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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 | import Mathlib.LinearAlgebra.Dual.Lemmas
import OAI.Computability.PerfectCompleteness.Algebra.BilinearGramCompressionLemmas
import OAI.Computability.PerfectCompleteness.Algebra.MatrixInverseConstantsLemmas
import OAI.Computability.PerfectCompleteness.Foundations.PartialTableInverse
namespace OAI
section
namespace PerfectCompleteness.FiniteRangeInverse
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Fourier.MatrixCharacters (F2)
open UniqueGamesTheorem.Fourier.MatrixRestrictions
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
open PerfectCompleteness.PartialTableInverse
attribute [local instance] linearMapFintype
universe u v w
variable {H : Type u} {K : Type v} {Y : Type w}
[AddCommGroup H] [Module F2 H] [AddCommGroup K] [Module F2 K]
[Fintype H] [Fintype K] [Fintype Y]
[FiniteDimensional F2 H] [FiniteDimensional F2 K]
def matrixStep (X : Module.Dual F2 H →ₗ[F2] K) (a : K) (h : H) :
Module.Dual F2 H →ₗ[F2] K :=
X + (Module.Dual.eval F2 H h).smulRight a
omit [Fintype H] [Fintype K] [FiniteDimensional F2 H] [FiniteDimensional F2 K] in
@[simp] theorem matrixStep_apply (X : Module.Dual F2 H →ₗ[F2] K) (a : K)
(h : H) (q : Module.Dual F2 H) : matrixStep X a h q = X q + q h • a := rfl
def agreement (f : (Module.Dual F2 H →ₗ[F2] K) → Option Y) : ℝ :=
𝔼 a : {a : K // a ≠ 0}, 𝔼 X : Module.Dual F2 H →ₗ[F2] K,
𝔼 h : H, definedEquality f X (matrixStep X a.val h)
omit [Fintype Y] [FiniteDimensional F2 K] in
theorem agreement_eq_nonzeroAgreement
(f : (Module.Dual F2 H →ₗ[F2] K) → Option Y) :
agreement f = nonzeroAgreement f := by
apply Finset.expect_congr rfl
intro a _
apply Finset.expect_congr rfl
intro X _
exact Fintype.expect_equiv (Module.evalEquiv F2 H).toEquiv
(fun h : H => definedEquality f X (matrixStep X a.val h))
(fun φ : Module.Dual F2 (Module.Dual F2 H) =>
definedEquality f X (X + φ.smulRight a.val)) (fun _ => rfl)
theorem dense_slice_of_parameters [Nontrivial K]
(r : Nat) (ρ η : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : UniqueGamesTheorem.Fourier.MatrixLevelBridge.levelCutoffConstant r ρ +
UniqueGamesTheorem.Appendix.RankLevelFilter.node (r + 1) < η)
(f : (Module.Dual F2 H →ₗ[F2] K) → Option Y) (hagreement : η ≤ agreement f) :
∃ (y : Y) (W : Submodule F2 (Module.Dual F2 H)) (C' : Submodule F2 K)
(T : Module.Dual F2 H →ₗ[F2] K),
order W C' ≤ r ∧
ρ < (𝔼 X : MatrixSlice.Slice W C' T, indicator f y X.val) := by
rw [agreement_eq_nonzeroAgreement] at hagreement
obtain ⟨y, W, C', T, horder, hdensity⟩ :=
exists_dense_slice_nonzero r ρ η hρ hρ1 hconstants f hagreement
refine ⟨y, W, C', T, horder, ?_⟩
rwa [MatrixSlice.expect_restrict_eq_slice] at hdensity
theorem inverse {η : ℝ} (hη : 0 < η) :
∃ r : Nat, 1 ≤ r ∧ ∃ ρ : ℝ, 0 < ρ ∧ ρ < 1 ∧
∀ (H : Type u) (K : Type v) (Y : Type w)
[AddCommGroup H] [Module F2 H] [AddCommGroup K] [Module F2 K]
[Fintype H] [Fintype K] [Fintype Y]
[FiniteDimensional F2 H] [FiniteDimensional F2 K] [Nontrivial K]
(f : (Module.Dual F2 H →ₗ[F2] K) → Option Y),
η ≤ agreement f →
∃ (y : Y) (W : Submodule F2 (Module.Dual F2 H)) (C' : Submodule F2 K)
(T : Module.Dual F2 H →ₗ[F2] K),
order W C' ≤ r ∧
ρ < (𝔼 X : MatrixSlice.Slice W C' T, indicator f y X.val) := by
obtain ⟨r, hr, ρ, hρ, hρ1, hconstants⟩ := MatrixInverseConstants.exists_cutoff_density hη
refine ⟨r, hr, ρ, hρ, hρ1, ?_⟩
intro H K Y _ _ _ _ _ _ _ _ _ _ f hagreement
exact dense_slice_of_parameters r ρ η hρ hρ1 hconstants f hagreement
end
end PerfectCompleteness.FiniteRangeInverse
end
end OAI
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