∀ y, 0 ≤ acceptance (kernel (d+2) b (4*resetS resetRho*(d+2))) (resetSuccess (d+2)) x y ∧ |
| acceptance (kernel (d+2) b (4*resetS resetRho*(d+2))) (resetSuccess (d+2)) x y ≤ 1) ∧ |
| (∀ y, markedKernel (kernel (d+2) b (4*resetS resetRho*(d+2))) (resetSuccess (d+2)) x (y,true) = |
| resetSuccess (d+2) / Fintype.card (Injection (d+2) b)) ∧ |
| (∀ y, (∑ z : Bool, markedKernel (kernel (d+2) b (4*resetS resetRho*(d+2))) |
| (resetSuccess (d+2)) x (y,z)) = kernel (d+2) b (4*resetS resetRho*(d+2)) x y) ∧ |
| (∑ z, markedKernel (kernel (d+2) b (4*resetS resetRho*(d+2))) (resetSuccess (d+2)) x z) = 1 := by |
| filter_upwards [reset_minorization resetRho (by norm_num [resetRho]) (by norm_num [resetRho])] with d hd |
| intro b hb x |
| let : Nonempty (Injection (d+2) b) := ⟨x⟩ |
| have hm := hd b hb |
| exact ⟨acceptance_bounds _ _ (resetSuccess_pos d) hm x, |
| marked_true _ _ (resetSuccess_pos d) hm x, |
| marked_marginal _ _ x, marked_sum _ _ (kernel_row_sum _ _ _) x⟩ |
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| end Thorp.Conditional.Reset |
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| end |
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| end OAI |
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