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Bounded table data and linear-size nominal coefficient records

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

proven

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    Natural Language Statement

    Lemma

    For fixed pin and channel budgets, the number of small tables is bounded independently of the primal dimension. Nominal pin bases can be padded to a fixed number of coefficient vectors; their bit count grows linearly in the primal dimension.

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

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

    1 coefficient_code proven

    2 coefficient_count proven

    3 table_count proven

    Lean source view on GitHub

    1import Lax342547.SmallTables
    2import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
    3import Mathlib.SetTheory.Cardinal.Finite
    4
    5/-!
    6---
    7title: Bounded table data and linear-size nominal coefficient records
    8type: lemma
    9---
    10For fixed pin and channel budgets, the number of small tables is bounded
    11independently of the primal dimension. Nominal pin bases can be padded to
    12a fixed number of coefficient vectors; their bit count grows linearly in
    13the primal dimension.
    14-/
    15
    16namespace Lax342547.TableCounts
    17
    18open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.TableSpaces Lax342547.SmallTables
    19
    20abbrev CoefficientCode (Comp B H : Type) (K : ℕ) :=
    21 (Comp × Bool) → Fin K → (Fin 2 × (B ⊕ H)) → Binary
    22
    23axiom table_count {Comp B H N : Type} [Fintype Comp] [Fintype B] [Fintype H]
    24 (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N) (K : ℕ)
    25 (hP : P.rank ≤ K) (hQ : Q.rank ≤ K) :
    26 Nat.card (Table P Q) ≤ 2 ^ (2 * Fintype.card Comp * (2 * K + 2 * Fintype.card H) ^ 2)
    27
    28axiom coefficient_code {Comp B H N : Type} [Fintype Comp] [Fintype B] [Fintype H]
    29 (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N) (K : ℕ) (hP : P.rank ≤ K) :
    30 ∃ code : CoefficientCode Comp B H K,
    31 ∀ a, Submodule.span Binary (Set.range (code a)) = P.space a
    32
    33axiom coefficient_count {Comp B H : Type} [Fintype Comp] [Fintype B] [Fintype H] (K : ℕ) :
    34 Nat.card (CoefficientCode Comp B H K) =
    35 2 ^ (4 * Fintype.card Comp * K * (Fintype.card B + Fintype.card H))
    36
    37end Lax342547.TableCounts
    38
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