Provedllmlarge-language-modelsformal-proofposition-encoding

Position zero has the canonical encoding

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{pe}}(\htmlData{mx=1}{r}, 0)} \mathrel{\htmlData{mx=4}{=}} \htmlData{mx=3}{\left( 0, 1 \right)}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
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, 0. Lean writes f x; papers write f(x). (LLMPE.pe r 0)
=Equality: pe(r, 0) and (0, 1) are the same. (LLMPE.pe r 0 = (0, 1))
(a, b)Tuple: The ordered pair (or tuple) of 0, 1. ((0, 1))

theorem pe-zero-position (r : ℝ) :
    LLMPE.pe r 0 = (0, 1)

pe r 0 = (0, 1): the origin is encoded by sine zero and cosine one.

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