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Fresh key directions are independent modulo table spaces

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

proven

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    Natural Language Statement

    Theorem

    One bounded exclusion set per endpoint works for all components and both signs. Any short linear combination of distinct fresh point rays that belongs to a table space has all its coefficients zero.

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

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

    1 fresh_keys proven

    2 fresh_points proven

    Lean source view on GitHub

    1import Lax342547.TableSpaces
    2import Lax342547.SparsePins
    3
    4/-!
    5---
    6title: Fresh key directions are independent modulo table spaces
    7type: theorem
    8---
    9One bounded exclusion set per endpoint works for all components and both
    10signs. Any short linear combination of distinct fresh point rays that
    11belongs to a table space has all its coefficients zero.
    12-/
    13
    14namespace Lax342547.FreshKeys
    15
    16open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.ProjectedPins Lax342547.TableSpaces
    17
    18axiom fresh_points {Base Comp : Type} [Fintype Base] [Fintype Comp] {b degree K R : ℕ}
    19 (U : Bool → Comp → Submodule Binary (SelectorCoordinates b degree × Option Base → Binary))
    20 (hdim : ∀ sign e, Module.finrank Binary (U sign e) ≤ K)
    21 (hdegree : (K + 1) * (R + 14) ≤ degree + 1) :
    22 ∃ B : Finset (Fin b → Binary),
    23 B.card ≤ 2 * Fintype.card Comp * K * (R + 14) ∧
    24 ∀ sign e (L : Finset (Fin b → Binary)) (c : (Fin b → Binary) → Binary)
    25 (z : (Fin b → Binary) → Base → Binary),
    26 L.card ≤ R + 14 → Disjoint L B →
    27 (∑ s ∈ L, c s • point (selectorEval (degree := degree)) s (z s)) ∈ U sign e →
    28 ∀ s ∈ L, c s = 0
    29
    30axiom fresh_keys {Base Comp H N : Type} [Fintype Base] [Fintype Comp] [Fintype H]
    31 {b degree K R : ℕ}
    32 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    33 (hK : P.rank ≤ K) (hdegree : (K + 1) * (R + 14) ≤ degree + 1) :
    34 ∃ B : Fin 2 → Finset (Fin b → Binary),
    35 (∀ i, (B i).card ≤ 2 * Fintype.card Comp * K * (R + 14)) ∧
    36 ∀ (a : Comp × Bool) (L : Fin 2 → Finset (Fin b → Binary))
    37 (c : Fin 2 → (Fin b → Binary) → Binary)
    38 (z : Fin 2 → (Fin b → Binary) → Base → Binary),
    39 (∀ i, (L i).card ≤ R + 14) → (∀ i, Disjoint (L i) (B i)) →
    40 (∑ i : Fin 2, ∑ s ∈ L i,
    41 c i s • primalEmbedding i (point (selectorEval (degree := degree)) s (z i s))) ∈
    42 tableSpace P a →
    43 ∀ i s, s ∈ L i → c i s = 0
    44
    45end Lax342547.FreshKeys
    46
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