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Bit rank and Walsh bounds for global vector-slot pairings

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

proven

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    Natural Language Statement

    Lemma

    Each of N bit coordinates carries the same cross-slot coefficient matrix. The resulting bilinear bit matrix has rank N times its rank, so a nonzero cross pattern has correlation at most 2^(-N/2).

    Concept map
    4 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 bit_evaluation proven

    2 bit_rank proven

    3 nonzero_pattern_bound proven

    Lean source view on GitHub

    1import Lax342547.BilinearWalsh
    2
    3/-!
    4---
    5title: Bit rank and Walsh bounds for global vector-slot pairings
    6type: lemma
    7---
    8Each of N bit coordinates carries the same cross-slot coefficient
    9matrix. The resulting bilinear bit matrix has rank N times its rank,
    10so a nonzero cross pattern has correlation at most 2^(-N/2).
    11-/
    12
    13namespace Lax342547.SlotWalsh
    14
    15open Lax342547.MomentSpace
    16
    17def bitMatrix {I J : Type} (C : Matrix I J Binary) (N : ℕ) :
    18 Matrix (Fin N × I) (Fin N × J) Binary :=
    19 fun x y => if x.1 = y.1 then C x.2 y.2 else 0
    20
    21axiom bit_evaluation {I J : Type} [Fintype I] [Fintype J]
    22 (C : Matrix I J Binary) (N : ℕ) (x : Fin N × I → Binary) (y : Fin N × J → Binary) :
    23 dotProduct x ((bitMatrix C N).mulVec y) = ∑ i, ∑ j, C i j * ∑ a, x (a, i) * y (a, j)
    24
    25axiom bit_rank {I J : Type} [Fintype J]
    26 (C : Matrix I J Binary) (N : ℕ) : (bitMatrix C N).rank = N * C.rank
    27
    28axiom nonzero_pattern_bound {I J : Type} [Fintype I] [Fintype J]
    29 [DecidableEq I] [DecidableEq J] (C : Matrix I J Binary) (hC : C ≠ 0) (N : ℕ)
    30 (f : (Fin N × I → Binary) → ℝ) (g : (Fin N × J → Binary) → ℝ)
    31 (hf : ∀ x, |f x| ≤ 1) (hg : ∀ y, |g y| ≤ 1) :
    32 |BilinearWalsh.average (bitMatrix C N) f g| ≤ 1 / Real.sqrt ((2 : ℝ) ^ N)
    33
    34end Lax342547.SlotWalsh
    35
    Show ProofShow ProofShow Proof
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