Provedllmlarge-language-modelsformal-proofposition-encoding

Sinusoidal components are bounded by one

For all $\htmlData{mx=1}{r, \mathit{pos} \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=5}{\left\lvert \htmlData{mx=4}{\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{pe}}(\htmlData{mx=1}{r}, \htmlData{mx=2}{\mathit{pos}})}_{1}} \right\rvert} \mathrel{\htmlData{mx=6}{\leq}} 1 \mathbin{\htmlData{mx=14}{\land}} \htmlData{mx=12}{\left\lvert \htmlData{mx=11}{\htmlData{mx=10}{\htmlData{mx=7}{\operatorname{pe}}(\htmlData{mx=8}{r}, \htmlData{mx=9}{\mathit{pos}})}_{2}} \right\rvert} \mathrel{\htmlData{mx=13}{\leq}} 1

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: r, pos range over real numbers: the statement must hold for every choice. ((r pos : ℝ))
|x|Absolute value: The size of pe(r, pos).1 without its sign. (|(LLMPE.pe r pos).1|)
≤At most: |pe(r, pos).1| is less than or equal to 1. (|(LLMPE.pe r pos).1| ≤ 1)
∧And: Both hold: |pe(r, pos).1| ≤ 1, and |pe(r, pos).2| ≤ 1. (|(LLMPE.pe r pos).1| ≤ 1 ∧ |(LLMPE.pe r…)
p₁Component: Component 1 of pe(r, pos) (the first). ((LLMPE.pe r pos).1)
pe: LLMPE.pe: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMPE.pe)
f(x)Function application: LLMPE.pe applied to r, pos. Lean writes f x; papers write f(x). (LLMPE.pe r pos)

theorem pe-bounded (r pos : ℝ) :
    |(LLMPE.pe r pos).1| ≤ 1 ∧ |(LLMPE.pe r pos).2| ≤ 1

|sin| ≤ 1 and |cos| ≤ 1 for every encoding pair.

How it was proved

Proved: 27 September 2026
By: Ajax, with the agent “ompmuse”
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: 13 lines of Lean