| import OAI.Computability.UniqueGames.Analysis.CompressionCountLemmas |
| import OAI.Computability.UniqueGames.Analysis.Identities |
| import OAI.Computability.UniqueGames.Analysis.RestrictionLemmas |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! |
| # Actual Fourier-projector partitions for Appendix A.4 and A.5 |
|
|
| The partitions use the proved unique geometric selectors on each actual |
| Fourier index. Their finite index types are the canonical counting types. |
| -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
| open UniqueGamesTheorem.Integration.BinaryLinear (F2) |
| open UniqueGamesTheorem.Fourier.MatrixFourier |
| open UniqueGamesTheorem.Appendix.Derivatives |
|
|
| namespace UniqueGamesTheorem.Appendix.OperatorPartitions |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| private theorem indicator_eq_sum_of_unique {I : Type*} [Fintype I] |
| (P : Prop) (R : I → Prop) (a : ℝ) |
| (hpart : P → ∃! i, R i) (hback : ∀ i, R i → P) : |
| (if P then a else 0) = ∑ i, if R i then a else 0 := by |
| classical |
| by_cases hp : P |
| · obtain ⟨i, hi, hu⟩ := hpart hp |
| rw [ite_eq_left hp] |
| symm |
| calc |
| (∑ j, if R j then a else 0) = if R i then a else 0 := by |
| apply Finset.sum_eq_single i |
| · intro j _ hji |
| exact ite_eq_right (fun hj => hji (hu j hj)) |
| · simp |
| _ = a := ite_eq_left hi |
| · rw [ite_eq_right hp] |
| symm |
| apply Finset.sum_eq_zero |
| intro i _ |
| exact ite_eq_right (fun hi => hp (hback i hi)) |
|
|
| variable {E F : Type*} |
| [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] |
| [FiniteDimensional F2 E] [FiniteDimensional F2 F] |
| [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] |
|
|
| theorem spectralProjector_partition {I : Type*} [Fintype I] |
| (P : (F →ₗ[F2] E) → Prop) (R : I → (F →ₗ[F2] E) → Prop) |
| (hpart : ∀ Y, P Y → ∃! i, R i Y) |
| (hback : ∀ i Y, R i Y → P Y) (f : (E →ₗ[F2] F) → ℝ) : |
| spectralProjector P f = ∑ i, spectralProjector (R i) f := by |
| apply function_eq_of_linearCoeff_eq |
| intro Y |
| rw [linearCoeff_spectralProjector, linearCoeff_sum] |
| simp only [linearCoeff_spectralProjector] |
| exact indicator_eq_sum_of_unique (P Y) (fun i => R i Y) |
| (linearCoeff f Y) (hpart Y) (fun i => hback i Y) |
|
|
| variable [Finite E] [Finite F] |
|
|
| theorem spectralProjector_restriction_projector |
| (A : Submodule F2 E) (B : Submodule F2 F) |
| [Fintype (B →ₗ[F2] (E ⧸ A))] |
| (P : (F →ₗ[F2] E) → Prop) |
| (Q : (B →ₗ[F2] (E ⧸ A)) → Prop) |
| (f : (E →ₗ[F2] F) → ℝ) (T : E →ₗ[F2] F) : |
| spectralProjector Q |
| (UniqueGamesTheorem.Fourier.MatrixRestrictions.restrict (spectralProjector P f) A B T) = |
| UniqueGamesTheorem.Fourier.MatrixRestrictions.restrict |
| (spectralProjector (fun Y => P Y ∧ Q (Restriction.compressFrequency A B Y)) f) |
| A B T := by |
| have hp : (fun Y => Q (Restriction.compressFrequency A B Y) ∧ P Y) = |
| (fun Y => P Y ∧ Q (Restriction.compressFrequency A B Y)) := by |
| funext Y |
| exact propext and_comm |
| rw [spectralProjector_restriction, spectralProjector_comp, hp] |
|
|
| open UniqueGamesTheorem.Appendix.LinearIdentities |
| open UniqueGamesTheorem.Appendix.RankAdditivity |
|
|
| abbrev A4Geometry (A : Submodule F2 E) (B : Submodule F2 F) := |
| A4IndexCount.Valid A B |
|
|
| abbrev A4GeometricIndex (A : Submodule F2 E) (B : Submodule F2 F) := |
| A4IndexCount.Index A B |
|
|
| def a4Mask (A : Submodule F2 E) (B : Submodule F2 F) |
| (i : A4GeometricIndex A B) (Y : F →ₗ[F2] E) : Prop := |
| Hybrid Y i.val.1 i.val.2.1 ∧ |
| RankBelow i.val.2.2 (compress Y i.val.1 i.val.2.1) |
|
|
| omit [FiniteDimensional F2 E] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] |
| [Finite E] [Finite F] in |
| theorem a4Mask_unique (A : Submodule F2 E) (B : Submodule F2 F) |
| (Y : F →ₗ[F2] E) (hY : A ≤ Y.range ∧ Y.ker ≤ B) : |
| ∃! i : A4GeometricIndex A B, a4Mask A B i Y := by |
| obtain ⟨p, hp, hu⟩ := (BinaryA4.a4_unique_selector Y A B).mp hY |
