Provedllmlarge-language-modelsformal-proofattention

The negated query negates the logit

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=8}{\htmlData{mx=0}{\operatorname{logit}}(\htmlData{mx=1}{d}, \htmlData{mx=6}{i \mapsto \htmlData{mx=5}{-\htmlData{mx=4}{\htmlData{mx=2}{q}(\htmlData{mx=3}{i})}}}, \htmlData{mx=7}{k})} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{-\htmlData{mx=13}{\htmlData{mx=9}{\operatorname{logit}}(\htmlData{mx=10}{d}, \htmlData{mx=11}{q}, \htmlData{mx=12}{k})}}

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, i ↦ −q(i), k) and −logit(d, q, k) are the same. (LLMAttention.logit d (fun i => -q i) k …)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending i to −q(i). Lean writes fun x => …; papers write x ↦ …. (fun i => -q i)
−xNegation: The negative of q(i). (-q i)

theorem attn-logit-neg-query (d : ℕ) (q k : Fin d → ℝ) :
    LLMAttention.logit d (fun i => -q i) k = -LLMAttention.logit d q k

logit d (−q) k = −logit d q k.

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: 19 lines of Lean

The negated query negates the logit

How it was proved

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