CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07‡ 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.

Unequal-depth equalities in a short exact sequence

Statement

In the setting of thm-depth-lemma, the following implications hold: $\begin{array}{lll} a < c \Rightarrow b = a, & a > c + 1 \Rightarrow b = c, \\ a < b \Rightarrow c = a - 1, & a > b \Rightarrow c = b, \\ b < c \Rightarrow a = b, & b > c \Rightarrow a = c + 1. \end{array}$ Only implications whose strict inequality is meaningful for the depth conventions are asserted.

Facts & Assumptions

Given: write $a, b, c$ for the depths of $A, B, C$ in the short exact sequence.

Proof

technique · direct

1.1

The first Depth Lemma inequality and either of the other two give $a < c \Rightarrow b = a$ and $a > c + 1 \Rightarrow b = c$ . Cyclically applying the same comparison to the remaining inequalities gives the other four displayed implications.

given

2.1

For example, if $a < c$ , then $b \geq a$ while $a \geq \min {b, c + 1}$ forces $a \geq b$ ; hence $b = a$ . If $a > c + 1$ , then $c \geq \min {a - 1, b}$ and $a - 1 > c$ force $c \geq b$ , while $b \geq c$ ; hence $b = c$ . If $a < b$ , then $c \geq a - 1$ , while $a \geq \min {b, c + 1}$ forces $a \geq c + 1$ ; hence $c = a - 1$ . The other three cases are the same comparisons.

step 1.1algebra∎

Depends on

The three Depth Lemma inequalities

Used by

Three sharp Depth Lemma inequalities Example

Dependency tree · two levels

4 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)