feat: add Higman infinite finitely-presented simple group eval problem#384
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Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Codex review of PR #384 confirmed faithfulness of the formal statement (existential shape correct; IsSimpleGroup includes Nontrivial so trivial-presentation witnesses are excluded; no vacuous-truth exploit) and caught two real errors. (1) My prior text claimed Higman's 1951 group was finitely presented and simple; in fact the 1951 paper constructs a finitely *generated* infinite simple group as a simple quotient of the four-generator "chain of squarings" group with no nontrivial finite quotients — the simple quotient need not be finitely presented. The first infinite finitely-presented simple groups are the Higman–Thompson groups G_{n,r} from Higman's 1974 monograph. Rewrote docstring + manifest accordingly. (2) Same HNN/Britton fix as #383: mathlib v4.30 DOES have HNN extensions and Britton's lemma in `Mathlib/GroupTheory/HNNExtension.lean'; corrected the "mathlib lacks X" list. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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This PR adds Higman's infinite finitely-presented simple group theorem as an eval problem: there exists an infinite, finitely presented, simple group — some
nand a finite relator setrels ⊆ FreeGroup (Fin n)for whichPresentedGroup relsis bothIsSimpleGroupandInfinite.Pure existence of a pathological finite presentation. The first such examples come from Graham Higman's 1974 monograph Finitely Presented Infinite Simple Groups — the Higman–Thompson groups
G_{n,r}(for integersn ≥ 2,r ≥ 1) as groups of piecewise-linear bijections of certain Cantor-like spaces. Higman's 1951 paper had earlier constructed a finitely generated infinite simple group as a simple quotient of the four-generator group⟨a, b, c, d | bab⁻¹ = a², cbc⁻¹ = b², dcd⁻¹ = c², ada⁻¹ = d²⟩(no nontrivial finite quotients) — but did not give a finite presentation for the simple quotient.Mathlib v4.30 has
FreeGroup,PresentedGroup,IsSimpleGroup,Infinite, the normal-closure substrate (Subgroup.IsNormalClosureFG), HNN extensions, and Britton's lemma (Mathlib/GroupTheory/HNNExtension.lean), but no Higman–Thompson construction, no Higman embedding theorem, and no construction of an infinite finitely-presented simple group on hand.§122 of Knill's Some Fundamental Theorems in Mathematics (https://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf).
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