{
  "artifactId": "artifact.library.mathlib.pythagorean-theorem.v001",
  "candidateOnly": false,
  "category": "Geometry and linear algebra",
  "claimBoundary": "This target indexes Mathlib's real inner-product-space equivalence: the squared norm of x+y is the sum of squared norms exactly when x and y are orthogonal. It is a vector identity, not a direct theorem about side lengths of a Euclidean triangle.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero {F : Type*} [SeminormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ inner ℝ x y = 0",
  "family": "Inner-product geometry",
  "id": "library.mathlib.pythagorean-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/pythagorean-theorem/",
    "entryData": "/data/library-theorems/pythagorean-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.pythagorean-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.pythagorean-theorem.v001.evidence.json",
    "source": "/sources/upstream/pythagorean-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero {F : Type*} [SeminormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F) : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ inner ℝ x y = 0",
    "concepts": [
      "orthogonality",
      "inner product",
      "norm",
      "vector sum",
      "equivalence"
    ],
    "plainLanguage": "For two vectors in a real inner-product space, the square of the length of their sum equals the sum of their squared lengths if and only if the vectors meet at a right angle.",
    "poster": {
      "alt": "An ivory editorial poster states the squared-norm identity if and only if the real inner product vanishes, beside a right-angle vector construction and three squared-norm tiles.",
      "byteSize": 2003478,
      "caption": "For real inner-product-space vectors, the Pythagorean norm identity is equivalent to orthogonality.",
      "description": "The poster foregrounds the two-way equivalence between a squared-norm sum and a zero real inner product, while separating that vector statement from a direct claim about triangle side lengths.",
      "derivatives": [
        {
          "byteSize": 77920,
          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-poster-v1-640.webp",
          "sha256": "sha256:6d56484439eeb2054b8ece607f4363e723d64d1d7a2c3f93f7d8373669fbac18",
          "width": 640
        },
        {
          "byteSize": 142302,
          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:7b4b9418757c2d1996bcfb3f69b24bc0dae0fa199a978cb6e0e773a5702a0ddd",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-22T01:04:16Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-poster-v1.png",
      "sha256": "sha256:83b6dd6225c0b61890f03cc6271d62f95517b21a5d0b9593199a512efc1c892b",
      "title": "Pythagorean theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "REAL INNER-PRODUCT SPACES · ORTHOGONALITY\nPYTHAGOREAN THEOREM\n‖x + y‖² = ‖x‖² + ‖y‖²\nif and only if\ninner(x, y) = 0\nTHE NORM IDENTITY EXACTLY CHARACTERIZES ORTHOGONALITY\nHOW TO READ IT\n1 · Choose vectors x and y\n2 · Compare the squared norm of x + y\n3 · Test whether the real inner product vanishes\n4 · Each condition implies the other\nEXACT SCOPE\nVector identity in a real inner-product space, not a direct theorem about triangle side lengths."
    },
    "visual": {
      "alt": "Two perpendicular green and cobalt vectors share an origin, their ivory diagonal reaches the opposite corner, and green-edged, cobalt-edged, and gold-edged squares record the squared norms.",
      "byteSize": 1020567,
      "caption": "Orthogonality is exactly the condition behind the squared-norm sum identity.",
      "description": "A right-angle vector construction and three area squares connect the norm of a vector sum with the norms of its two perpendicular summands.",
      "derivatives": [
        {
          "byteSize": 7338,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:9098df92531a43254870c4dbfa489dc01eb3457caf9267fa7c744b1aa27a56d1",
          "width": 640
        },
        {
          "byteSize": 16032,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:adf58f77cd1df0589ef5308e2b9f66fdaefdce5ab12acf84d53135db37ca6e2c",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-22T01:04:16Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/pythagorean-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:31dc2311d11d54638d0ed28939135c64dc3b4278f1031b301153bd172c407d09",
      "title": "Pythagorean theorem schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The Pythagorean relation is a defining bridge between distance and orthogonality. Mathlib's selected equivalence exposes the inner-product structure behind the familiar right-triangle identity."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Pythagorean theorem or Mathlib's declaration.",
    "The selected declaration is a vector equivalence, not a direct side-length theorem for drawn triangles.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.pythagorean-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "2 of 3 presentation reviews recorded",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "In a real inner-product space, the squared-norm identity for a vector sum holds exactly when the summands are orthogonal.",
  "targetId": "target.library.mathlib.pythagorean-theorem.v001",
  "title": "Pythagorean Theorem",
  "upstreamOrigin": {
    "declarationName": "norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/InnerProductSpace/Basic.lean#L554",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:5b95dccc6230b9251744b12292942814523bf36b7cb41bd50c31d00a99296f3e",
    "sourceByteLength": 46163,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/Analysis/InnerProductSpace/Basic.lean",
    "sourceLine": 554,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}