Provedllmlarge-language-modelsformal-proofattention

Raising the lower value raises the output

For all $\htmlData{mx=1}{s_{0}, s_{1}, v_{0}, v_{0}', v_{1} \in \htmlData{mx=0}{\mathbb{R}}}$ :

\begin{aligned} & \htmlData{mx=1}{v_{0}} \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{v_{0}'} \\ & \htmlData{mx=0}{\implies} \htmlData{mx=4}{v_{0}'} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{v_{1}} \\ & \htmlData{mx=0}{\implies} \htmlData{mx=12}{\htmlData{mx=7}{\operatorname{attend2}}(\htmlData{mx=8}{s_{0}}, \htmlData{mx=9}{s_{1}}, \htmlData{mx=10}{v_{0}}, \htmlData{mx=11}{v_{1}})} \mathrel{\htmlData{mx=19}{\leq}} \htmlData{mx=18}{\htmlData{mx=13}{\operatorname{attend2}}(\htmlData{mx=14}{s_{0}}, \htmlData{mx=15}{s_{1}}, \htmlData{mx=16}{v_{0}'}, \htmlData{mx=17}{v_{1}})} \end{aligned}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: s0, s1, v0, v0', v1 range over real numbers: the statement must hold for every choice. ((s0 s1 v0 v0' v1 : ℝ))
≤At most: v0 is less than or equal to v0'. (v0 ≤ v0')
⟹Implies: If v0' ≤ v1, then attend2(s0, s1, v0, v1) ≤ attend2(s0, s1, v0', v1). (Between types, A → B is the type of functions from A to B.) (v0' ≤ v1 → TransformerAttention.attend2…)
attend2: TransformerAttention.attend2: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.attend2)
f(x)Function application: TransformerAttention.attend2 applied to s0, s1, v0, v1. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 v0 v1)

theorem attn2-mono-v0 (s0 s1 v0 v0' v1 : ℝ) :
    v0 ≤ v0' → v0' ≤ v1 → TransformerAttention.attend2 s0 s1 v0 v1 ≤ TransformerAttention.attend2 s0 s1 v0' v1

For v0 ≤ v0' ≤ v1: attend2 with v0 is at most attend2 with v0'.

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: 19 lines of Lean

Raising the lower value raises the output

How it was proved

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