TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Farkas alternative for a finite linear system

Statement

Let $A \in R^{r \times n}$ and $b \in R^{r}$ . Exactly one of the following holds:

There is $x \in R^{n}$ with $x \geq 0$ and $A x = b$ .
There is $y \in R^{r}$ with $A^{T} y \geq 0$ and $b^{T} y < 0$ .

Inequalities between vectors are coordinatewise.

Facts & Assumptions

Given: The matrix and vector in the Statement and the standard Euclidean inner product.

[F1]

The cone generated by any finite list of Euclidean vectors is closed (A finitely generated cone in Euclidean space is closed).

[F2]

Every point has a unique nearest point in a nonempty closed convex Euclidean set (Every point has a unique nearest point in a nonempty closed Euclidean convex set).

[F3]

If $p$ is the nearest point to $b$ in a closed convex set $C$ , then $⟨ b - p, c - p ⟩ \leq 0$ for every $c \in C$ (Metric projection onto a closed convex set satisfies the variational inequality).

Proof

1.1F1F2givenconstructalgebra

If $r = 0$ , then $R^{r} = {0}$ and $b = 0$ . Alternative 1 holds with $x = 0 \in R^{n}$ , while alternative 2 is impossible because the only $y \in R^{0}$ is $0$ and $b^{T} y = 0$ . This handles the zero-dimensional ambient case. Now assume $r \geq 1$ . Let $C = {\sum_{j = 1}^{n} x_{j} A_{j} : x_{j} \geq 0}$ , where $A_{j}$ are the columns of $A$ . This is a nonempty convex cone and is closed by [F1]. If $b \in C$ , its nonnegative coefficients give alternative 1. If $b \notin C$ , let $p$ be its nearest point in $C$ by [F2] and put $z = b - p \neq 0$ .

2.1F3step 1.1algebra

Applying [F3] with $c = 0$ and $c = 2 p$ gives both $⟨ z, p ⟩ \geq 0$ and $⟨ z, p ⟩ \leq 0$ , hence $⟨ z, p ⟩ = 0$ . Applying [F3] to arbitrary $c \in C$ then gives $⟨ z, c ⟩ \leq 0$ . In particular $A_{j}^{T} z \leq 0$ for every column, and $⟨ z, b ⟩ = ⟨ z, p + z ⟩ = ∥ z ∥_{2}^{2} > 0$ . Thus $y = - z$ satisfies $A^{T} y \geq 0$ and $b^{T} y = - ∥ z ∥_{2}^{2} < 0$ , which is alternative 2.

3.1step 1.1algebra∎

The alternatives cannot both hold: if $A x = b$ with $x \geq 0$ and $A^{T} y \geq 0$ , then $b^{T} y = x^{T} A^{T} y \geq 0$ , contradicting $b^{T} y < 0$ . Step 1.1 produces one alternative in every case, so exactly one holds.

Depends on

Used by

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Boyd and Vandenberghe, Convex Optimization (standard reference, not scraped)
Nocedal and Wright, Numerical Optimization, second edition (standard reference, not scraped)