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.
The long exact homology sequence is natural
Statement
A morphism of short exact sequences of complexes induces a morphism between the associated long exact homology sequences.
Facts & Assumptions
Given: A morphism of short exact sequences of complexes.
Every chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).
The connecting square commutes under a morphism of short exact sequences (Naturality of the homology connecting morphism).
Proof
Each ordinary square in the two long exact sequences is induced by one of the three chain maps in the given ladder, so it commutes by [L1].
The only nonformal squares are the connecting ones, and they commute by [L2]. Therefore every square in the long exact ladder commutes.