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92416a2 | 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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Dynamics.ConditionalShuffle.InstrumentChunks
namespace OAI
noncomputable section
open Filter Topology
namespace Revealed.Physical
open scoped Classical
open Thorp Thorp.Conditional Thorp.Specht Revealed.Split Revealed.Instrument
lemma eight_block_coset_ratio (d : ℕ) :
2 * ((Fintype.card (State d) : ℝ) / Fintype.card (State (d+1+2))) < 1 := by
have hm : 1 ≤ 2^d := Nat.one_le_pow d 2 (by omega)
have hf : 2 * (2^d).factorial < (2^(d+1+2)).factorial := by
have h : 2 * (2^d).factorial < (2^d+2).factorial := by
calc
_ < (2^d+2) * (2^d).factorial := Nat.mul_lt_mul_of_pos_right (by omega) (Nat.factorial_pos _)
_ ≤ (2^d+2) * (2^d+1).factorial := Nat.mul_le_mul_left _ (Nat.factorial_le (by omega))
_ = (2^d+2).factorial := (Nat.factorial_succ _).symm
apply h.trans_le (Nat.factorial_le _)
simp only [pow_add]
norm_num
omega
have hc : (0 : ℝ) < Fintype.card (State (d+1+2)) := by exact_mod_cast Fintype.card_pos
rw [← mul_div_assoc, div_lt_one hc]
exact_mod_cast (by simpa only [State, Fintype.card_perm, card_position] using hf)
def fullConditionalRate (d : ℕ) : ℝ := 512 * conditionalRate d + (1/2 : ℝ) *
Real.sqrt ((96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))
lemma fullConditionalRate_tendsto : Tendsto fullConditionalRate atTop (nhds 0) := by
have he := Block.exceptional_cube_tendsto.comp (tendsto_add_atTop_nat 3)
have hh := (conditionalRate_tendsto.const_mul 512).add
(((he.const_mul ((96 : ℝ)^8)).sqrt).const_mul (1/2 : ℝ))
change Tendsto (fun d : ℕ => 512 * conditionalRate d + (1/2 : ℝ) *
Real.sqrt ((96 : ℝ)^8 * exceptionalReciprocalSum (2^(d+1+2)))) atTop (nhds 0)
simpa only [Function.comp_def, Nat.add_assoc, mul_zero, Real.sqrt_zero, add_zero] using hh
lemma physical_distance_bound {ι : Type} [Fintype ι] (d : ℕ) (hd : 1 ≤ d)
(hR : (permutationFamily (Position d)).reciprocalSum ≤ 3)
(e : Outside ι (Position (d+1+2)) (Position (d+2+2))) :
Instrument.distance (instrument (d+2+2)) e (3200*(d+2+2)) ≤ fullConditionalRate d := by
have hn : 16 ≤ Fintype.card (Position (d+1+2)) := by
rw [card_position]
calc
16 = 2^4 := by norm_num
_ ≤ 2^(d+1+2) := Nat.pow_le_pow_right (by norm_num) (by omega)
let : Nontrivial (Position (d+1+2)) := Fintype.one_lt_card_iff_nontrivial.mp (by omega)
let : Fintype (Outside ι (Position (d+1+2)) (Position (d+2+2))) := inferInstance
let I := instrument (ι:=ι) (α:=Position (d+1+2)) (d+2+2)
have hh := distance_eight_le (block I (400*(d+2+2))) hn
(Block.embedding (Block.eightEquiv d)) (eight_block_coset_ratio d)
(fun e p => block_kernel_dichotomy I
(physical_kernel_dichotomy (d+2+1)) e (400*(d+2+2)) p)
(16*conditionalRate d) (block_discarded_le d) e
rw [distance_block] at hh
have ht : 400*(d+2+2)*8 = 3200*(d+2+2) := by omega
rw [ht] at hh
apply hh.trans
unfold fullConditionalRate
have hp : (32*(permutationFamily (Position d)).reciprocalSum)^8 ≤ (96 : ℝ)^8 :=
pow_le_pow_left₀ (mul_nonneg (by norm_num) (permutationFamily (Position d)).reciprocalSum_nonneg)
(by linarith only [hR]) 8
have he : 0 ≤ exceptionalReciprocalSum (2^(d+1+2)) := by
exact Finset.sum_nonneg (fun p _ => by
change 0 ≤ (if 0 < defect p then inverseDegree p else 0)
split_ifs <;> first | exact inverseDegree_nonneg p | exact le_rfl)
rw [card_position]
have hs := Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_right hp he)
rw [show (32 : ℝ) * (16 * conditionalRate d) = 512 * conditionalRate d by ring]
exact add_le_add le_rfl (mul_le_mul_of_nonneg_left hs (by norm_num : (0 : ℝ) ≤ 1/2))
lemma physical_distance_eventually {ι : ℕ → Type} [∀ d, Fintype (ι d)] :
∀ᶠ d in atTop, ∀ e : Outside (ι d) (Position (d+1+2)) (Position (d+2+2)),
Instrument.distance (instrument (d+2+2)) e (3200*(d+2+2)) ≤ fullConditionalRate d := by
filter_upwards [Block.reciprocal_atMost_three, eventually_ge_atTop 1] with d hR hd e
exact physical_distance_bound d hd hR e
end Revealed.Physical
end
end OAI
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