Provedllmlarge-language-modelsformal-proofoptimization

Loss in distance coordinates

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{qloss}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=3}{c}, \htmlData{mx=6}{\htmlData{mx=4}{m} + \htmlData{mx=5}{d}})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=11}{\htmlData{mx=8}{a} \cdot \htmlData{mx=10}{\htmlData{mx=9}{d}^{2}}} + \htmlData{mx=12}{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, d range over real numbers: the statement must hold for every choice. ((a m c d : 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, m + d. Lean writes f x; papers write f(x). (LLMOptim.qloss a m c (m + d))
=Equality: qloss(a, m, c, m + d) and a · d^2 + c are the same. (LLMOptim.qloss a m c (m + d) = a * d ^ …)
+Addition: m plus d. (m + d)
·Multiplication: a times d^2. (a * d ^ 2)
aᵇPower: d raised to the power 2. (d ^ 2)

theorem opt-qloss-of-dist (a m c d : ℝ) :
    LLMOptim.qloss a m c (m + d) = a * d ^ 2 + c

qloss a m c (m + d) = a·d² + 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