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Monotonicity of span deficits

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

proven

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    Natural Language Statement

    Lemma

    Enlarging each mode space can only increase its addition-map kernel dimension and its span deficit.

    Concept map
    3 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.SpanDeficits
    2
    3/-!
    4---
    5title: Monotonicity of span deficits
    6type: lemma
    7---
    8Enlarging each mode space can only increase its addition-map kernel dimension and its span deficit.
    9-/
    10
    11namespace Lax342547.SpanDeficitMono
    12
    13open Lax342547.SpanDeficits
    14open scoped BigOperators
    15
    16axiom deficit_monotone {K V ι : Type} [Field K] [Fintype ι]
    17 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    18 (S T : ι → Submodule K V) (h : ∀ i, S i ≤ T i) :
    19 ((∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K V)) ≤
    20 ((∑ i, Module.finrank K (T i))-Module.finrank K (⨆ i, T i : Submodule K V))
    21
    22end Lax342547.SpanDeficitMono
    23
    Show Proof
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    From Mathlib

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