{ "artifactId": "artifact.library.mathlib.fundamental-theorem-calculus.v001", "candidateOnly": false, "category": "Analysis", "claimBoundary": "This page presents Mathlib's endpoint-difference form over the unordered closed interval Set.uIcc a b with an oriented interval integral. It requires pointwise differentiability there and IntervalIntegrable (deriv f); it is not every formulation of the Fundamental Theorem of Calculus.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem intervalIntegral.integral_deriv_eq_sub {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f : ℝ → E} (hderiv : ∀ x ∈ Set.uIcc a b, DifferentiableAt ℝ f x) (hint : IntervalIntegrable (deriv f) MeasureTheory.volume a b) : ∫ (y : ℝ) in a..b, deriv f y = f b - f a", "family": "Differentiation and integration", "id": "library.mathlib.fundamental-theorem-calculus.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/fundamental-theorem-calculus/", "entryData": "/data/library-theorems/fundamental-theorem-calculus.json", "evidence": "/proofs/artifact.library.mathlib.fundamental-theorem-calculus.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.fundamental-theorem-calculus.v001.evidence.json", "source": "/sources/upstream/fundamental-theorem-calculus/" }, "landmark": { "completeFormalType": "intervalIntegral.integral_deriv_eq_sub {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f : ℝ → E} (hderiv : ∀ x ∈ Set.uIcc a b, DifferentiableAt ℝ f x) (hint : IntervalIntegrable (deriv f) MeasureTheory.volume a b) : ∫ (y : ℝ) in a..b, deriv f y = f b - f a", "concepts": [ "derivatives", "interval integrals", "endpoint values", "normed spaces" ], "plainLanguage": "Let a function from the real line into a complete real normed vector space be differentiable throughout the closed interval between two endpoints. If its derivative is interval-integrable, then integrating that derivative from the first endpoint to the second gives the function's final value minus its initial value.", "poster": { "alt": "An ivory theorem poster shows an oriented interval from a to b, a derivative-accumulation ribbon, the endpoint difference formula, a reversed-orientation echo, and the theorem's completeness, differentiability, and integrability hypotheses.", "byteSize": 2666030, "caption": "Oriented integration of the derivative recovers the endpoint change, with every analytic hypothesis and the reversed-endpoint behavior kept visible.", "description": "The endpoint-difference equation dominates the upper poster. A ribbon carries accumulation from a to b and a smaller reversed copy shows the sign reversal. A hypothesis strip names completeness, the unordered closed interval differentiability condition, and interval integrability before a reading guide and exact-scope footer.", "derivatives": [ { "byteSize": 106730, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-poster-v1-640.webp", "sha256": "sha256:de703671a4d1e1a6ec6bfb9b7dff8ca37e7e4798ec5391661ccda6424ecfe621", "width": 640 }, { "byteSize": 230578, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-poster-v1-1024.webp", "sha256": "sha256:aca5697bf680c299bad609f11e0b0f7bdf706d143628e9e76d414c0b017e5638", "width": 1024 } ], "generatedAt": "2026-07-18T16:06:52.197Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-poster-v1.png", "sha256": "sha256:504754ae2694cb0bc3ca067a516442311e0745172ea6e4793d0963b80219329c", "title": "Fundamental Theorem of Calculus at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "ANALYSIS · ORIENTED INTERVAL INTEGRAL\nFUNDAMENTAL THEOREM OF CALCULUS\nf : ℝ → E\n∫ a..b deriv f = f(b) − f(a)\nE COMPLETE · REAL NORMED\nDIFFERENTIABLE ON uIcc a b\nderiv f INTERVAL-INTEGRABLE\nORIENTED FROM a TO b\nREVERSE a AND b · REVERSE THE SIGN\nHOW TO READ IT\n1 · Check the whole unordered closed interval\n2 · Integrate the derivative\n3 · Recover the endpoint change\nEXACT SCOPE\nThis endpoint-difference form keeps all stated analytic hypotheses." }, "visual": { "alt": "A smooth curve spans two endpoints above layered blue and green signed accumulation, ending at one vertical endpoint difference.", "byteSize": 2620663, "caption": "Accumulating the derivative across an oriented interval produces the function's endpoint change.", "description": "A smooth ivory curve runs between two marked endpoints. Layered cobalt and emerald regions below the baseline accumulate from left to right and visually meet a gold vertical difference at the final endpoint; a quieter reversed-orientation echo sits below.", "derivatives": [ { "byteSize": 13828, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-schematic-v1-640.webp", "sha256": "sha256:87c1c80934a66bde8c60b412c22c54047cdde2d57cf6ca52841dae084185fe2d", "width": 640 }, { "byteSize": 46500, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-schematic-v1-1200.webp", "sha256": "sha256:19ec1ea4ee22d25dcca1fd3b81d6a00af40dea019c189cfdb93c5b7ef9b4a7e9", "width": 1200 } ], "generatedAt": "2026-07-17T19:58:58Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fundamental-theorem-calculus/theorem-schematic-v1.png", "sha256": "sha256:f920f5b329c9e4dff586536454ab7c11e739569f2a88d7c9a7937d46506beb22", "title": "Derivative accumulation and endpoint change schematic", "width": 1536 }, "walkthrough": { "introduction": "The checked route proves the difficult real inequality through Vitali--Caratheodory approximation and closed-interval propagation, obtains the reverse inequality by negation, lifts the real identity through separating continuous dual functionals, handles interval orientation, and finally specializes the result to the derivative of a differentiable function.", "pages": [ { "alt": "A forest-green function ribbon rises between two gold endpoint pairs above a cobalt integral basin, with a thin vermilion margin separating the compared endpoint quantities.", "byteSize": 2976087, "caption": "The real-valued upper estimate is reduced to proving endpoint growth is at most the comparison integral plus every positive error margin.", "description": "For the hard real direction of FTC-2 on an ordered interval, `le_of_forall_pos_le_add` reduces the required inequality to one with an arbitrary positive epsilon margin. 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It is a central organizing principle of calculus and analysis." }, "nonClaims": [ "Proof Atlas did not originate the Fundamental Theorem of Calculus or Mathlib's declaration.", "The hypotheses in the exact formal type are material and are not replaced by an unconditional slogan.", "The page does not claim a new analytic argument." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.fundamental-theorem-calculus.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": "Under differentiability and integrability hypotheses, the oriented interval integral of a derivative equals the endpoint difference.", "targetId": "target.library.mathlib.fundamental-theorem-calculus.v001", "title": "Fundamental Theorem of Calculus", "upstreamOrigin": { "declarationName": "intervalIntegral.integral_deriv_eq_sub", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/MeasureTheory/Integral/IntervalIntegral/FundThmCalculus.lean#L1177", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:b10f02c6117ee45b34a8ec509c7fa7a949362e3bdf946fcd8fed72227e6593aa", "sourceByteLength": 75436, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/MeasureTheory/Integral/IntervalIntegral/FundThmCalculus.lean", "sourceLine": 1177, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }