Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27
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.

Additive categories are closed under passage to the opposite

Statement

If C is additive, then Cop is additive.

Facts & Assumptions

Given: An additive category C.

[L1]

The opposite of a preadditive category is preadditive (The opposite of a preadditive category is preadditive).

[L2]

An additive category is a preadditive category with finite biproducts (Additive category).

Proof

technique · direct
1.1

By [L2], C is preadditive, so Cop is preadditive by [L1].

L1L2
2.1

By [L2], C has finite biproducts. Passing to the opposite category swaps products and coproducts, so every finite biproduct diagram in C becomes a diagram in Cop that is again simultaneously a finite product and a finite coproduct. Thus Cop has finite biproducts.

L2step 1.1
3.1

Thus Cop is preadditive and has finite biproducts, so it is additive by [L2].

L2step 2.1∎

Depends on

Used by

Dependency tree · two levels

7 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