LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge 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.

Metric nonnegativity from the three axioms

Statement

For any real metric or pseudometric $d$ on a set $X$ in Real metric and pseudometric convention, $d (x, y) \geq 0$ for every $x, y \in X$ . Thus these three-axiom definitions already have values in $[0, \infty)$ , including the vacuous assertion when $X = \emptyset$ .

Facts & Assumptions

Given: The set $X$ and the finite real metric or pseudometric $d$ .

[F1]

Real metric and pseudometric convention supplies reflexivity, symmetry, the triangle inequality and ordered-field arithmetic.

Proof

1.1givenF1algebra

Fix $x, y \in X$ . Reflexivity, the triangle inequality with endpoints $x, x$ , and symmetry give $0 = d (x, x) \leq d (x, y) + d (y, x) = 2 d (x, y)$ .

2.1F1step 1.1algebra∎

The real number $2$ is positive, so dividing the inequality in step 1.1 by $2$ yields $0 \leq d (x, y)$ . This proves the universally quantified claim; if $X$ is empty there are no pairs to check.

Depends on

Real metric and pseudometric convention

Metric nonnegativity from the three axioms

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources