LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

A nonzero rank minor supplies the source coordinates for the constant-rank theorem

Statement

Let $k \geq 1$ , let $f : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , and suppose $rank D f (a) = r$ . After permuting source and target coordinates, if $r > 0$ the leading $r \times r$ minor of $D f (a)$ is nonzero and $Φ (x) = (f_{0} (x), \dots, f_{r - 1} (x), x_{r}, \dots, x_{m - 1})$ is a local $C^{k}$ diffeomorphism at $a$ . If $r = 0$ , the same conclusion holds with $Φ$ equal to the identity map. Empty coordinate blocks are omitted.

Facts & Assumptions

Given: The map $f$ , the point $a$ , and $r = rank D f (a)$ .

[L1]

Positive matrix rank is detected by a nonzero minor (A matrix has rank at least $r$ exactly when it has a nonzero $r$ -rowed minor, The rank of a derivative and constant-rank Euclidean maps).

[L2]

A real square matrix is invertible exactly when its determinant is nonzero; a $C^{1}$ map between equal-dimensional Euclidean open sets with invertible derivative at a point is a local $C^{1}$ diffeomorphism there, and a $C^{k}$ map has a $C^{k}$ local inverse (A finite square real matrix is invertible if and only if its determinant is nonzero, The Euclidean inverse function theorem, A local inverse of a $C^{k}$ regular map is $C^{k}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

Proof

technique · direct

1.1given

If $r = 0$ , take $Φ = {id}_{U}$ ; it is a $C^{k}$ diffeomorphism on every open neighbourhood of $a$ .

1.2givenL1choose

Suppose $r > 0$ . By [L1], choose a nonzero $r$ -rowed minor and permute coordinates so it is the leading minor. The derivative $D Φ (a)$ is block triangular with that $r \times r$ block and an identity block of size $m - r$ on its diagonal.

2.1step 1.2L2algebra

Its determinant is the nonzero leading minor, including the full-rank case $r = m$ where the identity block is empty. Thus [L2] makes $D Φ (a)$ invertible and $Φ$ a local $C^{k}$ diffeomorphism at $a$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 cover every possible rank and give the asserted source coordinates.

A nonzero rank minor supplies the source coordinates for the constant-rank theorem

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources