proofs
Provedadcgssupportingfw-gap

An LMO gap test certifies the inner variational inequality for every feasible comparator

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}}}}}, c,z,v∈EuclideanSpace⁡ R Fin⁡ n\htmlData{mx=6}{c, z, v \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 delta∈R\htmlData{mx=1}{\mathit{delta} \in \htmlData{mx=0}{\mathbb{R}}}, if v∈P\htmlData{mx=0}{v} \mathrel{\htmlData{mx=2}{\in}} \htmlData{mx=1}{P}, ∀w∈P, inner⁡(R,c,v)≤inner⁡(R,c,w)\htmlData{mx=12}{\forall w \in \htmlData{mx=11}{P},\ \htmlData{mx=4}{\htmlData{mx=0}{\operatorname{inner}}(\htmlData{mx=1}{\mathbb{R}}, \htmlData{mx=2}{c}, \htmlData{mx=3}{v})} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{\htmlData{mx=5}{\operatorname{inner}}(\htmlData{mx=6}{\mathbb{R}}, \htmlData{mx=7}{c}, \htmlData{mx=8}{w})}} and inner⁡(R,c,z−v)≤delta\htmlData{mx=6}{\htmlData{mx=0}{\operatorname{inner}}(\htmlData{mx=1}{\mathbb{R}}, \htmlData{mx=2}{c}, \htmlData{mx=5}{\htmlData{mx=3}{z} - \htmlData{mx=4}{v}})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\mathit{delta}}, then

∀w∈P, inner⁡(R,c,z−w)≤delta\htmlData{mx=10}{\forall w \in \htmlData{mx=9}{P},\ \htmlData{mx=6}{\htmlData{mx=0}{\operatorname{inner}}(\htmlData{mx=1}{\mathbb{R}}, \htmlData{mx=2}{c}, \htmlData{mx=5}{\htmlData{mx=3}{z} - \htmlData{mx=4}{w}})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\mathit{delta}}}
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)
x ∈ TVariables
c, z, v range over EuclideanSpace ℝ (Fin n): the statement must hold for every choice. ((c z v : EuclideanSpace ℝ (Fin n)))
∈Element of
v is a member of P. (v ∈ P)
∀For all
The statement after the comma holds for every w. (∀ w ∈ P, inner ℝ c v ≤ inner ℝ c w)
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)
inner
inner: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (inner)
≤At most
inner(ℝ, c, v) is less than or equal to inner(ℝ, c, w). (inner ℝ c v ≤ inner ℝ c w)
-Subtraction
z minus v. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (z - v)
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)

Algorithm 1, lines 13–15 and Appendix A.2 equation (15) of the uploaded arXiv:2601.20443v2. If v minimizes the linear functional defined by the quadratic subproblem gradient c and the tested gap <c,z-v> is at most delta, then the gap to every feasible w is at most delta. This proves the conversion from one oracle test to the universal FW-gap contract; it does not prove inner termination or an outer rate. Actual Euclidean inner product, arbitrary finite dimension.

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
16 lines of Lean
Sign in to see the proofThe proof, its signed report and its history.

More proved statements