LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)verified 2026-07-26 (claude-opus-5)

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 smallest admissible set exists

Statement

Let $(P, \le)$ be a chain-complete poset and $f : P \to P$ progressive (Chain-complete poset). Then there is a smallest $f$ -admissible subset $M \subseteq P$ (Admissible subset (Bourbaki–Witt)): $M$ is admissible, and $M \subseteq A$ for every admissible $A \subseteq P$ .

Facts & Assumptions

Given: A chain-complete poset $(P, \le)$ and a progressive map $f : P \to P$ .

[A1]

A subset $A \subseteq P$ is admissible when (C1) $f(x) \in A$ for every $x \in A$ , and (C2) $\sup C \in A$ for every chain $C \subseteq A$ (Admissible subset (Bourbaki–Witt)).

[L1]

Every chain of $P$ has a least upper bound in $P$ (Chain-complete poset).

Proof

technique · direct

1.1

$P$ satisfies (C1), because $f$ is a map from $P$ to $P$ , so $f(x) \in P$ for every $x \in P$ .

1.2

$P$ satisfies (C2), because every chain $C \subseteq P$ has a least upper bound in $P$ .

A1L1

2.1

So $P$ is admissible, and the collection $\mathcal{A}$ of all admissible subsets of $P$ is nonempty.

step 1.1step 1.2A1

3.1

Define $M = \bigcap \mathcal{A}$ , the intersection of all admissible subsets of $P$ .

step 2.1construct

4.1

Let $x \in M$ . For every $A \in \mathcal{A}$ we have $x \in A$ , hence $f(x) \in A$ by (C1); since this holds for every $A \in \mathcal{A}$ , $f(x) \in M$ , so $M$ satisfies (C1).

step 3.1A1

4.2

Let $C \subseteq M$ be a chain. For every $A \in \mathcal{A}$ we have $C \subseteq A$ , hence $\sup C \in A$ by (C2); since this holds for every $A \in \mathcal{A}$ , $\sup C \in M$ , so $M$ satisfies (C2).

step 3.1A1L1

4.3

If $A \subseteq P$ is admissible then $A \in \mathcal{A}$ , so $M \subseteq A$ because $M$ is the intersection of a collection containing $A$ .

step 3.1

5.1

$M$ is admissible.

step 4.1step 4.2A1

6.1

$M$ is admissible and contained in every admissible subset, so it is the smallest one.

step 5.1step 4.3∎

Remarks

Uniqueness is automatic: two smallest admissible sets each contain the other, so they are equal. This licenses writing $M$ for "the" smallest admissible set throughout Extremal element and its cut (Bourbaki–Witt) and the lemmas that follow.
$M$ is never empty. Condition (C2) applied to the empty chain puts $\bot = \sup\emptyset$ into every admissible set, so $\bot \in M$ .
Minimality is used exactly twice in what follows, in Everything in $M$ is comparable to an extremal element and in Every element of $M$ is extremal, and both times in the same shape: to prove that all of $M$ has some property, one shows that the elements of $M$ with that property again form an admissible set, which minimality then forces to be all of $M$ . That pattern is what replaces transfinite recursion in this proof of Bourbaki–Witt fixed point theorem. The remaining lemmas use only admissibility of $M$ , progressivity of $f$ , and the order axioms.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 5 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Mathlib, Order.BourbakiWitt (standard reference, not scraped)
Bourbaki-Witt Principle (Menemui Matematik 39(1), 2017) (standard reference, not scraped)
Bourbaki–Witt theorem (Wikipedia) (standard reference, not scraped)