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Image caps inside original retained cells

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

proven

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    Natural Language Statement

    Lemma

    Conditioning an original unary record cell costs only its own mass. The joint key and fresh-image cap gives the .95tN exponent, without intersecting cells over opposite parameters.

    Concept map
    4 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 conditional_image_cap proven

    2 conditional_law_nonneg proven

    3 conditional_law_total proven

    4 paper_cell_exponent proven

    5 paper_conditional_cap proven

    Lean source view on GitHub

    1import Lax342547.PushforwardWalsh
    2import Mathlib.Analysis.SpecialFunctions.Pow.Real
    3
    4/-!
    5---
    6title: Image caps inside original retained cells
    7type: lemma
    8---
    9Conditioning an original unary record cell costs only its own mass. The joint key and fresh-image cap gives the .95tN exponent, without intersecting cells over opposite parameters.
    10-/
    11
    12namespace Lax342547.RetainedImages
    13
    14open Lax342547.PushforwardWalsh
    15
    16noncomputable def cellMass {Ω : Type} [Fintype Ω] (ρ : Ω → ℝ) (C : Ω → Prop) : ℝ := by
    17 classical
    18 exact ∑ ω, if C ω then ρ ω else 0
    19
    20noncomputable def conditionalLaw {Ω : Type} [Fintype Ω] (ρ : Ω → ℝ) (C : Ω → Prop) : Ω → ℝ := by
    21 classical
    22 exact fun ω => (if C ω then ρ ω else 0) / cellMass ρ C
    23
    24axiom conditional_law_nonneg {Ω : Type} [Fintype Ω] (ρ : Ω → ℝ) (C : Ω → Prop)
    25 (hρ : ∀ ω, 0 ≤ ρ ω) (ω : Ω) : 0 ≤ conditionalLaw ρ C ω
    26
    27axiom conditional_law_total {Ω : Type} [Fintype Ω] (ρ : Ω → ℝ) (C : Ω → Prop)
    28 (hC : 0 < cellMass ρ C) : ∑ ω, conditionalLaw ρ C ω = 1
    29
    30axiom conditional_image_cap {Ω X : Type} [Fintype Ω]
    31 (ρ : Ω → ℝ) (C : Ω → Prop) (image : Ω → X) (joint : Ω → Prop)
    32 (τ cap : ℝ) (hρ : ∀ ω, 0 ≤ ρ ω) (hτ : 0 < τ)
    33 (hC : τ ≤ cellMass ρ C) (hCJ : ∀ ω, C ω → joint ω)
    34 (hcap : ∀ x, cellMass ρ (fun ω => joint ω ∧ image ω = x) ≤ cap) (x : X) :
    35 push (conditionalLaw ρ C) image x ≤ cap / τ
    36
    37axiom paper_cell_exponent (ζ k t : ℝ) (_hζ : 0 ≤ ζ)
    38 (hζsmall : ζ ≤ 1 / 1000) (hk : 2 * ζ * k ≤ 1 / 500) (ht : 1 ≤ t) :
    39 -(1 - 2 * ζ) * (k + t) + (k + 1 / 100) ≤ -(95 / 100 : ℝ) * t
    40
    41axiom paper_conditional_cap (ζ k t N : ℝ) (hζ : 0 ≤ ζ)
    42 (hζsmall : ζ ≤ 1 / 1000) (hk : 2 * ζ * k ≤ 1 / 500) (ht : 1 ≤ t) (hN : 0 ≤ N) :
    43 (2 : ℝ) ^ (-(1 - 2 * ζ) * (k + t) * N) /
    44 (2 : ℝ) ^ (-(k + 1 / 100) * N) ≤ (2 : ℝ) ^ (-(95 / 100 : ℝ) * t * N)
    45
    46end Lax342547.RetainedImages
    47
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