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A selected affine ray determines its moment block

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

proven

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    Natural Language Statement

    Lemma

    An explicit projection onto the quotient by an affine point ray. The Boolean moment diagonal identity removes the mixed first-row terms in a symmetric tensor killed by that quotient in both factors (§6).

    Concept map
    3 concepts
    100%
    Proven claimDefinitionThis 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 moment_ray proven

    2 projection_kernel proven

    Lean source view on GitHub

    1import Lax342547.BaseMoments
    2import Mathlib.LinearAlgebra.Matrix.ToLin
    3
    4/-!
    5---
    6title: A selected affine ray determines its moment block
    7type: lemma
    8---
    9An explicit projection onto the quotient by an affine point ray. The
    10Boolean moment diagonal identity removes the mixed first-row terms in
    11a symmetric tensor killed by that quotient in both factors (§6).
    12-/
    13
    14namespace Lax342547.SelectedMoments
    15
    16open Lax342547.MomentSpace Lax342547.BaseMoments
    17
    18def rayProjection {Base : Type} [Fintype Base] [DecidableEq Base]
    19 (z : Base → Binary) : Matrix (Option Base) (Option Base) Binary :=
    20 1 - Matrix.vecMulVec (baseEval z) (Pi.single none 1)
    21
    22axiom projection_kernel {Base : Type} [Fintype Base] [DecidableEq Base]
    23 (z : Base → Binary) :
    24 LinearMap.ker (rayProjection z).mulVecLin =
    25 Submodule.span Binary {baseEval z}
    26
    27axiom moment_ray {Base : Type} [Fintype Base] [DecidableEq Base]
    28 (z : Base → Binary) (Z : Matrix (Option Base) (Option Base) Binary)
    29 (hZ : IsBaseMoment Z)
    30 (h : rayProjection z * Z * (rayProjection z).transpose = 0) :
    31 Z = Z none none • baseMoment z
    32
    33end Lax342547.SelectedMoments
    34
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