| import OAI.Dynamics.ConditionalShuffle.Sequences |
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| namespace OAI |
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| noncomputable section |
| namespace Revealed.Instrument |
| open scoped Classical |
| open Thorp Thorp.Fourier Thorp.Specht Revealed.Disintegration |
| variable {E S Ω α β : Type} [fintype_S : Fintype S] [Fintype Ω] [nonempty_Ω : Nonempty Ω] |
| [Fintype α] [DecidableEq α] [nontrivial_α : Nontrivial α] [Fintype β] [DecidableEq β] |
| variable (I : Data E S (Equiv.Perm α) Ω) |
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| lemma kernels_dichotomy (h : ∀ e s, Trim.signDichotomy (kernel I e s)) |
| (e : E) (n : ℕ) (p : Fin n → S) : ∀ i, Trim.signDichotomy (kernels I e n p i) := by |
| let retained_fintype_S := fintype_S |
| let retained_nonempty_Ω := nonempty_Ω |
| let retained_nontrivial_α := nontrivial_α |
| induction n with |
| | zero => intro i; exact Fin.elim0 i |
| | succ n ih => |
| intro i |
| refine Fin.lastCases ?_ (fun j => ?_) i |
| · simpa only [kernels, Fin.snoc_last] using h (base I e n (Fin.init p)) (p (Fin.last n)) |
| · simpa only [kernels, Fin.snoc_castSucc] using ih (Fin.init p) j |
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| lemma block_kernel_dichotomy (h : ∀ e s, Trim.signDichotomy (kernel I e s)) |
| (e : E) (t : ℕ) (p : Fin t → S) : Trim.signDichotomy (kernel (block I t) e p) := by |
| change Trim.signDichotomy (conditional (pathJoint I e t) p) |
| by_cases hp : weights I e t p = 0 |
| · have hh : conditional (pathJoint I e t) p = fun _ => (Fintype.card (Equiv.Perm α) : ℝ)⁻¹ := by |
| funext g |
| simp [conditional, pathJoint_marginal, hp] |
| rw [hh] |
| exact Trim.signDichotomy_uniform |
| · rw [conditional_pathJoint I e t p hp] |
| exact Trim.signDichotomy_realProduct t _ (kernels_dichotomy I h e t p) |
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| theorem distance_eight_le (hn : 16 ≤ Fintype.card α) |
| (ψ : (Fin 8 → Equiv.Perm β) →* Equiv.Perm α) |
| (hc : 2 * ((Fintype.card (Equiv.Perm β) : ℝ) / Fintype.card (Equiv.Perm α)) < 1) |
| (hb : ∀ e s, Trim.signDichotomy (kernel I e s)) (C : ℝ) |
| (hC : ∀ e, ∑ s, probability I e s * Trim.discarded ψ (kernel I e s) ≤ C) (e : E) : |
| distance I e 8 ≤ 32 * C + (1/2 : ℝ) * Real.sqrt |
| ((32 * (permutationFamily β).reciprocalSum)^8 * exceptionalReciprocalSum (Fintype.card α)) := by |
| have hp (p : Fin 8 → S) := repaired_eight_tv hn ψ (kernels I e 8 p) |
| (kernels_nonneg I e 8 p) (kernels_sum I e 8 p) hc (kernels_dichotomy I hb e 8 p) |
| have ht := Finset.sum_le_sum (s := Finset.univ) (fun p _ => |
| mul_le_mul_of_nonneg_left (hp p) (weights_nonneg I e 8 p)) |
| have hcost := costs_sum_le I (fun e s => Trim.discarded ψ (kernel I e s)) C hC e 8 |
| simp_rw [costs_kernels] at hcost |
| change distance I e 8 ≤ _ at ht |
| simp only [mul_add, Finset.sum_add_distrib, ← Finset.sum_mul, weights_sum, |
| one_mul] at ht |
| have he : (∑ p : Fin 8 → S, weights I e 8 p * (4 * ∑ i, Trim.discarded ψ (kernels I e 8 p i))) = |
| 4 * ∑ p : Fin 8 → S, weights I e 8 p * ∑ i, Trim.discarded ψ (kernels I e 8 p i) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro p _ |
| ring |
| rw [he] at ht |
| norm_num only [Nat.cast_ofNat] at hcost |
| linarith |
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| end Revealed.Instrument |
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| end |
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| end OAI |
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