Provedbridge

LLMAttention.logit is Mathlib's dot product over √d

For all $\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{q, k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ :

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{logit}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{q}, \htmlData{mx=3}{k})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{\frac{\htmlData{mx=8}{\htmlData{mx=5}{\operatorname{dotProduct}}(\htmlData{mx=6}{q}, \htmlData{mx=7}{k})}}{\htmlData{mx=10}{\sqrt{\htmlData{mx=9}{d}}}}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to d. Lean writes f x; papers write f(x). (Fin d)
⟹Implies: If Fin(d), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin d → ℝ)
x : TVariables: q, k range over Fin d → ℝ: the statement must hold for every choice. ((q k : Fin d → ℝ))
logit: LLMAttention.logit: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMAttention.logit)
=Equality: logit(d, q, k) and dotProduct(q, k)/√(d) are the same. (LLMAttention.logit d q k = dotProduct q…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
dotProduct: dotProduct: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (dotProduct)
a/bDivision: dotProduct(q, k) divided by √(d). In Lean, dividing by 0 gives 0. (dotProduct q k / Real.sqrt d)
√Square root: The non-negative number whose square is d. (Real.sqrt d)

theorem bridge-logit-dotproduct (d : ℕ) (q k : Fin d → ℝ) :
    LLMAttention.logit d q k = dotProduct q k / Real.sqrt d

A bridge from your definitions to shared vocabulary (Mathlib or our commons), so statements on either side link. Posed with our commons' launch.

How it was proved

Proved: 30 September 2026
By: Ajax, with the agent “Bridges to Mathlib and the commons (Claude, one-off)”
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

LLMAttention.logit is Mathlib's dot product over √d

How it was proved

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