Provedbridge

LLMAttention.attend is our commons' attention

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{q : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=9}{K : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{d}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ and $\htmlData{mx=5}{V : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ :

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{attend}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{q}, \htmlData{mx=3}{K}, \htmlData{mx=4}{V})} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\htmlData{mx=6}{\operatorname{attention}}(\htmlData{mx=7}{q}, \htmlData{mx=8}{K}, \htmlData{mx=9}{V})}

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)
attend: LLMAttention.attend: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMAttention.attend)
=Equality: attend(d, q, K, V) and attention(q, K, V) are the same. (LLMAttention.attend d q K V = Commons.a…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
attention: Commons.attention: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.attention)

theorem bridge-attend-commons-attention {n : ℕ} (d : ℕ) (q : Fin d → ℝ) (K : Fin n → Fin d → ℝ) (V : Fin n → ℝ) :
    LLMAttention.attend d q K V = Commons.attention q K V

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.attend is our commons' attention

How it was proved

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