Provedllmlarge-language-modelsformal-proofoptimization

The surrogate loss expands as a monic quadratic in x

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{qloss}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=3}{c}, \htmlData{mx=4}{x})} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\htmlData{mx=16}{\htmlData{mx=9}{\htmlData{mx=6}{a} \cdot \htmlData{mx=8}{\htmlData{mx=7}{x}^{2}}} - \htmlData{mx=15}{\htmlData{mx=13}{\htmlData{mx=11}{2\htmlData{mx=10}{a}} \cdot \htmlData{mx=12}{m}} \cdot \htmlData{mx=14}{x}}} + (\htmlData{mx=22}{\htmlData{mx=20}{\htmlData{mx=17}{a} \cdot \htmlData{mx=19}{\htmlData{mx=18}{m}^{2}}} + \htmlData{mx=21}{c}})}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: a, m, c, x range over real numbers: the statement must hold for every choice. ((a m c x : ℝ))
qloss: LLMOptim.qloss: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMOptim.qloss)
f(x)Function application: LLMOptim.qloss applied to a, m, c, x. Lean writes f x; papers write f(x). (LLMOptim.qloss a m c x)
=Equality: qloss(a, m, c, x) and a · x^2 − 2 · a · m · x + (a · m^2 + c) are the same. (LLMOptim.qloss a m c x = a * x ^ 2 - 2 …)
·Multiplication: a times x^2. (a * x ^ 2)
-Subtraction: a · x^2 minus 2 · a · m · x. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (a * x ^ 2 - 2 * a * m * x)
aᵇPower: x raised to the power 2. (x ^ 2)
+Addition: a · m^2 plus c. (a * m ^ 2 + c)

theorem opt-qloss-quad-form (a m c x : ℝ) :
    LLMOptim.qloss a m c x = a * x ^ 2 - 2 * a * m * x + (a * m ^ 2 + c)

qloss a m c x = a·x² − 2·a·m·x + (a·m² + c).

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: 13 lines of Lean