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Recovery of low-rank Boolean moments from low degrees

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

proven

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    Natural Language Statement

    Lemma

    The rank obstruction in the second half of Lemma 5.2. A first nonzero selector moment of degree d produces an identity submatrix of size binomial(d, floor(d/2)). Consequently a Boolean moment matrix of rank at most r vanishes if all its moments of degree at most r vanish.

    Concept map
    2 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 eq_of_agree_low proven

    2 eq_zero_of_rank_le proven

    Lean source view on GitHub

    1import Lax342547.MomentSpace
    2import Mathlib.LinearAlgebra.Matrix.Rank
    3
    4/-!
    5---
    6title: Recovery of low-rank Boolean moments from low degrees
    7type: lemma
    8---
    9The rank obstruction in the second half of Lemma 5.2. A first nonzero
    10selector moment of degree d produces an identity submatrix of size
    11binomial(d, floor(d/2)). Consequently a Boolean moment matrix of rank at
    12most r vanishes if all its moments of degree at most r vanish.
    13-/
    14
    15namespace Lax342547.MomentRecovery
    16
    17open Lax342547.MomentSpace
    18
    19def IsSelectorHankel {Base : Type} {b degree : ℕ}
    20 (w : Matrix (SelectorCoordinates b degree × Base)
    21 (SelectorCoordinates b degree × Base) Binary) : Prop :=
    22 ∀ (S T U V : SelectorCoordinates b degree) (i j : Base),
    23 S.val ∪ T.val = U.val ∪ V.val → w (S, i) (T, j) = w (U, i) (V, j)
    24
    25axiom eq_zero_of_rank_le {Base : Type} [Fintype Base] {b degree r : ℕ}
    26 (w : Matrix (SelectorCoordinates b degree × Base)
    27 (SelectorCoordinates b degree × Base) Binary)
    28 (hH : IsSelectorHankel w) (hrank : w.rank ≤ r)
    29 (hlow : ∀ (S T : SelectorCoordinates b degree) (i j : Base),
    30 (S.val ∪ T.val).card ≤ r → w (S, i) (T, j) = 0) : w = 0
    31
    32axiom eq_of_agree_low {Base : Type} [Fintype Base] {b degree r₁ r₂ : ℕ}
    33 (w v : Matrix (SelectorCoordinates b degree × Base)
    34 (SelectorCoordinates b degree × Base) Binary)
    35 (hw : IsSelectorHankel w) (hv : IsSelectorHankel v)
    36 (hrw : w.rank ≤ r₁) (hrv : v.rank ≤ r₂)
    37 (hlow : ∀ (S T : SelectorCoordinates b degree) (i j : Base),
    38 (S.val ∪ T.val).card ≤ r₁ + r₂ → w (S, i) (T, j) = v (S, i) (T, j)) : w = v
    39
    40end Lax342547.MomentRecovery
    41
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