Provedllmlarge-language-modelsformal-proofdecoding

The greedy index is 0 or 1

For all $\htmlData{mx=1}{x, y \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{argmax2}}(\htmlData{mx=1}{x}, \htmlData{mx=2}{y})} \mathrel{\htmlData{mx=4}{=}} 0 \mathbin{\htmlData{mx=10}{\lor}} \htmlData{mx=8}{\htmlData{mx=5}{\operatorname{argmax2}}(\htmlData{mx=6}{x}, \htmlData{mx=7}{y})} \mathrel{\htmlData{mx=9}{=}} 1

What the notation means · or point at any symbol

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))
argmax2: LLMDecode.argmax2: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMDecode.argmax2)
f(x)Function application: LLMDecode.argmax2 applied to x, y. Lean writes f x; papers write f(x). (LLMDecode.argmax2 x y)
=Equality: argmax2(x, y) and 0 are the same. (LLMDecode.argmax2 x y = 0)
∨Or: At least one holds: argmax2(x, y) = 0, or argmax2(x, y) = 1. (LLMDecode.argmax2 x y = 0 ∨ LLMDecode.a…)

theorem dec-argmax2-cases (x y : ℝ) :
    LLMDecode.argmax2 x y = 0 ∨ LLMDecode.argmax2 x y = 1

A two-way greedy choice lands on one of the two indices.

How it was proved

Proved: 28 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: 15 lines of Lean