TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The chain rule for differentials of smooth maps

If $F : M \to N$ and $G : N \to P$ are smooth, then $d (G \circ F)_{p} = d G_{F (p)} \circ d F_{p}$ for every $p \in M$ .

Given: Smooth maps $F : M \to N$ and $G : N \to P$ and a point $p \in M$ .

[F1]

The differential is defined by pullback of target germs (The differential of a smooth map).

technique · direct

1.1F1given

Let $v \in T_{p} M$ and let $[h]$ be a smooth germ at $G (F (p))$ . Then $d (G \circ F)_{p} (v) ([h]) = v ([h \circ G \circ F])$ by [F1].

2.1F1step 1.1

Also $(d G_{F (p)} \circ d F_{p}) (v) ([h]) = d F_{p} (v) ([h \circ G]) = v ([h \circ G \circ F])$ by two uses of [F1].

3.1step 1.1step 2.1∎

The two linear maps agree on every $v$ and every $[h]$ , so they are equal.

Depends on