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One exposure controls both modes

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

proven

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    Natural Language Statement

    Lemma

    Finite greedy ordering of the product of column and row spaces supplies one dyadic exposed set with combined actual deficit less than one quarter of its remaining size.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.CombinedSpans
    2
    3/-!
    4---
    5title: One exposure controls both modes
    6type: lemma
    7---
    8Finite greedy ordering of the product of column and row spaces supplies one dyadic exposed set with combined actual deficit less than one quarter of its remaining size.
    9-/
    10
    11namespace Lax342547.SmallExposure
    12
    13open Lax342547.GreedySpans Lax342547.GreedyTails
    14open scoped BigOperators
    15
    16axiom simultaneous_small_exposure {K V W ι : Type} [Field K] [Fintype ι] [DecidableEq ι]
    17 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    18 [FiniteDimensional K V] [FiniteDimensional K W]
    19 (C : ι → Submodule K V) (R : ι → Submodule K W) (r : ℝ) (k n : ℕ)
    20 (hC : ∀ i, (Module.finrank K (C i) : ℝ) ≤ r)
    21 (hR : ∀ i, (Module.finrank K (R i) : ℝ) ≤ r)
    22 (hcard : Fintype.card ι = 2^k) (hnk : n ≤ k) (hn : 16*r < n) :
    23 ∃ l : List ι, l.Nodup ∧ l.toFinset = Finset.univ ∧ ∃ j < n,
    24 deficit C (spanAfter C ⊥ (l.take (l.length-2^(k-j)))) (l.drop (l.length-2^(k-j)))+
    25 deficit R (spanAfter R ⊥ (l.take (l.length-2^(k-j)))) (l.drop (l.length-2^(k-j))) <
    26 (2^(k-j) : ℝ)/4
    27
    28end Lax342547.SmallExposure
    29
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