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Exact-image bounds for independent affine columns

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

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    Natural Language Statement

    Lemma

    Every column may have its own affine translation space, provided they all contain one common subspace S. An injective nominal coefficient matrix of rank t then has at least t dim(S) uniform image bits. The estimate is valid for every t, with no bounded-rank assumption.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 dependence_bound proven

    2 evaluation_rank proven

    3 extension_dependence_bound proven

    4 image_bound proven

    Lean source view on GitHub

    1import Lax342547.FiniteLinearLaw
    2
    3/-!
    4---
    5title: Exact-image bounds for independent affine columns
    6type: lemma
    7---
    8Every column may have its own affine translation space, provided they
    9all contain one common subspace S. An injective nominal coefficient
    10matrix of rank t then has at least t dim(S) uniform image bits. The
    11estimate is valid for every t, with no bounded-rank assumption.
    12-/
    13
    14namespace Lax342547.AffineImages
    15
    16open Lax342547.MomentSpace
    17open scoped ENNReal
    18
    19variable {J T V : Type} [Fintype J] [Fintype T]
    20 [AddCommGroup V] [Module Binary V]
    21
    22def evaluation (S : J → Submodule Binary V) (C : Matrix J T Binary) :
    23 (∀ j, S j) →ₗ[Binary] (T → V) where
    24 toFun x t := ∑ j, C j t • (x j).val
    25 map_add' x y := by ext t; simp [smul_add, Finset.sum_add_distrib]
    26 map_smul' c x := by ext t; simp [Finset.smul_sum, smul_smul, mul_comm]
    27
    28def columnMap (v : J → V) : (J → Binary) →ₗ[Binary] V where
    29 toFun c := ∑ j, c j • v j
    30 map_add' c d := by simp [add_smul, Finset.sum_add_distrib]
    31 map_smul' s c := by simp [Finset.smul_sum, smul_smul]
    32
    33noncomputable def law [DecidableEq J] (S : J → Submodule Binary V) [∀ j, Fintype (S j)]
    34 (C : Matrix J T Binary) (a : J → V) : PMF (T → V) :=
    35 (PMF.uniformOfFintype (∀ j, S j)).map
    36 (fun x => (fun t => ∑ j, C j t • a j) + evaluation S C x)
    37
    38axiom evaluation_rank [FiniteDimensional Binary V]
    39 (S : J → Submodule Binary V) (S₀ : Submodule Binary V)
    40 (hS : ∀ j, S₀ ≤ S j) (C : Matrix J T Binary) (hC : Function.Injective C.mulVec) :
    41 Fintype.card T * Module.finrank Binary S₀ ≤
    42 Module.finrank Binary (LinearMap.range (evaluation S C))
    43
    44axiom image_bound [DecidableEq J] [FiniteDimensional Binary V]
    45 (S : J → Submodule Binary V) [∀ j, Fintype (S j)]
    46 (S₀ : Submodule Binary V) (hS : ∀ j, S₀ ≤ S j)
    47 (C : Matrix J T Binary) (hC : Function.Injective C.mulVec) (a : J → V) (y : T → V) :
    48 law S C a y ≤ 1 / (2 : ℝ≥0∞) ^ (Fintype.card T * Module.finrank Binary S₀)
    49
    50axiom dependence_bound [DecidableEq J] [FiniteDimensional Binary V]
    51 (S : J → Submodule Binary V) [∀ j, Fintype (S j)]
    52 (S₀ : Submodule Binary V) (hS : ∀ j, S₀ ≤ S j) (a : J → V) :
    53 (PMF.uniformOfFintype (∀ j, S j)).toOuterMeasure
    54 {x | ¬ Function.Injective (columnMap (fun j => a j + (x j).val))} ≤
    55 (2 : ℝ≥0∞) ^ Fintype.card J / 2 ^ Module.finrank Binary S₀
    56
    57axiom extension_dependence_bound {U : Type} [AddCommGroup U] [Module Binary U] [Fintype U]
    58 [DecidableEq J] [FiniteDimensional Binary V]
    59 (P : U →ₗ[Binary] V) (hP : Function.Injective P)
    60 (S : J → Submodule Binary V) [∀ j, Fintype (S j)]
    61 (S₀ : Submodule Binary V) (hS : ∀ j, S₀ ≤ S j) (a : J → V) :
    62 (PMF.uniformOfFintype (∀ j, S j)).toOuterMeasure
    63 {x | ¬ Function.Injective (fun z : U × (J → Binary) =>
    64 P z.1 + columnMap (fun j => a j + (x j).val) z.2)} ≤
    65 (Fintype.card U : ℝ≥0∞) * 2 ^ Fintype.card J / 2 ^ Module.finrank Binary S₀
    66
    67end Lax342547.AffineImages
    68
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