| rcases hp with ⟨hA, hB, hinner, hker, hrange, hrank⟩ |
| let i : A4GeometricIndex A B := ⟨p, hA, hB, hker, hrange⟩ |
| refine ⟨i, ⟨hinner, hrank⟩, ?_⟩ |
| intro j hj |
| apply Subtype.ext |
| apply hu |
| rcases j.property with ⟨hjA, hjB, hjker, hjrange⟩ |
| exact ⟨hjA, hjB, hj.1, hjker, hjrange, hj.2⟩ |
|
|
| omit [FiniteDimensional F2 E] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] |
| [Finite E] [Finite F] in |
| theorem a4Mask_implies_weak (A : Submodule F2 E) (B : Submodule F2 F) |
| (i : A4GeometricIndex A B) (Y : F →ₗ[F2] E) |
| (hi : a4Mask A B i Y) : A ≤ Y.range ∧ Y.ker ≤ B := by |
| rcases i.property with ⟨_, _, hker, hrange⟩ |
| exact a4_weak_selector_of_summand Y i.val.1 A B i.val.2.1 i.val.2.2 |
| hi.1 hker hrange hi.2 |
|
|
| omit [Finite E] [Finite F] in |
| theorem a4_projector_partition (A : Submodule F2 E) (B : Submodule F2 F) |
| [Fintype (A4GeometricIndex A B)] (f : (E →ₗ[F2] F) → ℝ) : |
| spectralProjector (fun Y => A ≤ Y.range ∧ Y.ker ≤ B) f = |
| ∑ i : A4GeometricIndex A B, spectralProjector (a4Mask A B i) f := by |
| exact spectralProjector_partition _ _ |
| (a4Mask_unique A B) (a4Mask_implies_weak A B) f |
|
|
| def A5Geometry (X : F →ₗ[F2] E) (A : Submodule F2 E) (B : Submodule F2 F) |
| (p : Submodule F2 E × Submodule F2 F) : Prop := |
| Disjoint p.1 X.range ∧ p.1 ⊔ X.range = A ∧ |
| p.2 ⊔ X.ker = ⊤ ∧ p.2 ⊓ X.ker = B |
|
|
| abbrev A5GeometricIndex (X : F →ₗ[F2] E) |
| (A : Submodule F2 E) (B : Submodule F2 F) := |
| {p : Submodule F2 E × Submodule F2 F // A5Geometry X A B p} |
|
|
| def a5Mask (X : F →ₗ[F2] E) (A : Submodule F2 E) (B : Submodule F2 F) |
| (i : A5GeometricIndex X A B) (Y : F →ₗ[F2] E) : Prop := |
| Hybrid Y i.val.1 i.val.2 ∧ |
| RankBelow (compress X i.val.1 i.val.2) (compress Y i.val.1 i.val.2) |
|
|
| def a5OuterMask (X : F →ₗ[F2] E) (A : Submodule F2 E) |
| (B : Submodule F2 F) (Y : F →ₗ[F2] E) : Prop := |
| RankBelow X Y ∧ Hybrid (compress Y X.range X.ker) |
| (A.map X.range.mkQ) (B.comap X.ker.subtype) |
|
|
| omit [FiniteDimensional F2 E] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] |
| [Finite E] [Finite F] in |
| theorem a5Mask_unique (X : F →ₗ[F2] E) |
| (A : Submodule F2 E) (B : Submodule F2 F) |
| (hI : X.range ≤ A) (hB : B ≤ X.ker) |
| (Y : F →ₗ[F2] E) (hY : a5OuterMask X A B Y) : |
| ∃! i : A5GeometricIndex X A B, a5Mask X A B i Y := by |
| obtain ⟨p, hp, hu⟩ := (a5_unique_selector X Y A B hI hB).mp hY |
| rcases hp with ⟨hinner, hdis, hA, hcover, hmeet, hlocal⟩ |
| let i : A5GeometricIndex X A B := ⟨p, hdis, hA, hcover, hmeet⟩ |
| refine ⟨i, ⟨hinner, hlocal⟩, ?_⟩ |
| intro j hj |
| apply Subtype.ext |
| apply hu |
| rcases j.property with ⟨hjdis, hjA, hjcover, hjmeet⟩ |
| exact ⟨hj.1, hjdis, hjA, hjcover, hjmeet, hj.2⟩ |
|
|
| omit [FiniteDimensional F2 E] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] |
| [Finite E] [Finite F] in |
| theorem a5Mask_implies_outer (X : F →ₗ[F2] E) |
| (A : Submodule F2 E) (B : Submodule F2 F) |
| (i : A5GeometricIndex X A B) (Y : F →ₗ[F2] E) |
| (hi : a5Mask X A B i Y) : a5OuterMask X A B Y := by |
| rcases i.property with ⟨hdis, hA, hcover, hmeet⟩ |
| exact ⟨rankBelow_of_compressed_rankBelow X Y i.val.1 i.val.2 |
| hi.1 hdis hcover hi.2, |
| reverse_outer_hybrid X Y i.val.1 A B i.val.2 |
| hi.1 hdis hA hcover hmeet hi.2⟩ |
|
|
| omit [Finite E] [Finite F] in |
| theorem a5_projector_partition (X : F →ₗ[F2] E) |
| (A : Submodule F2 E) (B : Submodule F2 F) |
| (hI : X.range ≤ A) (hB : B ≤ X.ker) |
| [Fintype (A5GeometricIndex X A B)] (f : (E →ₗ[F2] F) → ℝ) : |
| spectralProjector (a5OuterMask X A B) f = |
| ∑ i : A5GeometricIndex X A B, spectralProjector (a5Mask X A B i) f := by |
| exact spectralProjector_partition _ _ |
| (a5Mask_unique X A B hI hB) (a5Mask_implies_outer X A B) f |
|
|
| end UniqueGamesTheorem.Appendix.OperatorPartitions |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
