Alphabeta MathHow 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.
Covering sieves admit common refinements
Statement
On a small site, the intersection of two covering sieves on the same object is covering. Every sieve containing a covering sieve is covering.
Facts & Assumptions
The topology axioms and pullback of a sieve are as in Small sites and the plus construction.
Proof
Given: Covering sieves R and S on X, and a sieve T containing R.
Goal: Prove that R ∩ S and T cover X.
For each f in R, the pullback fS covers dom f by stability. Since R is a sieve, f(R ∩ S) = f*S. Transitivity applied to R shows that R ∩ S covers X.
For every f in R, f*T is maximal and hence covering. Transitivity applied to R shows that T covers X. Thus both assertions hold.
Depends on
Dependency tree · 1 level
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.