proofs
Provedalphamulti-device-learningpartitioning

An elementwise combine evaluates word by word

For all α∈Type⁡\htmlData{mx=1}{\alpha \in \htmlData{mx=0}{\operatorname{Type}}}, op:α→α→α\htmlData{mx=5}{\mathit{op} : \htmlData{mx=0}{\alpha} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\alpha} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\alpha}}, v:N→N→α\htmlData{mx=5}{v : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\mathbb{N}} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\alpha}}, i∈N\htmlData{mx=1}{i \in \htmlData{mx=0}{\mathbb{N}}} and t∈CombineTree⁡\htmlData{mx=1}{t \in \htmlData{mx=0}{\operatorname{CombineTree}}}:

eval⁡(t)(pointwise⁡(op),v,i)=eval⁡(t)(op,o↦v(o,i))\htmlData{mx=8}{\htmlData{mx=2}{\operatorname{eval}\left(\htmlData{mx=1}{t}\right)}(\htmlData{mx=5}{\htmlData{mx=3}{\operatorname{pointwise}}(\htmlData{mx=4}{\mathit{op}})}, \htmlData{mx=6}{v}, \htmlData{mx=7}{i})} \mathrel{\htmlData{mx=19}{=}} \htmlData{mx=18}{\htmlData{mx=11}{\operatorname{eval}\left(\htmlData{mx=10}{t}\right)}(\htmlData{mx=12}{\mathit{op}}, \htmlData{mx=17}{o \mapsto \htmlData{mx=16}{\htmlData{mx=13}{v}(\htmlData{mx=14}{o}, \htmlData{mx=15}{i})}})}
What the notation means · or point at any symbol
Type
Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (Nat)
CombineTree
CombineTree: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (CombineTree)
e.feval
eval of t: Lean's dot notation e.eval means eval applied to e. (t.eval)
=Equality
eval(pointwise(op), v, i) and eval(op, o ↦ v(o, i)) are the same. (t.eval (pointwise op) v i = t.eval op (…)
⟹Implies
If α, then α. (Between types, A → B is the type of functions from A to B.) (α → α)
pointwise
pointwise: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (pointwise)
f(x)Function application
pointwise applied to op. Lean writes f x; papers write f(x). (pointwise op)
x ↦ f(x)Function (λ-abstraction)
The function sending o to v(o, i). Lean writes fun x => …; papers write x ↦ …. (fun o => v o i)

Combining vectors elementwise over a tree equals combining each word over the same tree, for any operation. Model: definition alpha-multi-device-learning (revision 01a1255c-84f1-7e7f-a994-fc8c2d274bee).

How it was proved

Proved
10 October 2026
By
Ajax
Checked by
Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms
propext
Proof
20 lines of Lean
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