openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Analysis /BadCharacterCompactness.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.InwardVectorCharacters | |
| namespace OAI | |
| /-! # Compactness of the bad unit-character set | |
| The full positive-identity criterion makes the good region open. Its complement | |
| on the unit sphere is the compact set used in the convex tameness argument. | |
| -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open scoped MonoidAlgebra | |
| variable {Q M E : Type} [CommGroup Q] [AddCommGroup M] | |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| (ρ : Representation ℤ Q M) [Module.Finite ℤ[Q] ρ.asModule] | |
| (e : Q →* Multiplicative E) | |
| def positiveIdentityRegion : Set E := {v | ∃ a : ℤ[Q], ρ.asAlgebraHom a = 1 ∧ | |
| ∀ q ∈ a.coeff.support, 0 < (inwardVectorCharacter e v q).toAdd} | |
| def badUnitCharacters : Set E := {v | ‖v‖ = 1 ∧ | |
| ¬Module.Finite ℤ[characterNonnegativeMonoid (inwardVectorCharacter e v)] | |
| (characterMonoidRepresentation ρ (inwardVectorCharacter e v)).asModule} | |
| omit [Module.Finite ℤ[Q] ρ.asModule] in | |
| theorem positiveIdentityRegion_open : IsOpen (positiveIdentityRegion ρ e) := by | |
| have he : positiveIdentityRegion ρ e = ⋃ a : ℤ[Q], ⋃ (_h : ρ.asAlgebraHom a = 1), | |
| {v : E | ∀ q ∈ a.coeff.support, 0 < (inwardVectorCharacter e v q).toAdd} := by | |
| ext v | |
| simp only [positiveIdentityRegion, Set.mem_ofPred_eq, Set.mem_iUnion, exists_prop] | |
| rw [he] | |
| exact isOpen_iUnion fun a => isOpen_iUnion fun _ => | |
| isOpen_positiveIdentitySupport (inwardVectorCharacter e) (inwardVectorCharacter_continuous e) a 0 | |
| theorem badUnitCharacters_eq_sdiff | |
| (hspan : Submodule.span ℝ (Set.range (fun q => (e q).toAdd)) = ⊤) : | |
| badUnitCharacters ρ e = {v : E | ‖v‖ = 1} \ positiveIdentityRegion ρ e := by | |
| ext v | |
| constructor | |
| · rintro ⟨hv, hbad⟩ | |
| refine ⟨hv, ?_⟩ | |
| rintro ⟨a, hid, hpos⟩ | |
| exact hbad (finite_characterMonoidModule_of_positive_identity ρ | |
| (inwardVectorCharacter e v) a hpos hid) | |
| · rintro ⟨hv, hbad⟩ | |
| refine ⟨hv, ?_⟩ | |
| intro hfinite | |
| let := hfinite | |
| obtain ⟨q, hq⟩ := inwardVectorCharacter_exists_positive e hspan v (by | |
| intro hz | |
| simp [hz] at hv) | |
| obtain ⟨a, hpos, hid⟩ := positive_identity_of_finite_characterMonoidModule ρ | |
| (inwardVectorCharacter e v) q hq | |
| exact hbad ⟨a, hid, hpos⟩ | |
| theorem badUnitCharacters_compact [ProperSpace E] | |
| (hspan : Submodule.span ