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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.Computability.PerfectCompleteness.Algebra.EvaluationMatrix
import OAI.Computability.PerfectCompleteness.Algebra.LowerAffineSliceRank
import OAI.Computability.PerfectCompleteness.Algebra.ProductEvaluationRank
import OAI.Computability.PerfectCompleteness.Foundations.OddListExtraction
namespace OAI
namespace PerfectCompleteness.FunctionSpaceLowerRank
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
open UniqueGamesTheorem.Fourier.MatrixLevelBridge
open UniqueGamesTheorem.Appendix.RankLevelFilter
open PointwiseSpaces
section Transport
variable {K H V : Type*} [Field K]
[AddCommGroup H] [Module K H] [AddCommGroup V] [Module K V]
def transportForm (e : H ≃ₗ[K] V) (F : H →ₗ[K] Module.Dual K H) :
V →ₗ[K] Module.Dual K V :=
e.symm.toLinearMap.dualMap.comp (F.comp e.symm.toLinearMap)
@[simp] theorem transportForm_apply (e : H ≃ₗ[K] V)
(F : H →ₗ[K] Module.Dual K H) (u v : V) :
transportForm e F u v = F (e.symm u) (e.symm v) := rfl
theorem transportForm_rank (e : H ≃ₗ[K] V) (F : H →ₗ[K] Module.Dual K H) :
Module.finrank K (LinearMap.range (transportForm e F)) =
Module.finrank K (LinearMap.range F) := by
unfold transportForm
rw [LinearMap.range_comp,
LinearMap.range_comp_of_range_eq_top F e.symm.range]
exact e.symm.dualMap.finrank_map_eq (LinearMap.range F)
end Transport
section FunctionSpace
variable {Ω : Type*} (H : Submodule F2 (Ω → F2)) [FiniteDimensional F2 H]
{ℓ m : Nat}
def matrixForm (F : H →ₗ[F2] Module.Dual F2 H) :
Module.Dual F2 (Module.Dual F2 H) →ₗ[F2]
Module.Dual F2 (Module.Dual F2 (Module.Dual F2 H)) :=
transportForm (Module.evalEquiv F2 H) F
@[simp] theorem matrixForm_rank (F : H →ₗ[F2] Module.Dual F2 H) :
Module.finrank F2 (LinearMap.range (matrixForm H F)) =
Module.finrank F2 (LinearMap.range F) :=
transportForm_rank (Module.evalEquiv F2 H) F
theorem testedGram_matrixForm (F : H →ₗ[F2] Module.Dual F2 H)
(X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2)) :
LowerAffineSliceRank.testedGram (matrixForm H F) X =
(fun i j => F (EvaluationMatrix.rows H X i) (EvaluationMatrix.rows H X j)) := rfl
theorem testedGram_multiplicationForm (z : Module.Dual F2 (squareSpace H))
(X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2)) :
LowerAffineSliceRank.testedGram
(matrixForm H (OddListExtraction.multiplicationForm H z)) X =
(fun i j => z (OddListExtraction.productElement H
(EvaluationMatrix.rows H X i) (EvaluationMatrix.rows H X j))) := rfl
end FunctionSpace
open MixedSupport CanonicalKeys
open ProductEvaluation (productSpan displayedJoint labelFunctional)
variable {n ℓ m : Nat} {Z Y : Type*} [Fintype Y]
(slots : Fin n → Slot) (H : Submodule F2 (Assignment slots → F2))
[FiniteDimensional F2 H]
(other : Assignment slots → Z)
(hone : (1 : Assignment slots → F2) ∈ squareSpace H)
def displayedRows (X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2))
(_ : Unit) (i : Fin ℓ) : Assignment slots → F2 :=
(EvaluationMatrix.rows H X i).val
theorem displayedRows_products
(X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2))
(c : Unit) (i j : Fin ℓ) :
displayedRows slots H X c i * displayedRows slots H X c j ∈ squareSpace H :=
mul_mem_squareSpace H (EvaluationMatrix.rows H X i).property
(EvaluationMatrix.rows H X j).property
def testedProductSpan (X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2)) :
Submodule F2 (squareSpace H) :=
productSpan (squareSpace H) (displayedRows slots H X) hone
(displayedRows_products slots H X)
def IsLegalExtension (X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2))
(z : Module.Dual F2 (squareSpace H)) : Prop :=
∃ P : Label slots (displayedJoint (displayedRows slots H X) other),
z.comp (testedProductSpan slots H hone X).subtype =
labelFunctional slots (squareSpace H) (displayedRows slots H X) other hone
(displayedRows_products slots H X) P
theorem legal_extension_tested_rank_le_one
(X : Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2))
(z : Module.Dual F2 (squareSpace H))
(hz : IsLegalExtension slots H other hone X z) :
(LowerAffineSliceRank.testedGram
(matrixForm H (OddListExtraction.multiplicationForm H z)) X).rank ≤ 1 := by
obtain ⟨P, hP⟩ := hz
rw [testedGram_multiplicationForm]
exact ProductEvaluationRank.extension_productMatrix_rank_le_one slots
(squareSpace H) (displayedRows slots H X) other hone
(displayedRows_products slots H X) P z hP ()
variable [Fintype (Module.Dual F2 H)]
[Fintype (Module.Dual F2 (Module.Dual F2 H))]
[Fintype (Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2))]
[Fintype ((Fin ℓ → F2) →ₗ[F2] Module.Dual F2 H)]
theorem high_rank_agreement_lt_of_legal_extensions
(f : (Module.Dual F2 H →ₗ[F2] (Fin ℓ → F2)) → Option Y)
(output : Y → Module.Dual F2 (squareSpace H))
(legal : ∀ X y, f X = some y → IsLegalExtension slots H other hone X (output y))
(r s : Nat) (ρ η : ℝ) (hm : 0 < m)
(hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(hrows : 2 * m + r ≤ ℓ)
(herror : (2 : ℝ) ^ m / (2 : ℝ) ^ (s - 2 * r) +
(2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) ≤ ρ) :
PartialTableInverse.nonzeroAgreement
(MatrixErasure.erase f (fun X => ∃ y, f X = some y ∧
OddListExtraction.formRank H (OddListExtraction.multiplicationForm H (output y)) ≤ s)) < η := by
have tested : ∀ X y, f X = some y →
(LowerAffineSliceRank.testedGram
(matrixForm H (OddListExtraction.multiplicationForm H (output y))) X).rank ≤ 1 := by
intro X y h
exact legal_extension_tested_rank_le_one slots H other hone X (output y) (legal X y h)
have h := LowerAffineSliceRank.high_rank_agreement_lt f
(fun y => matrixForm H (OddListExtraction.multiplicationForm H (output y)))
tested r s ρ η hm hρ hρ1 hconstants hrows herror
simpa only [matrixForm_rank, OddListExtraction.formRank] using h
end
end PerfectCompleteness.FunctionSpaceLowerRank
end OAI
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