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Small actual greedy exposure tails

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

proven

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    Natural Language Statement

    Lemma

    Greedy exposure bounds the actual sum of projected mode dimensions minus their span increase by a dyadic tail deficit, yielding a small tail at an explicit scale.

    Concept map
    8 concepts
    100%
    Proven claimThis 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 greedy_deficit_bound proven

    2 greedy_tail_bound proven

    3 small_greedy_tail proven

    Lean source view on GitHub

    1import Lax342547.GreedySpans
    2import Lax342547.ListTails
    3import Lax342547.DyadicScale
    4import Mathlib.Algebra.Order.BigOperators.Group.List
    5
    6/-!
    7---
    8title: Small actual greedy exposure tails
    9type: lemma
    10---
    11Greedy exposure bounds the actual sum of projected mode dimensions minus their span increase by a dyadic tail deficit, yielding a small tail at an explicit scale.
    12-/
    13
    14namespace Lax342547.GreedyTails
    15
    16open Lax342547.GreedySpans Lax342547.ListTails Lax342547.DyadicTails
    17open scoped BigOperators
    18
    19noncomputable def deficit {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    20 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) : ℝ :=
    21 (l.map (fun i => gain W (S i))).sum-(increments S W l).sum
    22
    23axiom greedy_deficit_bound {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    24 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι)
    25 (hg : Greedy S W l) (hl : l ≠ []) :
    26 deficit S W l ≤ (l.length : ℝ)*value (increments S W l) 0-(increments S W l).sum
    27
    28axiom greedy_tail_bound {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    29 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι)
    30 (hg : Greedy S W l) (b : ℕ) (hb : 0 < b) (hbl : b ≤ l.length) :
    31 deficit S (spanAfter S W (l.take (l.length-b))) (l.drop (l.length-b)) ≤
    32 (b : ℝ)*tailDeficit (value (increments S W l)) l.length b
    33
    34axiom small_greedy_tail {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    35 [FiniteDimensional K V] (S : ι → Submodule K V) (W : Submodule K V) (l : List ι)
    36 (hg : Greedy S W l) (r : ℝ) (hr : ∀ i ∈ l, (Module.finrank K (S i) : ℝ) ≤ r)
    37 (k n : ℕ) (hlen : l.length = 2^k) (hnk : n ≤ k) (hn : 8*r < n) :
    38 ∃ j < n, deficit S (spanAfter S W (l.take (l.length-2^(k-j))))
    39 (l.drop (l.length-2^(k-j))) < (2^(k-j) : ℝ)/4
    40
    41end Lax342547.GreedyTails
    42
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