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Projection removes the exposed terms

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

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    Natural Language Statement

    Lemma

    The projection of the actual remaining sum equals the projection of the full sum because the exposed column span kills each exposed term.

    Concept map
    24 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.PartitionLists
    2import Lax342547.CoverProjection
    3import Lax342547.ListDeficits
    4
    5/-!
    6---
    7title: Projection removes the exposed terms
    8type: lemma
    9---
    10The projection of the actual remaining sum equals the projection of the full sum because the exposed column span kills each exposed term.
    11-/
    12
    13namespace Lax342547.ExposedSums
    14
    15open Lax342547.GreedySpans Lax342547.CoverProjection
    16open scoped BigOperators
    17
    18axiom remaining_projection_sum {K V W ι : Type} [Field K] [Fintype ι] [DecidableEq ι]
    19 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    20 (M : ι → V →ₗ[K] W) (l : List ι) (hl : l.Nodup) (hI : l.toFinset = Finset.univ) (t : ℕ) :
    21 let S := spanAfter (fun i => LinearMap.range (M i)) ⊥ (l.take t)
    22 let T := spanAfter (fun i => LinearMap.range (M i).dualMap) ⊥ (l.take t)
    23 projection (∑ i : (l.drop t).toFinset, M i.val) S T = projection (∑ i, M i) S T
    24
    25end Lax342547.ExposedSums
    26
    Show Proof
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    From Mathlib

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