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Exact common predictions on retained mass

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

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    Natural Language Statement

    Lemma

    An actual average-error bound retains a large set of units; Boolean polynomial nonvanishing turns their approximate predictions into one exact common function.

    Concept map
    18 concepts; 1 descendant hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 common_prediction_exact proven

    2 discrepancy_mass_union proven

    3 prediction_good_mass proven

    4 retained_common_prediction proven

    Lean source view on GitHub

    1import Lax342547.BooleanWeight
    2import Lax342547.MomentTails
    3import Lax342547.FiniteSampling
    4
    5/-!
    6---
    7title: Exact common predictions on retained mass
    8type: lemma
    9---
    10An actual average-error bound retains a large set of units; Boolean polynomial nonvanishing turns their approximate predictions into one exact common function.
    11-/
    12
    13namespace Lax342547.CommonExactification
    14
    15open Lax342547.MomentSpace Lax342547.SelectorInterpolation Lax342547.RetainedImages
    16open Lax342547.RelativeEntropy
    17open scoped BigOperators
    18
    19axiom discrepancy_mass_union {X : Type} [Fintype X]
    20 (α : X → ℝ) (f p q : X → Binary) (hα : ∀ x, 0 ≤ α x) :
    21 cellMass α (fun x => p x ≠ q x) ≤
    22 cellMass α (fun x => p x ≠ f x)+cellMass α (fun x => q x ≠ f x)
    23
    24axiom common_prediction_exact {b d : ℕ}
    25 (f p q : (Fin b → Binary) → Binary)
    26 (hp : p ∈ selectorSpan b d) (hq : q ∈ selectorSpan b d)
    27 (herr : cellMass (fun _ : Fin b → Binary => 1/(2 : ℝ)^b) (fun x => p x ≠ f x)+
    28 cellMass (fun _ : Fin b → Binary => 1/(2 : ℝ)^b) (fun x => q x ≠ f x) < 1/(2 : ℝ)^d) : p = q
    29
    30axiom prediction_good_mass {U X : Type} [Fintype U] [Fintype X]
    31 (β : U → ℝ) (α : X → ℝ) (f : X → Binary) (p : U → X → Binary) (ε a : ℝ)
    32 (hβ : Probability β) (hα : ∀ x, 0 ≤ α x) (ha : 0 < a)
    33 (herror : (∑ u, β u*cellMass α (fun x => p u x ≠ f x)) ≤ ε) :
    34 1-ε/a ≤ cellMass β (fun u => cellMass α (fun x => p u x ≠ f x) < a)
    35
    36axiom retained_common_prediction {U : Type} [Fintype U] {b d : ℕ}
    37 (β : U → ℝ) (f : (Fin b → Binary) → Binary)
    38 (p : U → (Fin b → Binary) → Binary) (ε a : ℝ)
    39 (hβ : Probability β) (hp : ∀ u, p u ∈ selectorSpan b d)
    40 (ha : 0 < a) (hε : ε < a) (hdegree : 2*a ≤ 1/(2 : ℝ)^d)
    41 (herror : (∑ u, β u*cellMass (fun _ : Fin b → Binary => 1/(2 : ℝ)^b)
    42 (fun x => p u x ≠ f x)) ≤ ε) :
    43 ∃ u₀ : U, 1-ε/a ≤ cellMass β (fun u => p u = p u₀)
    44
    45end Lax342547.CommonExactification
    46
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