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Counting component tensors with bounded total rank

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

proven

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    Natural Language Statement

    Lemma

    Actual rank budgets and matrix factors give the uniform candidate count with ambient exponent (dim I+dim J)*r, rather than paying r for each component separately.

    Concept map
    3 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 exists_rank_factors proven

    2 factor_card_bound proven

    3 total_rank_card proven

    Lean source view on GitHub

    1import Lax342547.LowRankCounting
    2import Mathlib.Data.Fintype.Card
    3
    4/-!
    5---
    6title: Counting component tensors with bounded total rank
    7type: lemma
    8---
    9Actual rank budgets and matrix factors give the uniform candidate count with ambient exponent (dim I+dim J)*r, rather than paying r for each component separately.
    10-/
    11
    12namespace Lax342547.TotalRankCount
    13
    14open Lax342547.MomentSpace
    15open scoped BigOperators
    16
    17abbrev RankBudgets (e : Type) [Fintype e] (r : ℕ) :=
    18 {a : e → Fin (r+1) // ∑ i, (a i).val ≤ r}
    19
    20abbrev Factors {e : Type} [Fintype e] (I J : Type) (r : ℕ) (a : RankBudgets e r) :=
    21 (∀ i : e, Matrix I (Fin (a.val i).val) Binary) ×
    22 (∀ i : e, Matrix (Fin (a.val i).val) J Binary)
    23
    24noncomputable def reconstruct {e I J : Type} [Fintype e] (r : ℕ) :
    25 (Σ a : RankBudgets e r, Factors I J r a) → e → Matrix I J Binary :=
    26 fun z i => z.2.1 i*z.2.2 i
    27
    28axiom exists_rank_factors {e I J : Type} [Fintype e] [Fintype I] [Fintype J]
    29 (r : ℕ) (A : e → Matrix I J Binary) (hA : ∑ i, (A i).rank ≤ r) :
    30 ∃ z : Σ a : RankBudgets e r, Factors I J r a, reconstruct r z = A
    31
    32axiom factor_card_bound {e I J : Type} [Fintype e] [Fintype I] [Fintype J]
    33 [DecidableEq e] [DecidableEq I] [DecidableEq J] (r : ℕ) (a : RankBudgets e r) :
    34 Fintype.card (Factors I J r a) ≤ 2^((Fintype.card I+Fintype.card J)*r)
    35
    36axiom total_rank_card {e I J : Type} [Fintype e] [Fintype I] [Fintype J]
    37 [DecidableEq e] [DecidableEq I] [DecidableEq J] (r : ℕ) :
    38 Fintype.card {A : e → Matrix I J Binary // ∑ i, (A i).rank ≤ r} ≤
    39 (r+1)^Fintype.card e*2^((Fintype.card I+Fintype.card J)*r)
    40
    41end Lax342547.TotalRankCount
    42
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    From Mathlib

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