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Actual deficits indexed by a distinct list

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

proven

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    Natural Language Statement

    Lemma

    For distinct remaining indices, the finite family dimension deficit is exactly the mapped list deficit used by greedy exposure.

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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 finset_span_eq proven

    2 list_mapped_deficit proven

    Lean source view on GitHub

    1import Lax342547.SumEnvelope
    2import Lax342547.MappedSpans
    3
    4/-!
    5---
    6title: Actual deficits indexed by a distinct list
    7type: lemma
    8---
    9For distinct remaining indices, the finite family dimension deficit is exactly the mapped list deficit used by greedy exposure.
    10-/
    11
    12namespace Lax342547.ListDeficits
    13
    14open Lax342547.SumEnvelope Lax342547.MappedSpans
    15open scoped BigOperators
    16
    17axiom finset_span_eq {K V ι : Type} [Field K] [DecidableEq ι]
    18 [AddCommGroup V] [Module K V] (S : ι → Submodule K V) (I : Finset ι) :
    19 (⨆ i : I, S i.val) = I.sup S
    20
    21axiom list_mapped_deficit {K V U ι : Type} [Field K] [DecidableEq ι]
    22 [AddCommGroup V] [Module K V] [AddCommGroup U] [Module K U]
    23 [FiniteDimensional K V] [FiniteDimensional K U]
    24 (S : ι → Submodule K V) (f : V →ₗ[K] U) (l : List ι) (hl : l.Nodup) :
    25 dimensionDeficit (fun i : l.toFinset => (S i.val).map f) = mappedDeficit S f l
    26
    27end Lax342547.ListDeficits
    28
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    Builds on
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    From Mathlib

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