Provedllmlarge-language-modelsformal-proofrope

RoPE at a fixed angle is injective

For all $\htmlData{mx=1}{\theta \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=1}{z_{1}, z_{2} \in \htmlData{mx=0}{\mathbb{C}}}$ :

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{rope}}(\htmlData{mx=1}{\theta}, \htmlData{mx=2}{z_{1}})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{rope}}(\htmlData{mx=5}{\theta}, \htmlData{mx=6}{z_{2}})} \mathrel{\htmlData{mx=12}{\implies}} \htmlData{mx=9}{z_{1}} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{z_{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
CComplex numbers: The complex numbers a + bi, where i² = −1. (ℂ)
x ∈ TVariables: z₁, z₂ range over complex numbers: the statement must hold for every choice. ((z₁ z₂ : ℂ))
rope: LLMRoPE.rope: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMRoPE.rope)
f(x)Function application: LLMRoPE.rope applied to θ, z₁. Lean writes f x; papers write f(x). (LLMRoPE.rope θ z₁)
=Equality: rope(θ, z₁) and rope(θ, z₂) are the same. (LLMRoPE.rope θ z₁ = LLMRoPE.rope θ z₂)
⟹Implies: If rope(θ, z₁) = rope(θ, z₂), then z₁ = z₂. (Between types, A → B is the type of functions from A to B.) (LLMRoPE.rope θ z₁ = LLMRoPE.rope θ z₂ →…)

theorem rope-injective (θ : ℝ) (z₁ z₂ : ℂ) :
    LLMRoPE.rope θ z₁ = LLMRoPE.rope θ z₂ → z₁ = z₂

Equal rotations imply equal inputs.

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