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The exact space of binary base moments

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

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    Natural Language Statement

    Lemma

    The first assertion of Lemma 5.1: the span of (1,z)(1,z)T(1,z)(1,z)^T consists exactly of the symmetric matrices satisfying Zii=Z0iZ_{ii}=Z_{0i}. The distinguished constant coordinate is nonenone.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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    1import Lax342547.MomentSpace
    2
    3/-!
    4---
    5title: The exact space of binary base moments
    6type: lemma
    7---
    8The first assertion of Lemma 5.1: the span of (1,z)(1,z)T(1,z)(1,z)^T consists
    9exactly of the symmetric matrices satisfying Zii=Z0iZ_{ii}=Z_{0i}.
    10The distinguished constant coordinate is `none`.
    11-/
    12
    13namespace Lax342547.BaseMoments
    14
    15open Lax342547.MomentSpace
    16
    17def baseMoment {Base : Type} (z : Base → Binary) : Matrix (Option Base) (Option Base) Binary :=
    18 fun i j => baseEval z i * baseEval z j
    19
    20def baseMomentSpace (Base : Type) : Submodule Binary (Matrix (Option Base) (Option Base) Binary) :=
    21 Submodule.span Binary (Set.range (baseMoment (Base := Base)))
    22
    23def IsBaseMoment {Base : Type} (Z : Matrix (Option Base) (Option Base) Binary) : Prop :=
    24 (∀ i j, Z i j = Z j i) ∧ ∀ i, Z i i = Z none i
    25
    26axiom mem_baseMomentSpace_iff {Base : Type} [Fintype Base]
    27 (Z : Matrix (Option Base) (Option Base) Binary) :
    28 Z ∈ baseMomentSpace Base ↔ IsBaseMoment Z
    29
    30end Lax342547.BaseMoments
    31
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