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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | import Mathlib
namespace OAI
namespace TorsionFreeZeroDivisors
/-- Ordinary group torsion-freeness, without requiring uniqueness of all roots. -/
def TorsionFree (G : Type) [Group G] : Prop :=
∀ (g : G) (n : ℕ), 0 < n → g ^ n = 1 → g = 1
/-- A connected finite Hausdorff 2-dimensional CW K(G,1). -/
def HasFiniteTwoDimensionalClassifyingSpace (G : Type) [Group G] : Prop :=
∃ (X : Type) (topX : TopologicalSpace X),
letI : TopologicalSpace X := topX
∃ (_hausdorff : T2Space X) (_connected : PathConnectedSpace X)
(cw : Topology.CWComplex (Set.univ : Set X)),
letI : Topology.CWComplex (Set.univ : Set X) := cw
Topology.CWComplex.Finite (Set.univ : Set X) ∧
(∀ n : ℕ, 2 < n → IsEmpty (Topology.CWComplex.cell (Set.univ : Set X) n)) ∧
Nonempty (Topology.CWComplex.cell (Set.univ : Set X) 2) ∧
∃ (x : X), Nonempty (G ≃* FundamentalGroup X x) ∧
∃ (E : Type) (topE : TopologicalSpace E),
letI : TopologicalSpace E := topE
ContractibleSpace E ∧
∃ p : E → X, IsCoveringMap p ∧ Function.Surjective p
/-- The entire main theorem, including its "Moreover" clause. -/
def MainTheorem : Prop :=
∃ (G : Type) (inst : Group G),
letI : Group G := inst
Group.IsFinitelyPresented G ∧ TorsionFree G ∧
HasFiniteTwoDimensionalClassifyingSpace G ∧
∃ α β : MonoidAlgebra (ZMod 2) G, α ≠ 0 ∧ β ≠ 0 ∧ α * β = 0
end TorsionFreeZeroDivisors
end OAI
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