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  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem ContractingWith.exists_fixedPoint {α : Type*} [EMetricSpace α] {K : NNReal} {f : α → α} [CompleteSpace α] (hf : ContractingWith K f) (x : α) (hx : edist x (f x) ≠ ⊤) : ∃ y, Function.IsFixedPt f y ∧ Filter.Tendsto (fun n ↦ f^[n] x) Filter.atTop (nhds y) ∧ ∀ n : ℕ, edist (f^[n] x) y ≤ edist x (f x) * (K : ENNReal) ^ n / (1 - (K : ENNReal))",
  "family": "Metric fixed-point theory",
  "id": "library.mathlib.banach-fixed-point-theorem.v001",
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    "completeFormalType": "theorem ContractingWith.exists_fixedPoint {α : Type*} [EMetricSpace α] {K : NNReal} {f : α → α} [CompleteSpace α] (hf : ContractingWith K f) (x : α) (hx : edist x (f x) ≠ ⊤) : ∃ y, Function.IsFixedPt f y ∧ Filter.Tendsto (fun n ↦ f^[n] x) Filter.atTop (nhds y) ∧ ∀ n : ℕ, edist (f^[n] x) y ≤ edist x (f x) * (K : ENNReal) ^ n / (1 - (K : ENNReal))",
    "concepts": [
      "contraction mapping",
      "fixed point",
      "complete metric space",
      "geometric convergence",
      "error bound"
    ],
    "plainLanguage": "Repeatedly apply a map that shrinks distances by a factor below one. In a complete space the iterates converge to a point left unchanged by the map, and the theorem quantifies how far each iterate can still be from that point.",
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      "caption": "The selected Mathlib declaration gives a fixed point, convergence from the chosen start, and an explicit error bound.",
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      "generatedAt": "2026-07-21T23:17:52Z",
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      "sha256": "sha256:63aa00889cb5b06edf65582b6924109c59927cc3616a6e48f7180f999a5a7926",
      "title": "Banach Fixed-Point Theorem — existence, convergence, and error",
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      "transcript": "ANALYSIS · CONTRACTIONS IN A COMPLETE SPACE\nBANACH FIXED-POINT THEOREM\nK < 1\nedist(f(u), f(v)) ≤ K · edist(u, v)\nedist(x, f(x)) < ∞\nx\nf(x)\nf²(x)\nf³(x)\nfⁿ(x)\ny\nThere exists y with f(y) = y\nf^[n](x) → y\nedist(f^[n](x), y) ≤ edist(x, f(x)) · K^n / (1 − K)\nHOW THE PROOF MOVES\n1 · Iterate the contraction from x\n2 · Bound successive gaps geometrically\n3 · Use completeness to obtain a limit y\n4 · Pass to f(y) = y and retain the error bound\nEXACT SCOPE\nExistence, convergence, and error bound. Uniqueness is not part of this selected declaration."
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      "title": "A contracting orbit converges",
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          "caption": "Contraction makes successive iterate gaps decay geometrically from the finite first displacement.",
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          "title": "Control successive iterate gaps",
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          "description": "cauchySeq_of_edist_le_geometric combines the shrinking successive-gap estimate, K<1, and the finite initial extended distance to prove that the iterate sequence is Cauchy.",
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          "title": "Turn the geometric bound into a Cauchy sequence",
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          "equation": "CauchySeq (fun n ↦ f^[n] x)",
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          "description": "isFixedPt_of_tendsto_iterate uses convergence and continuity of the Lipschitz map to prove f y = y. edist_le_of_edist_le_geometric_of_tendsto then retains the explicit bound on the distance from every iterate to y.",
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      "title": "From geometric contraction to a fixed point"
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    "whyLandmark": "Banach's theorem converts quantitative shrinking into existence and convergence. It underlies iterative numerical methods, differential and integral equations, and many constructive existence arguments."
  },
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    "The selected declaration does not itself state uniqueness of the fixed point.",
    "It assumes a complete extended metric space and finite first displacement from the chosen start.",
    "ProofAtlas is indexing an existing Mathlib theorem, not claiming a new proof."
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    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "2 of 3 presentation reviews recorded",
    "upstreamIndexed": "Pinned source bytes verified locally"
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  "targetId": "target.library.mathlib.banach-fixed-point-theorem.v001",
  "title": "Banach Fixed-Point Theorem",
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