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A small dyadic tail scale

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

proven

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    Natural Language Statement

    Lemma

    The actual dyadic recurrence and the total rank budget give a scale with tail deficit ratio less than one quarter.

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

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

    1 dyadic_small_tail proven

    2 single_scale proven

    Lean source view on GitHub

    1import Lax342547.DyadicTails
    2
    3/-!
    4---
    5title: A small dyadic tail scale
    6type: lemma
    7---
    8The actual dyadic recurrence and the total rank budget give a scale with tail deficit ratio less than one quarter.
    9-/
    10
    11namespace Lax342547.DyadicScale
    12
    13open Lax342547.DyadicTails
    14open scoped BigOperators
    15
    16axiom single_scale (f v : ℕ → ℝ) (n : ℕ) (r : ℝ)
    17 (hrec : ∀ j < n, 2*v j ≤ 2*(f j-f (j+1))+v (j+1))
    18 (hf0 : f 0 ≤ r) (hfn : 0 ≤ f n) (hv0 : 0 ≤ v 0) (hvn : v n ≤ f n)
    19 (hn : 8*r < n) : ∃ j < n, v j < 1/4
    20
    21axiom dyadic_small_tail (a : ℕ → ℝ) (ha : Antitone a) (ha0 : ∀ t, 0 ≤ a t)
    22 (k n : ℕ) (r : ℝ) (hnk : n ≤ k) (har : a 0 ≤ r) (hn : 8*r < n) :
    23 ∃ j < n, tailDeficit a (2^k) (2^(k-j)) < 1/4
    24
    25end Lax342547.DyadicScale
    26
    Show ProofShow Proof
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    From Mathlib

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