Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Order theory

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.

Exact theorem

Exact Mathlib statement

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

The theorem at a glance

Knaster–Tarski at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Mathlib packages the fixed points of a monotone self-map as a complete lattice, using nextFixed above ambient suprema and prevFixed below ambient infima. Explanatory diagram.
Detailed visual description

The source-bound poster presents the complete-lattice hypothesis, bundled monotone endomap, fixed-point complete-lattice instance, dual supremum and infimum constructions, and an exact-scope footer excluding one-point-only and Kleene-iteration readings.

Statement structure

Statement and scope

Statement map for Knaster–Tarski TheoremThe fixed points of a monotone self-map of a complete lattice carry their own complete-lattice structure. Claim boundary: 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 pinned upstream declaration is fixedPoints.completeLattice. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe fixed points of amonotone self-map of acomplete lattice carrytheir owncomplete-latticestructure.Claim boundaryThis target indexesMathlib'sstructure-producinginstance for thefixed-point subtype of abundled monotoneself-map of a completelattice. It gives thatsubtype aCompleteLatticeinstance; it is notmerely a one-fixed-pointexistence theorem anddoes not state theseparate Kleeneiteration result thatfollows later in thesource file.Pinned declarationmathlib ·fixedPoints.completeLatticeStatement map for Knaster–Tarski TheoremThe fixed points of a monotone self-map of a complete lattice carry their own complete-lattice structure. Claim boundary: 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 pinned upstream declaration is fixedPoints.completeLattice. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingThe fixed points of amonotone self-map of acomplete lattice carrytheir owncomplete-latticestructure.Claim boundaryThis target indexesMathlib'sstructure-producinginstance for thefixed-point subtype of abundled monotoneself-map of a completelattice. It gives thatsubtype aCompleteLatticeinstance; it is notmerely a one-fixed-pointexistence theorem anddoes not state theseparate Kleeneiteration result thatfollows later in thesource file.Pinned declarationmathlib ·fixedPoints.completeLattice

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Inside the ambient complete lattice, the map's fixed points retain both arbitrary supremum-like and infimum-like structure. Explanatory scientific diagram.
Detailed visual description

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.

Why it matters

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.

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

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

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.

Source and local evidence

Where the theorem comes from

Existing declaration
fixedPoints.completeLattice in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.knaster-tarski-theorem.v001
Source
Open the pinned upstream reference