Provedattentioninnovationexternal-review-3

Surrogate innovation: finite reward mean

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{r, a, b, v : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{B, c, p, o \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{a}(\htmlData{mx=4}{j})}}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{p}$ , $\htmlData{mx=11}{\sum_{j} \htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{a}(\htmlData{mx=4}{j})}} \cdot \htmlData{mx=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})}}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{p} \cdot \htmlData{mx=13}{o}}$ , $\htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{b}(\htmlData{mx=4}{j})}}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{B}$ and $\htmlData{mx=11}{\sum_{j} \htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{b}(\htmlData{mx=4}{j})}} \cdot \htmlData{mx=9}{\htmlData{mx=7}{v}(\htmlData{mx=8}{j})}}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{B} \cdot \htmlData{mx=13}{c}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\htmlData{mx=17}{\sum_{j} \htmlData{mx=16}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot (\htmlData{mx=15}{(\htmlData{mx=9}{\htmlData{mx=5}{\htmlData{mx=3}{a}(\htmlData{mx=4}{j})} - \htmlData{mx=8}{\htmlData{mx=6}{b}(\htmlData{mx=7}{j})}}) \cdot (\htmlData{mx=14}{\htmlData{mx=12}{\htmlData{mx=10}{v}(\htmlData{mx=11}{j})} - \htmlData{mx=13}{c}})})}} \mathrel{\htmlData{mx=23}{=}} \htmlData{mx=22}{\htmlData{mx=18}{p} \cdot (\htmlData{mx=21}{\htmlData{mx=19}{o} - \htmlData{mx=20}{c}})}

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)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
⟹Implies: If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables: r, a, b, v range over J → ℝ: the statement must hold for every choice. ((r a b v : J → ℝ))
x ∈ TVariables: B, c, p, o range over real numbers: the statement must hold for every choice. ((B c p o : ℝ))
∑Summation: Add up r(j) · a(j) for every j. (∑ j, r j * a j)
=Equality: ∑ j, r(j) · a(j) and p are the same. (∑ j, r j * a j = p)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: r(j) times a(j). (r j * a j)
-Subtraction: a(j) minus b(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (a j - b j)

theorem innov-surrogate-reward-identity {J : Type} [Fintype J] (r a b v : J → ℝ) (B c p o : ℝ) (hp : ∑ j, r j * a j = p) (ho : ∑ j, r j * a j * v j = p * o) (hB : ∑ j, r j * b j = B) (hc : ∑ j, r j * b j * v j = B * c) :
    ∑ j, r j * ((a j - b j) * (v j - c)) = p * (o - c)

Finite scalar weighted-mean identity under exact reference and surrogate aggregate assumptions. Does not formalize an infinite stopped process.

How it was proved

Proved: 6 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
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

Surrogate innovation: finite reward mean

How it was proved

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