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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 | import OAI.Computability.DepthThree.RestrictionProbability
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} [Fintype V] [instDecidableEqV : DecidableEq V]
theorem restrictionAvg_mask_le
{V : Type uDepth1}
[Fintype V]
[DecidableEq V]
{p : ℝ} (hp : 0 ≤ p) (hp1 : p ≤ 1)
(W : Restriction V → ℝ) (hW : ∀ σ, 0 ≤ W σ)
(good : Restriction V → Prop) :
restrictionAvg p (fun σ => if good σ then W σ else 0) ≤ restrictionAvg p W := by
apply restrictionAvg_mono hp hp1
intro σ
split_ifs
· exact le_rfl
· exact hW σ
theorem restrictionAvg_abs_le_split {p ε δ C : ℝ}
(hp : 0 ≤ p) (hp1 : p ≤ 1) (hε : 0 ≤ ε)
(q W : Restriction V → ℝ) (good : Restriction V → Prop)
(hW : ∀ σ, 0 ≤ W σ) (hmean : restrictionAvg p W ≤ C)
(hgood : ∀ σ, good σ → |q σ| ≤ ε * W σ)
(hunit : ∀ σ, |q σ| ≤ 1)
(hbad : restrictionAvg p (fun σ => if ¬ good σ then 1 else 0) ≤ δ) :
|restrictionAvg p q| ≤ ε * C + δ := by
have hmask : restrictionAvg p (fun σ => if good σ then W σ else 0) ≤ C :=
(restrictionAvg_mask_le hp hp1 W hW good).trans hmean
calc
|restrictionAvg p q| ≤ restrictionAvg p (fun σ => |q σ|) :=
restrictionAvg_abs_le hp hp1 q
_ ≤ restrictionAvg p (fun σ =>
ε * (if good σ then W σ else 0) + (if ¬ good σ then 1 else 0)) := by
apply restrictionAvg_mono hp hp1
intro σ
by_cases h : good σ
· simpa only [h, ite_true, not_true_eq_false, ite_false, add_zero]
using hgood σ h
· simpa only [h, ite_false, not_false_eq_true, ite_true, mul_zero, zero_add]
using hunit σ
_ = ε * restrictionAvg p (fun σ => if good σ then W σ else 0) +
restrictionAvg p (fun σ => if ¬ good σ then 1 else 0) := by
rw [restrictionAvg_add, restrictionAvg_const_mul]
_ ≤ ε * C + δ := add_le_add (mul_le_mul_of_nonneg_left hmask hε) hbad
theorem abs_finiteAvg_sign_indicator_le_one {Z : Type uDepth2} [Fintype Z]
(f J : Cube Z → Bool) :
|finiteAvg (fun x => sign (f x) * indicator (J x))| ≤ 1 := by
calc
_ ≤ finiteAvg (fun x => |sign (f x) * indicator (J x)|) := finiteAvg_abs_le _
_ ≤ finiteAvg (fun _ : Cube Z => (1 : ℝ)) := by
apply finiteAvg_mono
intro x
cases f x <;> cases J x <;> norm_num [sign, indicator]
_ = 1 := finiteAvg_const _
theorem restriction_correlation_transfer {p ε δ C : ℝ}
(hp : 0 ≤ p) (hp1 : p ≤ 1) (hε : 0 ≤ ε)
(f : Cube V → Bool) (H : CNF V) (b : ℕ)
(good : Restriction V → Prop) (W : Restriction V → ℝ)
(hW : ∀ σ, 0 ≤ W σ) (hmean : restrictionAvg p W ≤ C)
(hdom : ∀ σ,
|finiteAvg (fun z : Cube (Live σ) =>
sign (f (fill σ z)) * indicator (H.eval (fill σ z)))| ≤
W σ * corr b (fun z => sign (f (fill σ z))))
(hgood : ∀ σ, good σ → corr b (fun z => sign (f (fill σ z))) ≤ ε)
(hbad : restrictionAvg p (fun σ => if ¬ good σ then 1 else 0) ≤ δ) :
|finiteAvg (fun x => sign (f x) * indicator (H.eval x))| ≤ ε * C + δ := by
rw [← restrictionAvg_fill p (fun x => sign (f x) * indicator (H.eval x))]
apply restrictionAvg_abs_le_split hp hp1 hε _ W good hW hmean
· intro σ hσ
calc
_ ≤ W σ * corr b (fun z => sign (f (fill σ z))) := hdom σ
_ ≤ W σ * ε := mul_le_mul_of_nonneg_left (hgood σ hσ) (hW σ)
_ = ε * W σ := mul_comm _ _
· intro σ
calc
_ ≤ finiteAvg (fun z : Cube (Live σ) =>
|sign (f (fill σ z)) * indicator (H.eval (fill σ z))|) := finiteAvg_abs_le _
_ ≤ finiteAvg (fun _ : Cube (Live σ) => (1 : ℝ)) := by
apply finiteAvg_mono
intro z
cases f (fill σ z) <;> cases H.eval (fill σ z) <;> norm_num [sign, indicator]
_ = 1 := finiteAvg_const _
· exact hbad
theorem restriction_correlation_transfer_six {p ε : ℝ}
(hp : 0 ≤ p) (hp1 : p ≤ 1) (hε : 0 ≤ ε)
(f : Cube V → Bool) (H : CNF V) (b : ℕ)
(good : Restriction V → Prop) (W : Restriction V → ℝ)
(hW : ∀ σ, 0 ≤ W σ) (hmean : restrictionAvg p W ≤ 2)
(hdom : ∀ σ,
|finiteAvg (fun z : Cube (Live σ) =>
sign (f (fill σ z)) * indicator (H.eval (fill σ z)))| ≤
W σ * corr b (fun z => sign (f (fill σ z))))
(hgood : ∀ σ, good σ → corr b (fun z => sign (f (fill σ z))) ≤ ε)
(hbad : restrictionAvg p (fun σ => if ¬ good σ then 1 else 0) ≤ 4 * ε) :
|finiteAvg (fun x => sign (f x) * indicator (H.eval x))| ≤ 6 * ε := by
have h := restriction_correlation_transfer hp hp1 hε f H b good W hW hmean
hdom hgood hbad
linarith
end DepthThreeLowerBound
end
end OAI
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