LemmaStatement: AI-generatedProof: AI-generated

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.

Uniqueness of an initial object

PDF p. 5: Seminormalization in Lemma 2.4

Statement

Any two initial objects of a category are uniquely isomorphic. An object is initial if it has exactly one arrow to every object.

Facts & Assumptions

Given: Initial objects $I$ and $J$ in a category.

Goal: Prove there is a unique isomorphism $I \to J$ .

Proof

1.1given

Initiality supplies unique arrows $f : I \to J$ and $g : J \to I$ . Since each initial object has only one endomorphism, $g f = {id}_{I}$ and $f g = {id}_{J}$ .

2.1givenstep 1.1∎

Thus $f$ is an isomorphism with inverse $g$ ; it is unique because there is only one arrow from $I$ to $J$ .

Used by · 1

Seminormality and seminormalization