| import Mathlib |
| import OAI.RingTheory.DirectFiniteness.PrescribedPresentation |
| import OAI.RingTheory.DirectFiniteness.OddTorsion |
| import OAI.RingTheory.DirectFiniteness.Main |
|
|
| namespace OAI |
|
|
| /-! The prescribed finitely presented counterexample in characteristic two. -/ |
|
|
| namespace KaplanskyCounterexample |
|
|
| open Prescription |
|
|
| theorem thm_main : |
| (∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K 2), |
| ∃ (G : Type) (_ : Group G) (_ : Group.IsFinitelyPresented G), |
| (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : G), orderOf g = ℓ) ∧ |
| ∃ a_out b_out : MonoidAlgebra K G, a_out * b_out = 1 ∧ b_out * a_out ≠ 1) ∧ |
| (2 ≤ selectedH ∧ selectedH ≤ 217 ∧ Test selectedInput selectedPair ∧ |
| (∀ h, Succeeds h → selectedH ≤ h) ∧ |
| (∀ p, Test selectedInput p → |
| @LE.le _ (pairOrder selectedInput).toLE selectedPair p)) ∧ |
| Group.IsFinitelyPresented Output.G ∧ |
| (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : Output.G), orderOf g = ℓ) ∧ |
| Output.a * Output.b = 1 ∧ Output.b * Output.a ≠ 1 := by |
| refine ⟨?_, finite_search_terminates, Output.prescribed_finitelyPresented, |
| Output.prescribed_odd_prime_order, Output.prescribed_products⟩ |
| exact ⟨Output.Coeff, inferInstance, inferInstance, inferInstance, |
| Output.G, inferInstance, Output.prescribed_finitelyPresented, |
| Output.prescribed_odd_prime_order, Output.a, Output.b, Output.prescribed_products⟩ |
|
|
|
|
| / |
| the certificates for its prescribed construction. -/ |
| theorem finitelyPresented_counterexample : |
| ∃ (K : Type) (_ : Field K) (_ : Fintype K) (_ : CharP K 2), |
| ∃ (G : Type) (_ : Group G) (_ : Group.IsFinitelyPresented G), |
| (∃ (ℓ : ℕ) (_ : ℓ.Prime) (_ : Odd ℓ) (g : G), orderOf g = ℓ) ∧ |
| ∃ a_out b_out : MonoidAlgebra K G, a_out * b_out = 1 ∧ b_out * a_out ≠ 1 := |
| thm_main.1 |
|
|
| end KaplanskyCounterexample |
|
|
| end OAI |
|
|
