Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13
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.

Strong distribution convergence implies weak convergence

Statement

If a net of distributions converges strongly, it converges weakly to the same distribution. In particular this holds for sequences. The implication requires no choice axiom.

Facts & Assumptions

[F1]

Strong convergence is convergence uniformly on every bounded test set; weak convergence is pointwise convergence on tests, and every singleton test set is bounded (Weak and strong topologies on distributions).

Proof

Given: ui→u strongly.

1.1givenF1

Fix a test φ. By F1 the singleton B={φ} is bounded and ∣ui(φ)−u(φ)∣=pB(ui−u)→0.

2.1step 1.1F1

The test was arbitrary, so F1 identifies these scalar limits as weak convergence. This uses one given test at a time, without a simultaneous selection. For the zero test the seminorm is zero; on the empty domain the sole distribution is zero. No converse for arbitrary nets is asserted. □

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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