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Identification of the lifted tensor sum

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

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    Natural Language Statement

    Lemma

    The abstract block addition map is exactly the sum of the individually included tensors with their actual dual restrictions.

    Concept map
    7 concepts
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    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.BlockDeficits
    2
    3/-!
    4---
    5title: Identification of the lifted tensor sum
    6type: lemma
    7---
    8The abstract block addition map is exactly the sum of the individually included tensors with their actual dual restrictions.
    9-/
    10
    11namespace Lax342547.LiftedSum
    12
    13open Lax342547.SpanDeficits Lax342547.BlockDeficits
    14open scoped BigOperators
    15
    16axiom lifted_sum_apply {K V W ι : Type} [Field K] [Fintype ι]
    17 [AddCommGroup V] [Module K V] [AddCommGroup W] [Module K W]
    18 (S : ι → Submodule K W) (T : ι → Submodule K V)
    19 (M : ∀ i, Module.Dual K (T i) →ₗ[K] S i) (f : Module.Dual K V) :
    20 liftedSum S T M f = ∑ i, (M i ((T i).subtype.dualMap f)).val
    21
    22end Lax342547.LiftedSum
    23
    Show Proof
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