Provedllmlarge-language-modelsformal-prooflayernorm

Two-point mean via the explicit sum

For all $\htmlData{mx=4}{x : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,2} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ :

\htmlData{mx=3}{\sum_{i} \htmlData{mx=2}{\htmlData{mx=0}{x}(\htmlData{mx=1}{i})}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\htmlData{mx=5}{\htmlData{mx=4}{x}(0)} + \htmlData{mx=7}{\htmlData{mx=6}{x}(1)}}

What the notation means · or point at any symbol

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 2. Lean writes f x; papers write f(x). (Fin 2)
⟹Implies: If Fin(2), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 2 → ℝ)
∑Summation: Add up x(i) for every i. (∑ i, x i)
=Equality: ∑ i, x(i) and x(0) + x(1) are the same. (∑ i, x i = x 0 + x 1)
RReal numbers: The real numbers: every point on the number line. (ℝ)
+Addition: x(0) plus x(1). (x 0 + x 1)

theorem ln-mean-two-sum-form (x : Fin 2 → ℝ) :
    ∑ i, x i = x 0 + x 1

∑ i, x i = x 0 + x 1 over Fin 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: 13 lines of Lean