Provedalphabaguettebaguette-zquantizer

The dyadic quantizer is monotone

For all $\htmlData{mx=1}{k \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=1}{x, y \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=0}{x} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{y}$ , then

\htmlData{mx=4}{\left\lfloor \htmlData{mx=3}{\htmlData{mx=0}{x} \cdot \htmlData{mx=2}{2^{\htmlData{mx=1}{k}}}} \right\rfloor} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{\left\lfloor \htmlData{mx=8}{\htmlData{mx=5}{y} \cdot \htmlData{mx=7}{2^{\htmlData{mx=6}{k}}}} \right\rfloor}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: x, y range over real numbers: the statement must hold for every choice. ((x y : ℝ))
≤At most: x is less than or equal to y. (x ≤ y)
⌊x⌋Floor: The largest integer not above x · 2^k. (⌊x * 2 ^ k⌋)
·Multiplication: x times 2^k. (x * 2 ^ k)
aᵇPower: 2 raised to the power k. (2 ^ k)

theorem bz-dyadic-floor-monotone (k : ℕ) (x y : ℝ) (h : x ≤ y) :
    ⌊x * 2 ^ k⌋ ≤ ⌊y * 2 ^ k⌋

x ≤ y implies Q(x) ≤ Q(y): quantization never reorders values, so max-subtraction and comparisons survive it. Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 12 lines of Lean

The dyadic quantizer is monotone

How it was proved

More proved statements