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Counting tuples of symmetric matrices with bounded total rank

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

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    Natural Language Statement

    Theorem

    Separate the component rank tuple from the symmetric factorizations. There are at most (K+1)^|E| rank tuples, and each permitted tuple uses at most dK+K² binary entries. This is the finite profile count underlying the low-rank profile bound in Lemma 5.5, before substituting the concrete coordinate dimension.

    Concept map
    11 concepts; 2 descendants hidden
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 card_concrete_profiles proven

    2 card_symmetric_profiles proven

    Lean source view on GitHub

    1import Lax342547.SymmetricFactorization
    2import Lax342547.ConcreteCut
    3
    4/-!
    5---
    6title: Counting tuples of symmetric matrices with bounded total rank
    7type: theorem
    8---
    9Separate the component rank tuple from the symmetric factorizations.
    10There are at most (K+1)^|E| rank tuples, and each permitted tuple uses
    11at most dK+K² binary entries. This is the finite profile count underlying
    12the low-rank profile bound in Lemma 5.5, before substituting the concrete
    13coordinate dimension.
    14-/
    15
    16namespace Lax342547.ProfileCounting
    17
    18open Lax342547.MomentSpace
    19
    20axiom card_symmetric_profiles {I E : Type} [Fintype I] [Fintype E] (K : ℕ) :
    21 Nat.card {x : E → Matrix I I Binary //
    22 (∀ e, (x e).transpose = x e) ∧ ∑ e, (x e).rank ≤ K} ≤
    23 (K + 1) ^ Fintype.card E * 2 ^ (Fintype.card I * K + K * K)
    24
    25open Lax342547.ConcreteGeometry Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.TagGeometry
    26
    27axiom card_concrete_profiles (k n b degree K : ℕ) :
    28 Nat.card {x : Profile k n b degree // ∑ e, (x.val e).rank ≤ K} ≤
    29 (K + 1) ^ Fintype.card (Component (Tag k)) *
    30 2 ^ (Fintype.card (Coordinate k n b degree) * K + K * K)
    31
    32end Lax342547.ProfileCounting
    33
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