ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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 $f_{*} (Z) = Z / 6 = im (f)$ .

L3algebra

2.1

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 ${0, 2, 4}$ .

L1L2step 1.1∎

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)