Order theory · existing Mathlib declaration

Knaster–Tarski Theorem

For any order-preserving map from a complete lattice to itself, all points left unchanged by the map themselves form a complete lattice: every family of fixed points has both a fixed-point supremum and a fixed-point infimum.

complete lattice
monotone self-map
fixed-point subtype
least and greatest fixed points
arbitrary suprema and infima

The theorem at a glance

Knaster–Tarski at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

A dark green Knaster–Tarski poster states that fixedPoints f carries a CompleteLattice instance and pairs an order-preserving self-map diagram with the nextFixed supremum and prevFixed infimum constructions. — Mathlib packages the fixed points of a monotone self-map as a complete lattice, using nextFixed above ambient suprema and prevFixed below ambient infima.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Selected after original-pixel inspection of the theorem schematic scientific; theorem-specific structure, legibility, margins, and public claim boundary passed. — Inside the ambient complete lattice, the map's fixed points retain both arbitrary supremum-like and infimum-like structure.

Complementary intuition

A mathematical landmark

Knaster–Tarski turns monotonicity and completeness into an entire lattice of solutions. It is foundational in order theory and supports fixed-point semantics across logic, program verification, recursive definitions, and theoretical computer science.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

instance fixedPoints.completeLattice {α : Type u} [CompleteLattice α] (f : α →o α) : CompleteLattice (Function.fixedPoints f)

Selected evidence declaration text

instance fixedPoints.completeLattice {α : Type u} [CompleteLattice α] (f : α →o α) : CompleteLattice (Function.fixedPoints f)

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's structure-producing instance for the fixed-point subtype of a bundled monotone self-map of a complete lattice. It gives that subtype a CompleteLattice instance; it is not merely a one-fixed-point existence theorem and does not state the separate Kleene iteration result that follows later in the source file.

The selected declaration is an instance giving CompleteLattice structure to Function.fixedPoints f, not merely a proposition asserting existence of one fixed point.
It does not assert uniqueness, convergence, continuity, or an algorithm for computing fixed points.
It is not Mathlib's separate Kleene fixed-point theorem or an iteration formula.
ProofAtlas is indexing an existing Mathlib declaration, not claiming new mathematics or new proof evidence.

Provenance

Origin and evidence stay separate

Existing declaration: fixedPoints.completeLattice in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.knaster-tarski-theorem.v001
Source handling: Verified upstream reference; no mirrored source package