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Formal ordered product bits realize symmetric corrections

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

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    Natural Language Statement

    Lemma

    For every ordered pair there is at least one allowed product index. One such term realizes any off-diagonal correction to a symmetric matrix, while symmetrization keeps its diagonal fixed. These are formal bit prescriptions; existence of actual parameters satisfying them is a separate probabilistic assertion.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
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    1import Lax342547.MomentSpace
    2import Mathlib.Algebra.Module.ZMod
    3
    4/-!
    5---
    6title: Formal ordered product bits realize symmetric corrections
    7type: lemma
    8---
    9For every ordered pair there is at least one allowed product index. One
    10such term realizes any off-diagonal correction to a symmetric matrix,
    11while symmetrization keeps its diagonal fixed. These are formal bit
    12prescriptions; existence of actual parameters satisfying them is a
    13separate probabilistic assertion.
    14-/
    15
    16namespace Lax342547.BinaryPrescriptions
    17
    18open Lax342547.MomentSpace
    19
    20def productSum {I Term : Type} [Fintype Term] (L R : I → I → Term → Binary) (x y : I) : Binary :=
    21 ∑ t, L x y t * R x y t
    22
    23axiom realize {I Term : Type} [Fintype I] [Fintype Term]
    24 (allowed : I → I → Term → Prop) (hallowed : ∀ x y, ∃ t, allowed x y t)
    25 (T₀ M : Matrix I I Binary) (hT : ∀ x y, T₀ x y = T₀ y x)
    26 (hM : ∀ x y, M x y = M y x) (hdiag : ∀ x, M x x = T₀ x x) :
    27 ∃ L R : I → I → Term → Binary,
    28 (∀ x y t, ¬ allowed x y t → L x y t = 0 ∧ R x y t = 0) ∧
    29 ∀ x y, T₀ x y + productSum L R x y + productSum L R y x = M x y
    30
    31end Lax342547.BinaryPrescriptions
    32
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