File size: 3,555 Bytes
760f90e | 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 | import OAI.Computability.UniqueGames.Analysis.CompressionCountLemmas
namespace OAI
section
/-! Actual subspace cardinality bounds used in (5.12)--(5.13).
Every subspace is the image of a projection endomorphism. Thus the set of all
subspaces is a quotient of the endomorphism space. The exact finite-field
linear-map count is reused from `Appendix.CompressionCount`.
-/
namespace UniqueGamesTheorem.Fourier.MatrixSubspaceCount
open Module
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
section GeneralField
variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
/-- Taking the image of an endomorphism reaches every actual subspace. -/
theorem endomorphism_range_surjective :
Function.Surjective (fun f : V →ₗ[K] V => LinearMap.range f) := by
intro S
obtain ⟨T, hT⟩ := S.exists_isCompl
exact ⟨S.projection T hT, Submodule.range_projection hT⟩
theorem card_subspaces_le_endomorphisms [Finite V] :
Nat.card (Submodule K V) ≤ Nat.card (V →ₗ[K] V) := by
let : Finite (V →ₗ[K] V) := Finite.of_injective
(fun f : V →ₗ[K] V => (f : V → V)) DFunLike.coe_injective
exact Nat.card_le_card_of_surjective _ endomorphism_range_surjective
end GeneralField
section BinaryField
variable {C : Type*} [AddCommGroup C] [Module F2 C] [Module.Finite F2 C]
private theorem finite_color_space : Finite C :=
Finite.of_injective (Module.finBasis F2 C).equivFun
(Module.finBasis F2 C).equivFun.injective
private theorem finite_endomorphisms : Finite (C →ₗ[F2] C) := by
let : Finite C := finite_color_space
exact Finite.of_injective (fun f : C →ₗ[F2] C => (f : C → C)) DFunLike.coe_injective
private theorem finite_subspaces : Finite (Submodule F2 C) := by
let : Finite (C →ₗ[F2] C) := finite_endomorphisms
exact Finite.of_surjective _ endomorphism_range_surjective
theorem card_binary_subspaces_le :
Nat.card (Submodule F2 C) ≤ 2 ^ (finrank F2 C * finrank F2 C) := by
let : Finite C := finite_color_space
calc
_ ≤ Nat.card (C →ₗ[F2] C) := card_subspaces_le_endomorphisms
_ = _ := UniqueGamesTheorem.Appendix.CompressionCount.natCard_linearMap
theorem card_binary_subspaces_le_of_finrank_le (s : ℕ) (hs : finrank F2 C ≤ s) :
Nat.card (Submodule F2 C) ≤ 2 ^ (s * s) := by
exact card_binary_subspaces_le.trans
(Nat.pow_le_pow_right (by decide) (Nat.mul_le_mul hs hs))
theorem card_admissible_subspaces_le (P : Submodule F2 C → Prop) :
Nat.card {S : Submodule F2 C // P S} ≤ 2 ^ (finrank F2 C * finrank F2 C) := by
let : Finite (Submodule F2 C) := finite_subspaces
exact (Nat.card_le_card_of_injective Subtype.val Subtype.val_injective).trans
card_binary_subspaces_le
/-- Counting fixed colors together with actual admissible subspaces gives
precisely the `q * 2^(s*s)` denominator budget from (5.13). -/
theorem card_color_admissible_subspaces_le (P : Submodule F2 C → Prop) :
Nat.card (C × {S : Submodule F2 C // P S}) ≤
Nat.card C * 2 ^ (finrank F2 C * finrank F2 C) := by
rw [Nat.card_prod]
exact Nat.mul_le_mul_left _ (card_admissible_subspaces_le P)
theorem card_color_admissible_subspaces_le_of_finrank_le
(P : Submodule F2 C → Prop) (s : ℕ) (hs : finrank F2 C ≤ s) :
Nat.card (C × {S : Submodule F2 C // P S}) ≤ Nat.card C * 2 ^ (s * s) := by
exact (card_color_admissible_subspaces_le P).trans
(Nat.mul_le_mul_left _
(Nat.pow_le_pow_right (by decide) (Nat.mul_le_mul hs hs)))
end BinaryField
end UniqueGamesTheorem.Fourier.MatrixSubspaceCount
end
end OAI
|