proofs
Provedalphamulti-device-learningordered-combine

Only non-associativity makes the combine tree's shape matter

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→α\htmlData{mx=3}{v : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\alpha}} and s,t∈CombineTree⁡\htmlData{mx=1}{s, t \in \htmlData{mx=0}{\operatorname{CombineTree}}}, if ∀a,b,c, op(op(a,b),c)=op(a,op(b,c))\htmlData{mx=15}{\forall a, b, c,\ \htmlData{mx=6}{\htmlData{mx=0}{\mathit{op}}(\htmlData{mx=4}{\htmlData{mx=1}{\mathit{op}}(\htmlData{mx=2}{a}, \htmlData{mx=3}{b})}, \htmlData{mx=5}{c})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=7}{\mathit{op}}(\htmlData{mx=8}{a}, \htmlData{mx=12}{\htmlData{mx=9}{\mathit{op}}(\htmlData{mx=10}{b}, \htmlData{mx=11}{c})})}} and leaves⁡(s)=leaves⁡(t)\htmlData{mx=2}{\operatorname{leaves}\left(\htmlData{mx=1}{s}\right)} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\operatorname{leaves}\left(\htmlData{mx=4}{t}\right)}, then

eval⁡(s)(op,v)=eval⁡(t)(op,v)\htmlData{mx=5}{\htmlData{mx=2}{\operatorname{eval}\left(\htmlData{mx=1}{s}\right)}(\htmlData{mx=3}{\mathit{op}}, \htmlData{mx=4}{v})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{\htmlData{mx=8}{\operatorname{eval}\left(\htmlData{mx=7}{t}\right)}(\htmlData{mx=9}{\mathit{op}}, \htmlData{mx=10}{v})}
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)
∀For all
The statement after the comma holds for every a, b, c. (∀ a b c, op (op a b) c = op a (op b c))
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)
x ∈ TVariables
s, t range over CombineTree: the statement must hold for every choice. ((s t : CombineTree))
e.fleaves
leaves of s: Lean's dot notation e.leaves means leaves applied to e. (s.leaves)
e.feval
eval of s: Lean's dot notation e.eval means eval applied to e. (s.eval)
⟹Implies
If α, then α. (Between types, A → B is the type of functions from A to B.) (α → α)
=Equality
op(op(a, b), c) and op(a, op(b, c)) are the same. (op (op a b) c = op a (op b c))
f(x)Function application
op applied to a, b. Lean writes f x; papers write f(x). (op a b)

Two combine trees with the same leaf sequence compute the same value under any associative operation; so a hierarchy and the star's chain can differ only through the operation's non-associativity (as binary64 addition), which is why Alpha names the tree in the numerical contract. 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
Quot.sound, propext
Proof
45 lines of Lean
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