Provedllmlarge-language-modelsformal-proofattention

Attention is jointly homogeneous in scores and values (values side)

For all $\htmlData{mx=1}{n, d \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ , $\htmlData{mx=5}{q : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=9}{K : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{d}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ 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=10}{\htmlData{mx=0}{\operatorname{attend}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{q}, \htmlData{mx=3}{K}, \htmlData{mx=9}{j \mapsto \htmlData{mx=8}{\htmlData{mx=4}{c} \cdot \htmlData{mx=7}{\htmlData{mx=5}{V}(\htmlData{mx=6}{j})}}})} \mathrel{\htmlData{mx=19}{=}} \htmlData{mx=18}{\htmlData{mx=11}{c} \cdot \htmlData{mx=17}{\htmlData{mx=12}{\operatorname{attend}}(\htmlData{mx=13}{d}, \htmlData{mx=14}{q}, \htmlData{mx=15}{K}, \htmlData{mx=16}{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)
x ∈ TVariables: n, d range over natural numbers: the statement must hold for every choice. ((n d : ℕ))
RReal numbers: The real numbers: every point on the number line. (ℝ)
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, j ↦ c · V(j)) and c · attend(d, q, K, V) are the same. (LLMAttention.attend d q K (fun j => c *…)
x ↦ f(x)Function (λ-abstraction): The function sending j to c · V(j). Lean writes fun x => …; papers write x ↦ …. (fun j => c * V j)
·Multiplication: c times V(j). (c * V j)

theorem attn-attend-smul-both (n d : ℕ) (c : ℝ) (q : Fin d → ℝ) (K : Fin n → Fin d → ℝ) (V : Fin n → ℝ) :
    LLMAttention.attend d q K (fun j => c * V j) = c * LLMAttention.attend d q K V

attend d q K (fun j => c * V j) = c * attend d q K V (restated confidently).

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: 15 lines of Lean

Attention is jointly homogeneous in scores and values (values side)

How it was proved

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