proofs
Provedadcgssupportingfeasibility

Both AdCGS outer averaging updates remain in the compact convex feasible set

For all n∈N\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}, P∈Set⁡ EuclideanSpace⁡ R Fin⁡ n\htmlData{mx=8}{P \in \htmlData{mx=7}{\htmlData{mx=0}{\operatorname{Set}}\,\htmlData{mx=6}{\htmlData{mx=1}{\operatorname{EuclideanSpace}}\,\htmlData{mx=2}{\mathbb{R}}\,\htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{n}}}}}, hP∈Convex⁡ R P\htmlData{mx=4}{\mathit{hP} \in \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{Convex}}\,\htmlData{mx=1}{\mathbb{R}}\,\htmlData{mx=2}{P}}}, x,y,z∈EuclideanSpace⁡ R Fin⁡ n\htmlData{mx=6}{x, y, z \in \htmlData{mx=5}{\htmlData{mx=0}{\operatorname{EuclideanSpace}}\,\htmlData{mx=1}{\mathbb{R}}\,\htmlData{mx=4}{\htmlData{mx=2}{\operatorname{Fin}}\,\htmlData{mx=3}{n}}}} and b,tau∈R\htmlData{mx=1}{b, \mathit{tau} \in \htmlData{mx=0}{\mathbb{R}}}, if x∈P\htmlData{mx=0}{x} \mathrel{\htmlData{mx=2}{\in}} \htmlData{mx=1}{P}, y∈P\htmlData{mx=0}{y} \mathrel{\htmlData{mx=2}{\in}} \htmlData{mx=1}{P}, z∈P\htmlData{mx=0}{z} \mathrel{\htmlData{mx=2}{\in}} \htmlData{mx=1}{P}, 0≤b0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{b}, b≤1\htmlData{mx=0}{b} \mathrel{\htmlData{mx=1}{\leq}} 1 and 0≤tau0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\mathit{tau}}, then

(1−b) y+b z∈P∧(tau1+tau) x+(11+tau) z∈P\begin{aligned} & \htmlData{mx=8}{\htmlData{mx=4}{(\htmlData{mx=2}{1 - \htmlData{mx=1}{b}}) \, \htmlData{mx=3}{y}} + \htmlData{mx=7}{\htmlData{mx=5}{b} \, \htmlData{mx=6}{z}}} \mathrel{\htmlData{mx=10}{\in}} \htmlData{mx=9}{P} \\ & \htmlData{mx=0}{\land} \htmlData{mx=22}{\htmlData{mx=16}{\left(\htmlData{mx=14}{\frac{\htmlData{mx=11}{\mathit{tau}}}{\htmlData{mx=13}{1 + \htmlData{mx=12}{\mathit{tau}}}}}\right) \, \htmlData{mx=15}{x}} + \htmlData{mx=21}{\left(\htmlData{mx=19}{\frac{1}{\htmlData{mx=18}{1 + \htmlData{mx=17}{\mathit{tau}}}}}\right) \, \htmlData{mx=20}{z}}} \mathrel{\htmlData{mx=24}{\in}} \htmlData{mx=23}{P} \end{aligned}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Set
Set: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Set)
Convex
Convex: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Convex)
x ∈ TVariables
x, y, z range over EuclideanSpace ℝ (Fin n): the statement must hold for every choice. ((x y z : EuclideanSpace ℝ (Fin n)))
∈Element of
x is a member of P. (x ∈ P)
≤At most
0 is less than or equal to b. (0 ≤ b)
•Scalar multiplication
y scaled by the number 1 − b. ((1 - b) • y)
+Addition
(1 − b) • y plus b • z. ((1 - b) • y + b • z)
∧And
Both hold: (1 − b) • y + b • z ∈ P, and (tau/(1 + tau)) • x + (1/(1 + tau)) • z ∈ P. ((1 - b) • y + b • z ∈ P ∧ (tau / (1 + t…)
-Subtraction
1 minus b. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - b)
EuclideanSpace
EuclideanSpace: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (EuclideanSpace)
RReal numbers
The real numbers: every point on the number line. (ℝ)
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 n. Lean writes f x; papers write f(x). (Fin n)
a/bDivision
tau divided by 1 + tau. In Lean, dividing by 0 gives 0. (tau / (1 + tau))

Section 2, immediately after equation (2), and Algorithm 1 lines 4–5. Convexity alone suffices; compactness is not needed for this local preservation lemma. This certifies the two affine updates given feasible inputs and admissible beta/tau, rather than assuming their output membership.

How it was proved

Proved
30 September 2026
By
Ajax
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
17 lines of Lean
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