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Nonzero tensor count from span deficits

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

proven

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    Natural Language Statement

    Lemma

    For the actual sum of maps, column and row envelopes bound the number of nonzero terms by the sum rank plus their two actual dimension deficits.

    Concept map
    9 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 natural_deficit_cast proven

    2 nonzero_sum_envelope proven

    3 span_dimension_sum_bound proven

    Lean source view on GitHub

    1import Lax342547.SumRank
    2import Lax342547.SpanDeficitMono
    3import Mathlib.Data.Real.Basic
    4
    5/-!
    6---
    7title: Nonzero tensor count from span deficits
    8type: lemma
    9---
    10For the actual sum of maps, column and row envelopes bound the number of nonzero terms by the sum rank plus their two actual dimension deficits.
    11-/
    12
    13namespace Lax342547.SumEnvelope
    14
    15open Lax342547.SpanDeficits
    16open scoped BigOperators
    17
    18noncomputable def dimensionDeficit {K V ι : Type} [Field K] [Fintype ι]
    19 [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    20 (S : ι → Submodule K V) : ℝ :=
    21 (∑ i, (Module.finrank K (S i) : ℝ))-Module.finrank K (⨆ i, S i : Submodule K V)
    22
    23axiom span_dimension_sum_bound {K V ι : Type} [Field K] [Fintype ι]
    24 [AddCommGroup V] [Module K V] [FiniteDimensional K V] (S : ι → Submodule K V) :
    25 Module.finrank K (⨆ i, S i : Submodule K V) ≤ ∑ i, Module.finrank K (S i)
    26
    27axiom natural_deficit_cast {K V ι : Type} [Field K] [Fintype ι]
    28 [AddCommGroup V] [Module K V] [FiniteDimensional K V] (S : ι → Submodule K V) :
    29 (((∑ i, Module.finrank K (S i))-Module.finrank K (⨆ i, S i : Submodule K V) : ℕ) : ℝ) =
    30 dimensionDeficit S
    31
    32axiom nonzero_sum_envelope {K V W ι : Type} [Field K] [Fintype ι]
    33 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    34 [FiniteDimensional K V] [FiniteDimensional K W]
    35 (M : ι → V →ₗ[K] W) (S : ι → Submodule K W) (T : ι → Submodule K (Module.Dual K V))
    36 (hS : ∀ i, LinearMap.range (M i) ≤ S i) (hT : ∀ i, LinearMap.range (M i).dualMap ≤ T i) : by
    37 classical
    38 exact ((Finset.univ.filter (fun i => M i ≠ 0)).card : ℝ) ≤
    39 Module.finrank K (LinearMap.range (∑ i, M i))+dimensionDeficit S+dimensionDeficit T
    40
    41end Lax342547.SumEnvelope
    42
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