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Two-sided Gram normalization for individually injective frames

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

proven

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    Natural Language Statement

    Lemma

    Lemma 4.2, stated with exact finite probabilities and ratios of powers of two. Subtraction in the extended nonnegative reals is truncated at zero; when p+q≤N this is the displayed lower bound in the paper.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 column_failure proven

    2 normalization_lower proven

    3 normalization_quarter proven

    4 normalization_upper proven

    Lean source view on GitHub

    1import Lax342547.GramColumns
    2
    3/-!
    4---
    5title: Two-sided Gram normalization for individually injective frames
    6type: lemma
    7---
    8Lemma 4.2, stated with exact finite probabilities and ratios of powers of
    9two. Subtraction in the extended nonnegative reals is truncated at zero;
    10when p+q≤N this is the displayed lower bound in the paper.
    11-/
    12
    13namespace Lax342547.GramNormalization
    14
    15open Lax342547.MomentSpace
    16open scoped ENNReal
    17
    18noncomputable def probability {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    19 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary) : ℝ≥0∞ :=
    20 (PMF.uniformOfFintype (Matrix N I Binary × Matrix N J Binary)).toOuterMeasure
    21 {z | Function.Injective z.1.mulVec ∧ Function.Injective z.2.mulVec ∧ z.1.transpose * z.2 = G}
    22
    23axiom column_failure {J N : Type} [Fintype J] [Fintype N] [DecidableEq J] [DecidableEq N] :
    24 (PMF.uniformOfFintype (Matrix N J Binary)).toOuterMeasure {A | ¬ Function.Injective A.mulVec} ≤
    25 (2 : ℝ≥0∞) ^ Fintype.card J / 2 ^ Fintype.card N
    26
    27axiom normalization_upper {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    28 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary) :
    29 probability (N := N) G ≤ 1 / (2 : ℝ≥0∞) ^ (Fintype.card I * Fintype.card J)
    30
    31axiom normalization_lower {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    32 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary) :
    33 (1 / (2 : ℝ≥0∞) ^ (Fintype.card I * Fintype.card J)) *
    34 (1 - (2 : ℝ≥0∞) ^ Fintype.card I / 2 ^ Fintype.card N) *
    35 (1 - (2 : ℝ≥0∞) ^ (Fintype.card I + Fintype.card J) / 2 ^ Fintype.card N) ≤
    36 probability (N := N) G
    37
    38axiom normalization_quarter {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    39 [DecidableEq I] [DecidableEq J] [DecidableEq N] (G : Matrix I J Binary)
    40 (hN : Fintype.card I + Fintype.card J + 1 ≤ Fintype.card N) :
    41 1 / (2 : ℝ≥0∞) ^ (Fintype.card I * Fintype.card J + 2) ≤ probability (N := N) G
    42
    43end Lax342547.GramNormalization
    44
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