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Common predictions from actual point tensors

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

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    Natural Language Statement

    Lemma

    Degree-two actual point-tensor evaluations yield an exact common marked prediction after an explicitly bounded mass loss.

    Concept map
    22 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 marked_point_degree proven

    2 retained_point_prediction proven

    Lean source view on GitHub

    1import Lax342547.CommonExactification
    2import Lax342547.PointQuadratics
    3import Lax342547.BooleanWeight
    4
    5/-!
    6---
    7title: Common predictions from actual point tensors
    8type: lemma
    9---
    10Degree-two actual point-tensor evaluations yield an exact common marked prediction after an explicitly bounded mass loss.
    11-/
    12
    13namespace Lax342547.PointPrediction
    14
    15open Lax342547.MomentSpace Lax342547.SelectorInterpolation Lax342547.RetainedImages
    16open Lax342547.RelativeEntropy
    17open scoped BigOperators
    18
    19axiom marked_point_degree {b : ℕ} {Label Coord : Type} [Fintype Coord]
    20 (p : Label → Coord → Binary) (s : Label)
    21 (φ : Module.Dual Binary (Matrix (Coord × Option (Fin b)) (Coord × Option (Fin b)) Binary))
    22 (α : Binary) (q : (Fin b → Binary) → Binary) (hq : q ∈ selectorSpan b 2) :
    23 (fun x => φ (pointMoment p s x)+α*q x) ∈ selectorSpan b 2
    24
    25axiom retained_point_prediction {U Label Coord : Type} [Fintype U] [Fintype Coord] {b : ℕ}
    26 (β : U → ℝ) (f : (Fin b → Binary) → Binary) (p : Label → Coord → Binary) (s : Label)
    27 (φ : U → Module.Dual Binary (Matrix (Coord × Option (Fin b)) (Coord × Option (Fin b)) Binary))
    28 (α : U → Binary) (q : (Fin b → Binary) → Binary) (ε : ℝ)
    29 (hβ : Probability β) (hq : q ∈ selectorSpan b 2) (hε : ε < 1/16)
    30 (herror : (∑ u, β u*cellMass (fun _ : Fin b → Binary => 1/(2 : ℝ)^b)
    31 (fun x => φ u (pointMoment p s x)+α u*q x ≠ f x)) ≤ ε) :
    32 ∃ u₀ : U, 1-16*ε ≤ cellMass β (fun u =>
    33 (fun x => φ u (pointMoment p s x)+α u*q x) =
    34 (fun x => φ u₀ (pointMoment p s x)+α u₀*q x))
    35
    36end Lax342547.PointPrediction
    37
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