ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 first isomorphism theorem for $(x, y, z) \mapsto (x + y, y + z)$

Example

For $T : F^{3} ⟶ F^{2}, T (x, y, z) = (x + y, y + z),$ one has $\ker T = F (- 1, 1, - 1)$ and $im T = F^{2}$ . The induced map $\tilde{T} : F^{3} / F (- 1, 1, - 1) ⟶ F^{2}, (x, y, z) + \ker T ⟼ (x + y, y + z)$ is an isomorphism, with inverse $(a, b) \mapsto (a, 0, b) + \ker T$ .

Facts & Assumptions

Given: The displayed coordinate map $T$ .

[L1]

The first isomorphism theorem gives a unique isomorphism $V / \ker T \to im T$ sending $v + \ker T$ to $T (v)$ (First isomorphism theorem for vector spaces: $V / \ker T$ is isomorphic to $im T$ ).

Verification

technique · computation

1.1algebra

Solving $x + y = 0$ and $y + z = 0$ gives $(x, y, z) = y (- 1, 1, - 1)$ , so the kernel is the stated line.

1.2algebra

For every $(a, b) \in F^{2}$ , $T (a, 0, b) = (a, b)$ , so $T$ is surjective.

2.1step 1.1step 1.2L1∎

Fact [L1] gives the induced isomorphism with the displayed formula, and step 1.2 shows that sending $(a, b)$ to $(a, 0, b) + \ker T$ is its two-sided inverse.

Depends on

First isomorphism theorem for vector spaces: $V/\ker T$ is isomorphic to $\operatorname{im}T$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.

The first isomorphism theorem for (x,y,z)↦(x+y,y+z)(x,y,z)↦(x+y,y+z)