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Actual projected span deficit

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

proven

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    Natural Language Statement

    Lemma

    The projected-space dimension deficit equals the greedy tail statistic, so the small-tail conclusion concerns actual quotient spaces.

    Concept map
    10 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 map_finset_sup proven

    2 projected_deficit_eq proven

    3 small_projected_tail proven

    Lean source view on GitHub

    1import Lax342547.ProjectedSpans
    2import Lax342547.GreedyTails
    3
    4/-!
    5---
    6title: Actual projected span deficit
    7type: lemma
    8---
    9The projected-space dimension deficit equals the greedy tail statistic, so the small-tail conclusion concerns actual quotient spaces.
    10-/
    11
    12namespace Lax342547.ActualDeficits
    13
    14open Lax342547.GreedySpans Lax342547.GreedyTails
    15open scoped BigOperators
    16
    17noncomputable def projectedDeficit {K V ι : Type} [Field K] [DecidableEq ι]
    18 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    19 (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) : ℝ :=
    20 (l.map (fun i => (Module.finrank K ((S i).map W.mkQ) : ℝ))).sum-
    21 Module.finrank K (l.toFinset.sup (fun i => (S i).map W.mkQ) : Submodule K (V ⧸ W))
    22
    23axiom map_finset_sup {K V U ι : Type} [Field K] [DecidableEq ι]
    24 [AddCommGroup V] [Module K V] [AddCommGroup U] [Module K U]
    25 (S : ι → Submodule K V) (f : V →ₗ[K] U) (I : Finset ι) :
    26 (I.sup S).map f = I.sup (fun i => (S i).map f)
    27
    28axiom projected_deficit_eq {K V ι : Type} [Field K] [DecidableEq ι]
    29 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    30 (S : ι → Submodule K V) (W : Submodule K V) (l : List ι) :
    31 projectedDeficit S W l = deficit S W l
    32
    33axiom small_projected_tail {K V ι : Type} [Field K] [DecidableEq ι]
    34 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    35 (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, projectedDeficit 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.ActualDeficits
    42
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