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Projected nonzero terms in the actual remaining sum

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

proven

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    Natural Language Statement

    Lemma

    The rank-deficit inequality applied to actual column quotients and row restrictions bounds nonzero remaining terms by the projected sum rank and both original-space exposure deficits.

    Concept map
    21 concepts
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    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.ListDeficits
    2import Lax342547.CoverProjection
    3
    4/-!
    5---
    6title: Projected nonzero terms in the actual remaining sum
    7type: lemma
    8---
    9The rank-deficit inequality applied to actual column quotients and row restrictions bounds nonzero remaining terms by the projected sum rank and both original-space exposure deficits.
    10-/
    11
    12namespace Lax342547.ProjectedCounts
    13
    14open Lax342547.CoverProjection Lax342547.GreedyTails
    15open scoped BigOperators
    16
    17axiom projected_nonzero_count {K V W ι : Type} [Field K] [DecidableEq ι]
    18 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    19 [FiniteDimensional K V] [FiniteDimensional K W]
    20 (M : ι → V →ₗ[K] W) (S : Submodule K W) (T : Submodule K (Module.Dual K V))
    21 (l : List ι) (hl : l.Nodup) : by
    22 classical
    23 exact ((Finset.univ.filter (fun i : l.toFinset => projection (M i.val) S T ≠ 0)).card : ℝ) ≤
    24 Module.finrank K (LinearMap.range (projection (∑ i : l.toFinset, M i.val) S T))+
    25 deficit (fun i => LinearMap.range (M i)) S l+
    26 deficit (fun i => LinearMap.range (M i).dualMap) T l
    27
    28end Lax342547.ProjectedCounts
    29
    Show Proof
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    From Mathlib

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