{
  "artifactId": "artifact.library.mathlib.well-ordering-theorem.v001",
  "candidateOnly": false,
  "category": "Logic and foundations",
  "claimBoundary": "This target indexes Mathlib's theorem that every type α admits some LinearOrder whose strict relation is well founded. It is an existential result only: the selected declaration does not provide a computable or canonical order, preserve or extend a relation already on α, or identify the chosen order with a particular ordinal.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem exists_wellOrder (α : Type u) : ∃ (_ : LinearOrder α), WellFoundedLT α",
  "family": "Choice and well-ordering",
  "id": "library.mathlib.well-ordering-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/well-ordering-theorem/",
    "entryData": "/data/library-theorems/well-ordering-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.well-ordering-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.well-ordering-theorem.v001.evidence.json",
    "source": "/sources/upstream/well-ordering-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem exists_wellOrder (α : Type u) : ∃ (_ : LinearOrder α), WellFoundedLT α",
    "concepts": [
      "well-orders",
      "linear orders",
      "well-founded relations",
      "cardinals",
      "pullback orders",
      "choice principles"
    ],
    "plainLanguage": "For any type α, regardless of its size or prior structure, there is some way to compare every pair of elements linearly such that the resulting strict order is well founded.",
    "poster": {
      "alt": "An ivory and forest-green theorem poster shows the same arbitrary collection before and after placement on one open-ended total-order path, states the exact existential LinearOrder and WellFoundedLT conclusion, and excludes canonicity, computability, and relation extension.",
      "byteSize": 2200137,
      "caption": "Every type admits some linear order whose strict relation is well founded.",
      "description": "The poster pairs the exact Mathlib endpoint with a restrained before-and-after diagram. Heterogeneous elements begin without relations and reappear along one selected open-ended order, while the exact-scope panel keeps the result existential and disclaims canonicity, computability, and extension of an existing relation.",
      "derivatives": [
        {
          "byteSize": 69138,
          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-poster-v1-640.webp",
          "sha256": "sha256:ec145f4eeb1111bf980707ecc5c629ec879460919ac83df1228122117a8a6ce4",
          "width": 640
        },
        {
          "byteSize": 140004,
          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:f500b5878a27024f0431619938edb9abf3eef333764f7343320078872b5423fa",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-25T16:25:07-04:00",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-poster-v1.png",
      "sha256": "sha256:9bd41c250b4ec8b1e9e3e9706002e669d1be020897854991eb63958ffaae9b24",
      "title": "Well-Ordering Theorem — existence without canonicity",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "FOUNDATIONS · WELL-ORDERING PRINCIPLE\nWELL-ORDERING THEOREM\nFor every type α\n∃ (_ : LinearOrder α), WellFoundedLT α\nSOME LINEAR ORDER HAS A WELL-FOUNDED STRICT ORDER\nEXACT SCOPE\nExistence only · not canonical · not computable\nDoes not extend a pre-existing relation"
    },
    "visual": {
      "alt": "Exactly twelve distinct ivory tokens appear in a three-by-four array and reappear once each on one gold order rail, with one-to-one connector lines.",
      "byteSize": 2596545,
      "caption": "The same sample elements are organized into one selected strict order without adding, losing, or merging any element.",
      "description": "A finite twelve-element sample illustrates how one underlying collection can be equipped with a selected well-order. Every distinct token is preserved exactly once; the sample does not restrict the theorem to finite types or claim a canonical construction.",
      "derivatives": [
        {
          "byteSize": 24778,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-schematic-v3-640.webp",
          "sha256": "sha256:1d7f4ea475fd53c28fdabc674d3225eadfb836931ee2f3fd4c4e76e20e5d700f",
          "width": 640
        },
        {
          "byteSize": 85834,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-schematic-v3-1200.webp",
          "sha256": "sha256:a08042e95a6fb1895ad864c9d74abcc4252197d96a1457fcb5107d6d5ec1d67c",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T20:55:00-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/well-ordering-theorem/theorem-schematic-v3.png",
      "sha256": "sha256:3b6ce2418c6415734b2d078255bf224b804ed145b1d514f43923f80b8c242661",
      "title": "One underlying collection, one selected well-order",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The well-ordering theorem is a central choice principle: it makes transfinite methods available on an arbitrary type after choosing a suitable order. Mathlib's declaration captures the exact existence of a linear order with well-founded strict comparison."
  },
  "nonClaims": [
    "Proof Atlas did not originate the well-ordering theorem or Mathlib's declaration.",
    "The theorem asserts existence; it does not choose a canonical, unique, or computationally recoverable order.",
    "The selected declaration does not say that its order extends or preserves any pre-existing relation on α.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.well-ordering-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": "Every type admits a linear order whose strict relation is well founded.",
  "targetId": "target.library.mathlib.well-ordering-theorem.v001",
  "title": "Well-Ordering Theorem",
  "upstreamOrigin": {
    "declarationName": "exists_wellOrder",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/SetTheory/Cardinal/Order.lean#L529",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:7a219ff0395f8eb23002f7aa3f3ae4eef3f5ad9d727a5aea2dbb1645ea2f83a0",
    "sourceByteLength": 26881,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/SetTheory/Cardinal/Order.lean",
    "sourceLine": 529,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}