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9dbc64e | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Computability.DepthThree.FiniteProbabilityBounds
import OAI.Computability.DepthThree.RestrictionTails
import OAI.Computability.DepthThree.RestrictionCorrelation
import Mathlib.Tactic.Linarith
namespace OAI
universe uDepth1 uDepth2
noncomputable section
open scoped BigOperators Classical
namespace DepthThreeLowerBound
variable {V : Type uDepth1} {Ω : Type uDepth2} [Fintype V] [instDecidableEqV : DecidableEq V]
[Fintype Ω] [Nonempty Ω]
theorem exists_small_restriction_failure
{V : Type uDepth1}
{Ω : Type uDepth2}
[Fintype V]
[DecidableEq V]
[Fintype Ω]
[Nonempty Ω]
{p δ η : ℝ}
(hp : 0 ≤ p) (hp1 : p ≤ 1) (hδ : 0 ≤ δ)
(window : Restriction V → Prop) (good : Ω → Restriction V → Prop)
(hfixed : ∀ σ, window σ → finiteProb (fun ω => ¬ good ω σ) ≤ δ)
(hwindow : restrictionAvg p (fun σ => if ¬ window σ then 1 else 0) ≤ η) :
∃ ω : Ω,
restrictionAvg p (fun σ => if ¬ (window σ ∧ good ω σ) then 1 else 0) ≤ η + δ := by
have hpoint : ∀ σ : Restriction V,
finiteAvg (fun ω : Ω => if ¬ (window σ ∧ good ω σ) then (1 : ℝ) else 0) ≤
(if ¬ window σ then 1 else 0) + δ := by
intro σ
by_cases hσ : window σ
· have hfixed' : finiteAvg (fun ω : Ω => if ¬ good ω σ then (1 : ℝ) else 0) ≤ δ := by
calc
_ = finiteProb (fun ω => ¬ good ω σ) := by
apply finiteAvg_congr
intro ω
exact (ite_eq_ite _ _ _).mpr trivial
_ ≤ δ := hfixed σ hσ
simpa only [hσ, true_and, not_true_eq_false, ite_false, zero_add] using hfixed'
· simpa only [hσ, false_and, not_false_eq_true, ite_true, finiteAvg_const]
using (le_add_of_nonneg_right hδ : (1 : ℝ) ≤ 1 + δ)
apply finiteAvg_exists_le_of_le
calc
_ = restrictionAvg p (fun σ =>
finiteAvg (fun ω : Ω => if ¬ (window σ ∧ good ω σ) then 1 else 0)) :=
finiteAvg_restrictionAvg p _
_ ≤ restrictionAvg p (fun σ => (if ¬ window σ then 1 else 0) + δ) :=
restrictionAvg_mono hp hp1 hpoint
_ = restrictionAvg p (fun σ => if ¬ window σ then 1 else 0) + δ := by
rw [restrictionAvg_add, restrictionAvg_const]
_ ≤ η + δ := add_le_add_left hwindow δ
def restrictionWindow (p : ℝ) (σ : Restriction V) : Prop :=
(p * Fintype.card V) / 2 ≤ (Fintype.card (Live σ) : ℝ) ∧
(Fintype.card (Live σ) : ℝ) ≤ 2 * (p * Fintype.card V)
def goodRestriction (p ε : ℝ) (b : ℕ) (f : Cube V → Bool)
(σ : Restriction V) : Prop :=
restrictionWindow p σ ∧ corr b (fun z => sign (f (fill σ z))) ≤ ε
theorem goodRestriction_corr
{V : Type uDepth1}
[Fintype V]
[DecidableEq V]
{p ε : ℝ} {b : ℕ} {f : Cube V → Bool}
{σ : Restriction V} (h : goodRestriction p ε b f σ) :
corr b (fun z => sign (f (fill σ z))) ≤ ε := h.2
theorem exists_good_parameter {p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1)
(b : ℕ) (f : Ω → Cube V → Bool)
(hfixed : ∀ σ : Restriction V, restrictionWindow p σ →
finiteProb (fun ω =>
¬ corr b (fun z => sign (f ω (fill σ z))) ≤
(2 : ℝ) ^ (-(p * Fintype.card V) / 16)) ≤
2 * (2 : ℝ) ^ (-(p * Fintype.card V) / 16)) :
∃ ω : Ω,
restrictionAvg p (fun σ => if ¬ goodRestriction p
((2 : ℝ) ^ (-(p * Fintype.card V) / 16)) b (f ω) σ then 1 else 0) ≤
4 * (2 : ℝ) ^ (-(p * Fintype.card V) / 16) := by
have hε : 0 ≤ (2 : ℝ) ^ (-(p * Fintype.card V) / 16) :=
Real.rpow_nonneg (by norm_num) _
have hwindow : restrictionAvg p (fun σ : Restriction V =>
if ¬ restrictionWindow p σ then 1 else 0) ≤
2 * (2 : ℝ) ^ (-(p * Fintype.card V) / 16) := by
calc
_ = restrictionAvg p (fun σ : Restriction V =>
if ¬ ((p * Fintype.card V) / 2 ≤ (Fintype.card (Live σ) : ℝ) ∧
(Fintype.card (Live σ) : ℝ) ≤ 2 * (p * Fintype.card V)) then 1 else 0) := by
apply congrArg (restrictionAvg p)
funext σ
exact (ite_eq_ite _ _ _).mpr trivial
_ ≤ _ := restrictionAvg_outside_window (V := V) p hp hp1
obtain ⟨ω, hω⟩ := exists_small_restriction_failure hp hp1
(mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hε)
(restrictionWindow p)
(fun ω σ => corr b (fun z => sign (f ω (fill σ z))) ≤
(2 : ℝ) ^ (-(p * Fintype.card V) / 16))
hfixed hwindow
refine ⟨ω, ?_⟩
have heq : restrictionAvg p (fun σ => if ¬ goodRestriction p
((2 : ℝ) ^ (-(p * Fintype.card V) / 16)) b (f ω) σ then 1 else 0) =
restrictionAvg p (fun σ => if ¬ (restrictionWindow p σ ∧
corr b (fun z => sign (f ω (fill σ z))) ≤
(2 : ℝ) ^ (-(p * Fintype.card V) / 16)) then 1 else 0) := by
apply congrArg (restrictionAvg p)
funext σ
exact (ite_eq_ite _ _ _).mpr trivial
rw [heq]
linarith
end DepthThreeLowerBound
end
end OAI
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