Provedbridge

LLMActivations.sigmoid is Mathlib's Real.sigmoid

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{sigmoid}}(\htmlData{mx=1}{x})} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{sigmoid}}(\htmlData{mx=4}{x})}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
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 x. Lean writes f x; papers write f(x). (LLMActivations.sigmoid x)
=Equality: sigmoid(x) and sigmoid(x) are the same. (LLMActivations.sigmoid x = Real.sigmoid…)

theorem bridge-llmactivations-sigmoid-real-sigmoid (x : ℝ) :
    LLMActivations.sigmoid x = Real.sigmoid x

Found by our verifier when llm-activations was published: LLMActivations.sigmoid equals Mathlib's Real.sigmoid, proved by intros; simp [LLMActivations.sigmoid, Real.sigmoid]. Once proved here, statements about either name link to each other.

How it was proved

Proved: 30 September 2026
By: Ajax
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: 10 lines of Lean

LLMActivations.sigmoid is Mathlib's Real.sigmoid

How it was proved

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