Provedllmlarge-language-modelsformal-proofattention

logits at an index is the logit at that key

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 $\htmlData{mx=3}{j \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{logits}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{q}, \htmlData{mx=3}{K}, \htmlData{mx=4}{j})} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{\htmlData{mx=6}{\operatorname{logit}}(\htmlData{mx=7}{d}, \htmlData{mx=8}{q}, \htmlData{mx=11}{\htmlData{mx=9}{K}(\htmlData{mx=10}{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)
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)
=Equality: logits(d, q, K, j) and logit(d, q, K(j)) are the same. (LLMAttention.logits d q K j = LLMAttent…)
RReal numbers: The real numbers: every point on the number line. (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)

theorem attn-logits-at-index (n d : ℕ) (q : Fin d → ℝ) (K : Fin n → Fin d → ℝ) (j : Fin n) :
    LLMAttention.logits d q K j = LLMAttention.logit d q (K j)

logits d q K j = logit d q (K j) (definitional unfolding at j).

How it was proved

Proved: 28 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: 8 lines of Lean

logits at an index is the logit at that key

How it was proved

More proved statements