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Finite linear images and their uniform-law density bounds

Lax342547.FiniteLinearLaw · concepts/Lax342547/FiniteLinearLaw.lean · lax-342547

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    Natural Language Statement

    Theorem

    Surjective linear observation cannot increase the codimension of a subspace. Over the binary field, an image of codimension at most c has point masses bounded by 2^c times the full-space uniform point mass. These are the projection and domination steps used for mixer row bits.

    Concept map
    2 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 codimension_map_le proven

    2 family_codimension proven

    3 uniform_affine_image_bound proven

    4 uniform_image_bound proven

    5 uniform_image_of_surjective proven

    Lean source view on GitHub

    1import Lax342547.MomentSpace
    2import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
    3import Mathlib.LinearAlgebra.Quotient.Basic
    4import Mathlib.Probability.Distributions.Uniform
    5import Mathlib.Probability.ProbabilityMassFunction.Constructions
    6import Mathlib.LinearAlgebra.Dimension.Constructions
    7
    8/-!
    9---
    10title: Finite linear images and their uniform-law density bounds
    11type: theorem
    12---
    13Surjective linear observation cannot increase the codimension of a
    14subspace. Over the binary field, an image of codimension at most c has
    15point masses bounded by 2^c times the full-space uniform point mass.
    16These are the projection and domination steps used for mixer row bits.
    17-/
    18
    19namespace Lax342547.FiniteLinearLaw
    20
    21def familyMap {K T : Type} [Field K] {V W : T → Type}
    22 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    23 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)]
    24 (f : ∀ i, V i →ₗ[K] W i) : (∀ i, V i) →ₗ[K] (∀ i, W i) :=
    25 LinearMap.pi (fun i => (f i).comp (LinearMap.proj i))
    26
    27axiom family_codimension {K T : Type} [Field K] [Fintype T] {V W : T → Type}
    28 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    29 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)]
    30 [∀ i, FiniteDimensional K (V i)] [∀ i, FiniteDimensional K (W i)]
    31 (f : ∀ i, V i →ₗ[K] W i) (c : T → ℕ)
    32 (hc : ∀ i, Module.finrank K (W i) ≤ Module.finrank K (LinearMap.range (f i)) + c i) :
    33 Module.finrank K (∀ i, W i) ≤ Module.finrank K (LinearMap.range (familyMap f)) + ∑ i, c i
    34
    35axiom codimension_map_le {K V W : Type} [Field K] [AddCommGroup V] [Module K V]
    36 [AddCommGroup W] [Module K W] [FiniteDimensional K V] [FiniteDimensional K W]
    37 (f : V →ₗ[K] W) (hf : Function.Surjective f) (S : Submodule K V) :
    38 Module.finrank K W - Module.finrank K (S.map f) ≤ Module.finrank K V - Module.finrank K S
    39
    40open Lax342547.MomentSpace
    41open scoped ENNReal
    42
    43axiom uniform_image_bound {V W : Type} [AddCommGroup V] [Module Binary V]
    44 [AddCommGroup W] [Module Binary W] [Fintype V] [Fintype W]
    45 [FiniteDimensional Binary V] [FiniteDimensional Binary W]
    46 (f : V →ₗ[Binary] W) (c : ℕ)
    47 (hc : Module.finrank Binary W ≤ Module.finrank Binary (LinearMap.range f) + c) (y : W) :
    48 (PMF.uniformOfFintype V).map f y ≤ 2 ^ c * PMF.uniformOfFintype W y
    49
    50axiom uniform_image_of_surjective {V W : Type} [AddCommGroup V] [Module Binary V]
    51 [AddCommGroup W] [Module Binary W] [Fintype V] [Fintype W]
    52 [FiniteDimensional Binary V] [FiniteDimensional Binary W]
    53 (f : V →ₗ[Binary] W) (hf : Function.Surjective f) :
    54 (PMF.uniformOfFintype V).map f = PMF.uniformOfFintype W
    55
    56axiom uniform_affine_image_bound {V W : Type} [AddCommGroup V] [Module Binary V]
    57 [AddCommGroup W] [Module Binary W] [Fintype V]
    58 [FiniteDimensional Binary V]
    59 (f : V →ₗ[Binary] W) (a y : W) (r : ℕ)
    60 (hr : r ≤ Module.finrank Binary (LinearMap.range f)) :
    61 (PMF.uniformOfFintype V).map (fun x => a + f x) y ≤ 1 / (2 : ℝ≥0∞) ^ r
    62
    63end Lax342547.FiniteLinearLaw
    64
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