Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Actual dyadic deficit recurrence

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    For nonnegative decreasing span increments, tail deficits are nonnegative and satisfy the dyadic averaging recurrence.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 dyadic_recurrence proven

    2 tail_deficit_le proven

    3 tail_deficit_nonneg proven

    Lean source view on GitHub

    1import Lax342547.DyadicDeficits
    2import Mathlib.Algebra.BigOperators.Field
    3
    4/-!
    5---
    6title: Actual dyadic deficit recurrence
    7type: lemma
    8---
    9For nonnegative decreasing span increments, tail deficits are nonnegative and satisfy the dyadic averaging recurrence.
    10-/
    11
    12namespace Lax342547.DyadicTails
    13
    14open scoped BigOperators
    15
    16noncomputable def tailSum (a : ℕ → ℝ) (s b : ℕ) : ℝ := ∑ t ∈ Finset.range b, a (s-b+t)
    17noncomputable def tailDeficit (a : ℕ → ℝ) (s b : ℕ) : ℝ := a (s-b)-tailSum a s b/b
    18
    19axiom tail_deficit_nonneg (a : ℕ → ℝ) (ha : Antitone a) (s b : ℕ) (hb : 0 < b) :
    20 0 ≤ tailDeficit a s b
    21
    22axiom tail_deficit_le (a : ℕ → ℝ) (ha : ∀ t, 0 ≤ a t) (s b : ℕ) :
    23 tailDeficit a s b ≤ a (s-b)
    24
    25axiom dyadic_recurrence (a : ℕ → ℝ) (ha : Antitone a) (s b : ℕ)
    26 (hb : 0 < b) (hbs : 2*b ≤ s) :
    27 2*tailDeficit a s (2*b) ≤ 2*(a (s-2*b)-a (s-b))+tailDeficit a s b
    28
    29end Lax342547.DyadicTails
    30
    Show ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…