feat: add Schauder fixed-point theorem eval problem#316
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§60 of Knill's "Some Fundamental Theorems in Mathematics" (additional statement; the boxed main theorem is Lefschetz–Hopf, classification (c)). The Banach-space generalization of Brouwer (Schauder, 1930): every continuous self-map of a nonempty compact convex subset of a real Banach space has a fixed point. Mathlib has the supporting normed-space / convex / compactness machinery and the Banach contraction principle (strictly weaker), but no Schauder fixed-point theorem and no open mathlib PR for it. The Sperner → Brouwer → Schauder dependency chain is in motion (Sperner partly landed; Brouwer in flight via mathlib4 #36770 for invariance of domain); active downstream demand from PDE formalization efforts (Nelson Spence, 2026-03-06 Zulip thread). Stateable with zero new definitions. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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This PR adds an eval problem for the Schauder fixed-point theorem
(Juliusz Schauder, 1930): every continuous self-map of a nonempty
compact convex subset of a real Banach space has a fixed point. §60
of Knill's Some Fundamental Theorems in Mathematics (additional
statement; the boxed main theorem of §60 is Lefschetz–Hopf, classified
(c) and not stated).
Mathlib has
NormedAddCommGroup,NormedSpace,CompleteSpace,IsCompact,Convex ℝ,ContinuousOn,MapsTo, and the Banachcontraction principle (
ContractingWith.exists_fixedPoint, a strictlyweaker fixed-point theorem). But no Schauder fixed-point theorem
(
grep -ri 'Schauder.*fixed\|fixed.*Schauder' Mathlib/returnsnothing) and no open mathlib PR for it. The Sperner → Brouwer →
Schauder dependency chain is partially in motion in mathlib (Sperner
foundations partly landed; open PR
leanprover-community/mathlib4#36770 uses
Brouwer to prove invariance of domain). Active downstream demand from
PDE formalization efforts (Nelson Spence's 2026-03-06 Zulip thread
requesting Schauder / Schaefer / Leray–Schauder machinery for
elliptic-existence formalizations).
Stateable with zero new definitions.
🤖 Prepared with Claude Code