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.
in : the sumset has five elements and the bound is tight
Example
Let .
Facts & Assumptions
Given: the set in .
Cauchy-Davenport gives (Cauchy–Davenport: for prime and nonempty , ).
Verification
The sumset is since the nine sums reduce to modulo .
Therefore , so the lower bound of [L1] is attained.
The other branch of the theorem is also visible: if and , then .
Depends on
- Cauchy–Davenport: for $p$ prime and nonempty $A,B\subseteq\mathbb{Z}/p$, $\lvert A+B\rvert\ge\min\{p,\lvert A\rvert+\lvert B\rvert-1\}$
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- For $n\ge 1$, every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$, so $\lvert\mathbb{Z}/n\rvert=n$; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
22 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
- O. Pikhurko, An Introduction to Combinatorics, §11.1 (standard reference, not scraped)