Provedllmlarge-language-modelsformal-proofattention

attend in named form

For all $\htmlData{mx=1}{n, d \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{q : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=9}{K : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{d}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ 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=19}{=}} \htmlData{mx=18}{\sum_{j} \htmlData{mx=17}{\htmlData{mx=13}{\htmlData{mx=6}{\operatorname{sm}}(\htmlData{mx=11}{\htmlData{mx=7}{\operatorname{logits}}(\htmlData{mx=8}{d}, \htmlData{mx=9}{q}, \htmlData{mx=10}{K})}, \htmlData{mx=12}{j})} \cdot \htmlData{mx=16}{\htmlData{mx=14}{V}(\htmlData{mx=15}{j})}}}

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))
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 ∑ j, sm(logits(d, q, K), j) · V(j) are the same. (LLMAttention.attend d q K V = ∑ j, LLMS…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up sm(logits(d, q, K), j) · V(j) for every j. (∑ j, LLMSoftmax.sm (LLMAttention.logits…)
sm: LLMSoftmax.sm: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.sm)
·Multiplication: sm(logits(d, q, K), j) times V(j). (LLMSoftmax.sm (LLMAttention.logits d q …)
logits: LLMAttention.logits: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMAttention.logits)

theorem attn-attend-def (n d : ℕ) (q : Fin d → ℝ) (K : Fin n → Fin d → ℝ) (V : Fin n → ℝ) :
    LLMAttention.attend d q K V = ∑ j, LLMSoftmax.sm (LLMAttention.logits d q K) j * V j

LLMAttention.attend d q K V = ∑ j, LLMSoftmax.sm (LLMAttention.logits d q K) j * V j.

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

attend in named form

How it was proved

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