ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Images and preimages of submodules form a concrete Galois connection

Example

Let $f : Z \to Z / 6$ be reduction modulo $6$ . Then direct images and inverse images of submodules are the familiar image and preimage operations on subgroups, and they satisfy the Galois-connection inequality $f_{*} B \leq C ⟺ B \leq f^{*} C$ .

Facts & Assumptions

Given: The homomorphism $f : Z \to Z / 6$ .

[L1]

Direct and inverse images form a Galois connection (Direct and inverse image of subobjects form a Galois connection).

[L2]

Inverse images preserve meets and direct images preserve joins (Inverse image preserves meets and direct image preserves joins).

[L3]

Kernels and ordinary images are the special cases of inverse and direct images (Kernel and image are the inverse and direct images along a morphism).

Verification

technique · direct

1.1L3algebra

For the subgroup $2 Z \leq Z$ , the direct image is $f_{*} (2 Z) = {0, 2, 4} \leq Z / 6$ . For the subgroup $C = {0, 2, 4} \leq Z / 6$ , the inverse image is $f^{*} C = 2 Z$ . Also [L3] identifies $f^{*} (0) = 6 Z = \ker (f)$ and $f_{*} (Z) = Z / 6 = im (f)$ .

2.1L1L2step 1.1∎

Thus $f_{*} (2 Z) \leq C$ and $2 Z \leq f^{*} C$ are the same true statement in this example, exactly as [L1] predicts. Likewise [L2] is visible here: the meet of ${0, 2, 4}$ with $0$ pulls back to $2 Z \cap 6 Z = 6 Z$ , and the join of $2 Z$ with $6 Z$ pushes forward to ${0, 2, 4}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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.

Sources

Saunders Mac Lane, Categories for the Working Mathematician, Section VIII.3 (standard reference, not scraped)