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Unselected sparse vectors cannot conceal fresh key coefficients

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

proven

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    Natural Language Statement

    Theorem

    If an unselected sparse vector differs from a short fresh key combination by a projected pin, each key's base-vector coefficient is zero. This also covers inactive keys whose base vector is already zero.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.QuotientExtractors
    2
    3/-!
    4---
    5title: Unselected sparse vectors cannot conceal fresh key coefficients
    6type: theorem
    7---
    8If an unselected sparse vector differs from a short fresh key combination
    9by a projected pin, each key's base-vector coefficient is zero. This
    10also covers inactive keys whose base vector is already zero.
    11-/
    12
    13namespace Lax342547.SparseKeyCoefficients
    14
    15open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.ProjectedPins
    16open Lax342547.PinLabelExclusions Lax342547.SparsePins
    17
    18axiom coefficients_zero {Base Comp H N I : Type} [Fintype I] {b degree R : ℕ}
    19 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    20 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R)
    21 (i : Fin 2) (a : Comp × Bool) (L : Finset (Fin b → Binary)) (hL : L.card ≤ R)
    22 (v : (Fin b → Binary) → Option Base → Binary)
    23 (label : I → Fin b → Binary) (hlabel : Function.Injective label)
    24 (base : I → Option Base → Binary) (hcard : Fintype.card I ≤ 14)
    25 (hfresh : ∀ j, label j ∉ A i)
    26 (hdisjoint : Disjoint L (Finset.univ.image label)) (c : I → Binary)
    27 (h : (∑ s ∈ L, labelTensor (selectorEval (degree := degree)) s (v s)) -
    28 (∑ j, c j • labelTensor (selectorEval (degree := degree)) (label j) (base j)) ∈ projected P i a) :
    29 ∀ j, c j • base j = 0
    30
    31end Lax342547.SparseKeyCoefficients
    32
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