{
  "artifactId": "artifact.library.mathlib.intermediate-value-theorem.v001",
  "candidateOnly": false,
  "category": "Topology and analysis",
  "claimBoundary": "This target indexes Mathlib's ordered interval-image inclusion for a continuous function on [a,b]: every value between f(a) and f(b) occurs when a is at most b. The selected orientation uses Icc(f a)(f b), so it does not separately state the reversed-endpoint form.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem intermediate_value_Icc {α : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type*} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b)) : Set.Icc (f a) (f b) ⊆ f '' Set.Icc a b",
  "family": "Continuous interval maps",
  "id": "library.mathlib.intermediate-value-theorem.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/intermediate-value-theorem/",
    "entryData": "/data/library-theorems/intermediate-value-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.intermediate-value-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.intermediate-value-theorem.v001.evidence.json",
    "source": "/sources/upstream/intermediate-value-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem intermediate_value_Icc {α : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type*} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b)) : Set.Icc (f a) (f b) ⊆ f '' Set.Icc a b",
    "concepts": [
      "intermediate values",
      "continuity",
      "closed intervals",
      "connectedness",
      "image inclusion"
    ],
    "plainLanguage": "If f is continuous from a to b and f(a) is below f(b), then no value between those endpoint outputs can be skipped: each one equals f(x) for some x between a and b.",
    "poster": {
      "alt": "Editorial theorem poster states the ordered closed-interval hypotheses, shows a continuous curve meeting an intermediate horizontal level, and explains the four-step preimage conclusion.",
      "byteSize": 2778233,
      "caption": "The selected Mathlib declaration states the increasing endpoint-value orientation as an interval-image inclusion.",
      "description": "The retained poster presents the domain order, continuity hypothesis, image inclusion, pointwise reading, and boundary separating the reversed and unordered forms.",
      "derivatives": [
        {
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          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-poster-v1-640.webp",
          "sha256": "sha256:9150eb21635b3a248bca2996273ae374fef5aee0dd9817ba1edccd4699a0e984",
          "width": 640
        },
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-poster-v1-1024.webp",
          "sha256": "sha256:4c7cd06253568af8aaebc0730667c9cda058018d445b8fb96d6f057d332b4682",
          "width": 1024
        }
      ],
      "generatedAt": "2026-07-21T23:03:23Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-poster-v1.png",
      "sha256": "sha256:1e309a544f100a3717281f5fb8a881b6805c05a19020ba529f13939c590e8e8a",
      "title": "Intermediate Value Theorem — exact interval-image form",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "TOPOLOGY · CONTINUOUS INTERVAL MAPS\nINTERMEDIATE VALUE THEOREM\na ≤ b\nf is continuous on [a, b]\n[f(a), f(b)] ⊆ f([a, b])\nEVERY BETWEEN-VALUE IS HIT\nChoose t with f(a) ≤ t ≤ f(b)\nThen some x in [a, b] satisfies f(x) = t\nHOW TO READ IT\n1 · Fix the closed interval [a, b]\n2 · Follow a continuous function f\n3 · Choose a value between the endpoint values\n4 · Find a preimage inside the interval\nEXACT SCOPE\nSelected endpoint orientation. Reversed and unordered forms are separate declarations."
    },
    "visual": {
      "alt": "A continuous warm-ivory curve travels between two green endpoint markers while an antique-gold horizontal level meets the curve above a point inside the interval.",
      "byteSize": 2318528,
      "caption": "Continuity forces each value between f(a) and f(b) to occur at some point of the closed interval.",
      "description": "The theorem turns continuity on Icc a b and the endpoint order into an inclusion of the entire output interval inside the image of the input interval.",
      "derivatives": [
        {
          "byteSize": 17330,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:50793e42b14a516a34cb17fca0f8d105c2e5162909f8c28d20c8ba12b2d84fe6",
          "width": 640
        },
        {
          "byteSize": 59256,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:21ce05caf972614c02f0603fc619be85855f56392bef7d2fe5d4f75addced7f5",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-21T22:48:00Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/intermediate-value-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:e85f2adf15c8ce5c0d2b337324447c83b6c7b298ded9bdd016ce54799f7db8f3",
      "title": "Every level between the endpoints is reached",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The Intermediate Value Theorem captures the no-jumps consequence of continuity. It is central to real analysis, root existence, connectedness, and numerical bracketing arguments."
  },
  "nonClaims": [
    "The selected declaration fixes the endpoint-value orientation from f(a) to f(b).",
    "Reversed and unordered endpoint-value forms are separate Mathlib declarations.",
    "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.intermediate-value-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": "A continuous function on a closed interval attains every value between its endpoint values in the selected increasing endpoint orientation.",
  "targetId": "target.library.mathlib.intermediate-value-theorem.v001",
  "title": "Intermediate Value Theorem",
  "upstreamOrigin": {
    "declarationName": "intermediate_value_Icc",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Topology/Order/IntermediateValue.lean#L543",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:cfa4c897696242691c5545e03d4b9794a1aa8e35c3e7abf9b36b30dfd666f260",
    "sourceByteLength": 43657,
    "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6",
    "sourceFile": "Mathlib/Topology/Order/IntermediateValue.lean",
    "sourceLine": 543,
    "verificationKind": "git_worktree"
  },
  "publicPresentationReview": {
    "status": "reviewed"
  }
}