Provedllmlarge-language-modelsformal-proofdecoding

Greedy tie-break picks the first index

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

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{argmax2}}(\htmlData{mx=1}{x}, \htmlData{mx=2}{x})} \mathrel{\htmlData{mx=4}{=}} 0

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (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, x. Lean writes f x; papers write f(x). (LLMDecode.argmax2 x x)
=Equality: argmax2(x, x) and 0 are the same. (LLMDecode.argmax2 x x = 0)

theorem dec-argmax2-tie (x : ℝ) :
    LLMDecode.argmax2 x x = 0

argmax2 x x = 0.

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: 12 lines of Lean