Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The pullback and pushout theorems

Statement

In an abelian category:

  1. pullbacks of epimorphisms are epimorphisms;
  2. pushouts of monomorphisms are monomorphisms;
  3. in a pullback square, the induced map on kernels of the parallel arrows is an isomorphism;
  4. a commuting square is cartesian exactly when the associated short sequence is exact;
  5. a cartesian square over an epimorphism is also cocartesian.

Proof

technique · direct
1.1

The first claim is [L1]'s pullback-of-epimorphism theorem. The second claim is its pushout-of-monomorphism dual. The third claim is its kernel-comparison theorem.

L1given
1.2

The fourth claim is [L1]'s exact-square criterion, and the fifth claim is the cartesian-implies-cocartesian theorem over an epimorphism.

L1given
2.1

Hence the pullback and pushout results actually used by the diagram-lemma proofs are exactly the previously published theorems listed above.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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