Provedllmlarge-language-modelsformal-proofsigmoid

Sigmoid at zero

\htmlData{mx=1}{\htmlData{mx=0}{\operatorname{sigmoid}}(0)} \mathrel{\htmlData{mx=3}{=}} \htmlData{mx=2}{\frac{1}{2}}

What the notation means · or point at any symbol

sigmoid: LLMActivations.sigmoid: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMActivations.sigmoid)
f(x)Function application: LLMActivations.sigmoid applied to 0. Lean writes f x; papers write f(x). (LLMActivations.sigmoid 0)
=Equality: sigmoid(0) and 1/2 are the same. (LLMActivations.sigmoid 0 = 1 / 2)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

theorem act-sigmoid-zero :
    LLMActivations.sigmoid 0 = 1 / 2

sigmoid 0 = 1/2 (instance).

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: 14 lines of Lean