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Dimension deficits after an arbitrary mode map

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

proven

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    Natural Language Statement

    Lemma

    Actual mapped mode dimensions equal their increments modulo the map kernel; mapped span deficits therefore equal the corresponding exposure deficits.

    Concept map
    11 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 mapped_deficit_eq proven

    2 mapped_dimension_gain proven

    Lean source view on GitHub

    1import Lax342547.ActualDeficits
    2
    3/-!
    4---
    5title: Dimension deficits after an arbitrary mode map
    6type: lemma
    7---
    8Actual mapped mode dimensions equal their increments modulo the map kernel; mapped span deficits therefore equal the corresponding exposure deficits.
    9-/
    10
    11namespace Lax342547.MappedSpans
    12
    13open Lax342547.GreedySpans Lax342547.GreedyTails
    14open scoped BigOperators
    15
    16axiom mapped_dimension_gain {K V U : Type} [Field K] [AddCommGroup V] [Module K V]
    17 [AddCommGroup U] [Module K U] [FiniteDimensional K V]
    18 (f : V →ₗ[K] U) (S : Submodule K V) :
    19 (Module.finrank K (S.map f) : ℝ) = gain (LinearMap.ker f) S
    20
    21noncomputable def mappedDeficit {K V U ι : Type} [Field K] [DecidableEq ι]
    22 [AddCommGroup V] [Module K V] [AddCommGroup U] [Module K U] [FiniteDimensional K V]
    23 (S : ι → Submodule K V) (f : V →ₗ[K] U) (l : List ι) : ℝ :=
    24 (l.map (fun i => (Module.finrank K ((S i).map f) : ℝ))).sum-
    25 Module.finrank K (l.toFinset.sup (fun i => (S i).map f) : Submodule K U)
    26
    27axiom mapped_deficit_eq {K V U ι : Type} [Field K] [DecidableEq ι]
    28 [AddCommGroup V] [Module K V] [AddCommGroup U] [Module K U] [FiniteDimensional K V]
    29 (S : ι → Submodule K V) (f : V →ₗ[K] U) (l : List ι) :
    30 mappedDeficit S f l = deficit S (LinearMap.ker f) l
    31
    32end Lax342547.MappedSpans
    33
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