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Sparse pin exclusions with arbitrary base coefficients

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

proven

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    Natural Language Statement

    Lemma

    The annihilator argument needs more than independence of fresh point rays: an expansion may mix selected fresh labels with other labels in the exclusion. The uniform cover bounds every nonzero base-vector label coefficient of every short projected pin expansion.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 covers_exists proven

    2 fresh_coefficient proven

    3 keys_exclusion proven

    Lean source view on GitHub

    1import Lax342547.KeySpans
    2import Lax342547.SparsePins
    3
    4/-!
    5---
    6title: Sparse pin exclusions with arbitrary base coefficients
    7type: lemma
    8---
    9The annihilator argument needs more than independence of fresh point
    10rays: an expansion may mix selected fresh labels with other labels in
    11the exclusion. The uniform cover bounds every nonzero base-vector label
    12coefficient of every short projected pin expansion.
    13-/
    14
    15namespace Lax342547.PinLabelExclusions
    16
    17open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.ProjectedPins
    18open Lax342547.SparsePins Lax342547.KeySpans
    19
    20variable {Base Comp H N : Type} {b degree : ℕ}
    21
    22def Covers
    23 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    24 (A : Fin 2 → Finset (Fin b → Binary)) (R : ℕ) : Prop :=
    25 ∀ i a (L : Finset (Fin b → Binary)) (v : (Fin b → Binary) → Option Base → Binary),
    26 L.card ≤ R + 14 →
    27 (∑ s ∈ L, labelTensor (selectorEval (degree := degree)) s (v s)) ∈ projected P i a →
    28 ∀ s ∈ L, v s ≠ 0 → s ∈ A i
    29
    30axiom covers_exists [Fintype Base] [Fintype Comp] [Fintype H] {K R : ℕ}
    31 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    32 (hK : P.rank ≤ K) (hdegree : (K + 1) * (R + 14) ≤ degree + 1) :
    33 ∃ A : Fin 2 → Finset (Fin b → Binary),
    34 (∀ i, (A i).card ≤ 2 * Fintype.card Comp * K * (R + 14)) ∧ Covers P A R
    35
    36axiom fresh_coefficient {R : ℕ}
    37 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    38 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R)
    39 (i : Fin 2) (a : Comp × Bool) (L : Finset (Fin b → Binary))
    40 (v : (Fin b → Binary) → Option Base → Binary) (hL : L.card ≤ R + 14)
    41 (hmem : (∑ t ∈ L, labelTensor (selectorEval (degree := degree)) t (v t)) ∈ projected P i a)
    42 (s : Fin b → Binary) (hs : s ∈ L) (hfresh : s ∉ A i) : v s = 0
    43
    44axiom keys_exclusion {R : ℕ}
    45 (P : Pin (Comp × Bool) (Fin 2 × ((SelectorCoordinates b degree × Option Base) ⊕ H)) N)
    46 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Covers P A R) : Excludes P A
    47
    48end Lax342547.PinLabelExclusions
    49
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