{ "artifactId": "artifact.library.mathlib.knaster-tarski-theorem.v001", "candidateOnly": false, "category": "Order theory", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "instance fixedPoints.completeLattice {α : Type u} [CompleteLattice α] (f : α →o α) : CompleteLattice (Function.fixedPoints f)", "family": "Order-theoretic fixed-point theory", "id": "library.mathlib.knaster-tarski-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/knaster-tarski-theorem/", "entryData": "/data/library-theorems/knaster-tarski-theorem.json", "evidence": "/proofs/artifact.library.mathlib.knaster-tarski-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.knaster-tarski-theorem.v001.evidence.json", "source": "/sources/upstream/knaster-tarski-theorem/" }, "landmark": { "completeFormalType": "instance fixedPoints.completeLattice {α : Type u} [CompleteLattice α] (f : α →o α) : CompleteLattice (Function.fixedPoints f)", "concepts": [ "complete lattice", "monotone self-map", "fixed-point subtype", "least and greatest fixed points", "arbitrary suprema and infima" ], "plainLanguage": "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.", "poster": { "alt": "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.", "byteSize": 2669724, "caption": "Mathlib packages the fixed points of a monotone self-map as a complete lattice, using nextFixed above ambient suprema and prevFixed below ambient infima.", "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.", "derivatives": [ { "byteSize": 89054, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:c1e9d80cc03fd6f4e5c52b4311e44438ad14d99a331c11e7511028df9c075d2a", "width": 640 }, { "byteSize": 209888, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:4e1cd20cbfee166ff240a22ab03852c4b947339b1958733e02d50de4a4c34fed", "width": 1024 } ], "generatedAt": "2026-07-25T16:27:34-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-poster-v1.png", "sha256": "sha256:e555917f3059d06a7617a213fe9eaa9bf9702c7564f5d3f34e6db2d48d5a3926", "title": "Knaster–Tarski at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "ORDER THEORY · FIXED POINTS\nKNASTER–TARSKI\nα A COMPLETE LATTICE\nf : α →o α\nfixedPoints f CARRIES A CompleteLattice INSTANCE\n\nFIXED POINTS FORM THEIR OWN COMPLETE LATTICE\n\nSUPREMA\nAMBIENT SUPREMUM → nextFixed\n\nINFIMA\nAMBIENT INFIMUM → prevFixed\n\nEXACT SCOPE\nA MONOTONE SELF-MAP ON A COMPLETE LATTICE.\nTHIS IS AN INSTANCE, NOT MERELY ONE FIXED-POINT EXISTENCE CLAIM.\nNOT KLEENE ITERATION." }, "visual": { "alt": "An ivory complete lattice on deep green contains a luminous gold fixed-point sublattice with top, bottom, joins, and meets, while emerald order-preserving arrows and dual closure arcs indicate the monotone self-map and fixed-point bounds.", "byteSize": 3355558, "caption": "Inside the ambient complete lattice, the map's fixed points retain both arbitrary supremum-like and infimum-like structure.", "description": "The low-text schematic shows one ambient complete lattice, an order-preserving emerald endomap, and a highlighted gold family of fixed points closed into a complete sublattice through dual upper and lower constructions.", "derivatives": [ { "byteSize": 54702, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:5d3206915e512ee8d46e40b2d4fce5d8f695208552b2236e26f0405bdf6f112f", "width": 640 }, { "byteSize": 218436, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:5bf79364f623be3f4610d988965a4ef7c6d00e953cd5336b660c352ed4711e19", "width": 1200 } ], "generatedAt": "2026-07-25T16:27:34-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/knaster-tarski-theorem/theorem-schematic-v1.png", "sha256": "sha256:64053448a41f0b5d8ad70a0c36268845384ca9243b1f136103a8568b0752e0d7", "title": "Fixed points form a complete lattice", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "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." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.knaster-tarski-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "The fixed points of a monotone self-map of a complete lattice carry their own complete-lattice structure.", "targetId": "target.library.mathlib.knaster-tarski-theorem.v001", "title": "Knaster–Tarski Theorem", "upstreamOrigin": { "declarationName": "fixedPoints.completeLattice", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Order/FixedPoints.lean#L252", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:51112ea02c58e0a3a0d32b0c855592bcea65a878370832e20fbe6ab2f2604d43", "sourceByteLength": 10061, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Order/FixedPoints.lean", "sourceLine": 252, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }