Provedalphamulti-device-learningpartitioning
An elementwise combine evaluates word by word
For all , , , and :
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)
theorem alpha-elementwise-combine-wordwise {α : Type} (op : α → α → α) (v : Nat → Nat → α) (i : Nat) (t : CombineTree) :
t.eval (pointwise op) v i = t.eval op (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
Sign in to see the proofThe proof, its signed report and its history.
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