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Greedy independence of subspaces

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

proven

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    Natural Language Statement

    Lemma

    Disjointness from accumulated pin spans constructs independent subspaces with an explicit cumulative dimension budget.

    Concept map
    14 concepts; 8 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 accumulated_dimension proven

    2 finite_span_dimension proven

    3 independent_snoc proven

    Lean source view on GitHub

    1import Lax342547.FiniteSampling
    2import Mathlib.LinearAlgebra.DFinsupp
    3import Mathlib.Data.Fin.Tuple.Basic
    4
    5/-!
    6---
    7title: Greedy independence of subspaces
    8type: lemma
    9---
    10Disjointness from accumulated pin spans constructs independent subspaces with an explicit cumulative dimension budget.
    11-/
    12
    13namespace Lax342547.GreedySpaces
    14
    15open Lax342547.MomentSpace
    16open scoped BigOperators
    17
    18axiom independent_snoc {K V : Type} [Field K] [AddCommGroup V] [Module K V]
    19 {n : ℕ} (U : Fin n → Submodule K V) (W : Submodule K V)
    20 (hU : iSupIndep U) (hW : Disjoint W (⨆ i,U i)) : iSupIndep (Fin.snoc U W)
    21
    22axiom finite_span_dimension {K V ι : Type} [Field K] [AddCommGroup V] [Module K V]
    23 [FiniteDimensional K V] (U : ι → Submodule K V) (S : Finset ι) :
    24 Module.finrank K ↥(S.sup U) ≤ ∑ i ∈ S,Module.finrank K (U i)
    25
    26axiom accumulated_dimension {K V Ω : Type} [Field K] [AddCommGroup V] [Module K V]
    27 [FiniteDimensional K V] (U : Ω → Submodule K V) (r n : ℕ) (s : Fin n → Ω)
    28 (hU : ∀ x,Module.finrank K (U x) ≤ r) :
    29 Module.finrank K ↥(⨆ i : Fin n,U (s i)) ≤ r*n
    30
    31end Lax342547.GreedySpaces
    32
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