Provedllmlarge-language-modelsformal-proofattention

A common shift preserves the tie

For all $\htmlData{mx=1}{s, c \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=3}{\htmlData{mx=1}{s} + \htmlData{mx=2}{c}}, \htmlData{mx=6}{\htmlData{mx=4}{s} + \htmlData{mx=5}{c}})} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\frac{1}{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: s, c range over real numbers: the statement must hold for every choice. ((s c : ℝ))
weight0: TransformerAttention.weight0: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.weight0)
f(x)Function application: TransformerAttention.weight0 applied to s + c, s + c. Lean writes f x; papers write f(x). (TransformerAttention.weight0 (s + c) (s…)
=Equality: weight0(s + c, s + c) and 1/2 are the same. (TransformerAttention.weight0 (s + c) (s…)
+Addition: s plus c. (s + c)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

theorem attn2-shifted-tie-half (s c : ℝ) :
    TransformerAttention.weight0 (s + c) (s + c) = 1 / 2

weight0 (s+c) (s+c) = 1/2.

How it was proved

Proved: 27 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