Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

Vertical composites of natural transformations satisfy naturality

Statement

The vertical composite of two natural transformations satisfies the naturality equation.

Facts & Assumptions

Given: Natural transformations α:F⇒G\alpha:F\Rightarrow G and β:G⇒H\beta:G\Rightarrow H, and a morphism f:A→Bf:A\to B.

[L1]

Vertical composition is componentwise, and α,β\alpha,\beta satisfy their naturality equations (Identity natural transformation and vertical composition).

Proof

technique · direct
1.1

By naturality, Gf∘αA=αB∘FfGf\circ\alpha_A=\alpha_B\circ Ff and Hf∘βA=βB∘GfHf\circ\beta_A=\beta_B\circ Gf.

givenL1
2.1

Therefore Hf∘(βA∘αA)=(βB∘Gf)∘αA=βB∘(αB∘Ff)=(βB∘αB)∘FfHf\circ(\beta_A\circ\alpha_A)=(\beta_B\circ Gf)\circ\alpha_A=\beta_B\circ(\alpha_B\circ Ff)=(\beta_B\circ\alpha_B)\circ Ff.

step 1.1L1
3.1

This is the naturality equation for the componentwise family β∘α\beta\circ\alpha.

step 2.1L1∎

Depends on

Used by

Cited to discharge well-definedness by Identity natural transformation and vertical composition.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources