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2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.Computability.PerfectCompleteness.Foundations.PartialTableInverse
namespace OAI
section
namespace PerfectCompleteness.MatrixErasure
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Fourier.MatrixCharacters
open PerfectCompleteness.PartialTableInverse
variable {Ω Y : Type*}
def erase (f : Ω → Option Y) (T : Ω → Prop) (X : Ω) : Option Y :=
if T X then none else f X
def uniformMass [Fintype Ω] (T : Ω → Prop) : ℝ :=
𝔼 X, if T X then (1 : ℝ) else 0
@[simp] theorem erase_eq_none_of (f : Ω → Option Y) (T : Ω → Prop)
{X : Ω} (hX : T X) : erase f T X = none := by
simp [erase, hX]
theorem erase_eq_of_not (f : Ω → Option Y) (T : Ω → Prop)
{X : Ω} (hX : ¬T X) : erase f T X = f X := by
simp [erase, hX]
theorem erase_eq_some_iff (f : Ω → Option Y) (T : Ω → Prop) (X : Ω) (y : Y) :
erase f T X = some y ↔ ¬T X ∧ f X = some y := by
by_cases hX : T X <;> simp [erase, hX]
theorem indicator_erase_le (f : Ω → Option Y) (T : Ω → Prop) (y : Y) (X : Ω) :
indicator (erase f T) y X ≤ indicator f y X := by
by_cases hX : T X
· simp only [indicator, erase, ite_eq_left hX, reduceCtorEq, ite_false]
split_ifs <;> norm_num
· simp [indicator, erase, hX]
theorem definedEquality_erase (f : Ω → Option Y) (T : Ω → Prop) (X X' : Ω) :
definedEquality (erase f T) X X' =
if T X ∨ T X' then 0 else definedEquality f X X' := by
by_cases hX : T X <;> by_cases hX' : T X' <;>
simp [definedEquality, erase, hX, hX']
theorem definedEquality_le_one (f : Ω → Option Y) (X X' : Ω) :
definedEquality f X X' ≤ 1 := by
unfold definedEquality
split_ifs <;> norm_num
theorem definedEquality_nonneg (f : Ω → Option Y) (X X' : Ω) :
0 ≤ definedEquality f X X' := by
unfold definedEquality
split_ifs <;> norm_num
theorem definedEquality_erase_le (f : Ω → Option Y) (T : Ω → Prop) (X X' : Ω) :
definedEquality (erase f T) X X' ≤ definedEquality f X X' := by
rw [definedEquality_erase]
split_ifs
· exact definedEquality_nonneg f X X'
· exact le_rfl
theorem definedEquality_le_erase_add (f : Ω → Option Y) (T : Ω → Prop) (X X' : Ω) :
definedEquality f X X' ≤ definedEquality (erase f T) X X' +
(if T X then (1 : ℝ) else 0) + (if T X' then (1 : ℝ) else 0) := by
have hbound := definedEquality_le_one f X X'
by_cases hX : T X <;> by_cases hX' : T X' <;>
simp [definedEquality_erase, hX, hX'] <;> linarith
theorem uniformMass_nonneg [Fintype Ω] (T : Ω → Prop) : 0 ≤ uniformMass T := by
apply Finset.expect_nonneg
intro X _
split_ifs <;> norm_num
theorem expect_add_right [AddGroup Ω] [Fintype Ω] (g : Ω → ℝ) (v : Ω) :
(𝔼 X, g (X + v)) = (𝔼 X, g X) :=
Fintype.expect_equiv (Equiv.addRight v) _ _ (fun _ => rfl)
variable {E F : Type*}
[AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F]
[Fintype F] [Fintype (E →ₗ[F2] F2)] [Fintype (E →ₗ[F2] F)]
omit [Fintype F] in
theorem step_second_marginal (g : (E →ₗ[F2] F) → ℝ) (a : F) :
(𝔼 X, 𝔼 h : E →ₗ[F2] F2, g (X + h.smulRight a)) = (𝔼 X, g X) := by
rw [Finset.expect_comm]
simp_rw [expect_add_right]
exact Fintype.expect_const _
theorem nonzero_second_marginal [Nontrivial F] (g : (E →ₗ[F2] F) → ℝ) :
(𝔼 a : {a : F // a ≠ 0},
𝔼 X, 𝔼 h : E →ₗ[F2] F2, g (X + h.smulRight a.val)) = (𝔼 X, g X) := by
obtain ⟨a, ha⟩ := exists_ne (0 : F)
let : Nonempty {a : F // a ≠ 0} := ⟨⟨a, ha⟩⟩
simp_rw [step_second_marginal]
exact Fintype.expect_const _
omit [Fintype F] in
theorem stepAgreement_erase_bound (f : (E →ₗ[F2] F) → Option Y)
(T : (E →ₗ[F2] F) → Prop) (a : F) :
stepAgreement f a ≤ stepAgreement (erase f T) a + 2 * uniformMass T := by
have havg : stepAgreement f a ≤
𝔼 X, 𝔼 h : E →ₗ[F2] F2,
(definedEquality (erase f T) X (X + h.smulRight a) +
(if T X then (1 : ℝ) else 0) +
(if T (X + h.smulRight a) then (1 : ℝ) else 0)) := by
apply Finset.expect_le_expect
intro X _
apply Finset.expect_le_expect
intro h _
exact definedEquality_le_erase_add f T X (X + h.smulRight a)
simp_rw [Finset.expect_add_distrib, Fintype.expect_const] at havg
rw [step_second_marginal (fun X => if T X then (1 : ℝ) else 0) a] at havg
change stepAgreement f a ≤
stepAgreement (erase f T) a + uniformMass T + uniformMass T at havg
linarith
theorem nonzeroAgreement_erase [Nontrivial F] (f : (E →ₗ[F2] F) → Option Y)
(T : (E →ₗ[F2] F) → Prop) :
nonzeroAgreement f - 2 * uniformMass T ≤ nonzeroAgreement (erase f T) := by
obtain ⟨a, ha⟩ := exists_ne (0 : F)
let : Nonempty {a : F // a ≠ 0} := ⟨⟨a, ha⟩⟩
have havg : nonzeroAgreement f ≤
𝔼 a : {a : F // a ≠ 0}, (stepAgreement (erase f T) a.val + 2 * uniformMass T) := by
unfold nonzeroAgreement
apply Finset.expect_le_expect
intro a _
exact stepAgreement_erase_bound f T a.val
simp only [Finset.expect_add_distrib, Fintype.expect_const] at havg
change nonzeroAgreement f ≤ nonzeroAgreement (erase f T) + 2 * uniformMass T at havg
linarith
end
end PerfectCompleteness.MatrixErasure
end
end OAI
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