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Greedy independent failure samples

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

proven

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    Natural Language Statement

    Lemma

    Actual sequential failure tests produce independent pin spaces and retain a uniform amplified lower probability.

    Concept map
    16 concepts; 7 descendants hidden
    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 accepted_spaces_independent proven

    2 greedy_failure_lower_probability proven

    Lean source view on GitHub

    1import Lax342547.SequentialTests
    2import Lax342547.GreedySpaces
    3import Lax342547.FiniteSampling
    4import Mathlib.LinearAlgebra.DFinsupp
    5import Mathlib.Data.Fin.Tuple.Basic
    6
    7/-!
    8---
    9title: Greedy independent failure samples
    10type: lemma
    11---
    12Actual sequential failure tests produce independent pin spaces and retain a uniform amplified lower probability.
    13-/
    14
    15namespace Lax342547.IndependentSampling
    16
    17open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.RetainedImages Lax342547.FiniteSampling
    18open scoped BigOperators
    19
    20axiom accepted_spaces_independent {K V Ω : Type} [Field K] [AddCommGroup V] [Module K V]
    21 (U : Ω → Submodule K V) (F : Ω → Prop) (n : ℕ) (s : Fin n → Ω)
    22 (hs : Lax342547.SequentialTests.accepts
    23 (fun j t x => F x ∧ Disjoint (U x) (⨆ i : Fin j,U (t i))) n s) :
    24 iSupIndep (fun i => U (s i))
    25
    26axiom greedy_failure_lower_probability {K V Ω : Type} [Field K] [AddCommGroup V] [Module K V]
    27 [Fintype Ω] (U : Ω → Submodule K V) (μ : Ω → ℝ) (F : Ω → Prop)
    28 (η δ : ℝ) (n : ℕ) (hμ : Probability μ) (hgap : δ ≤ η)
    29 (hF : η ≤ cellMass μ F)
    30 (havoid : ∀ j,j < n → ∀ t : Fin j → Ω,
    31 cellMass μ (fun x => ¬ Disjoint (U x) (⨆ i : Fin j,U (t i))) ≤ δ) :
    32 (η-δ)^n ≤ cellMass (productLaw (fun _ : Fin n => μ))
    33 (fun s => Lax342547.SequentialTests.accepts
    34 (fun j t x => F x ∧ Disjoint (U x) (⨆ i : Fin j,U (t i))) n s)
    35
    36end Lax342547.IndependentSampling
    37
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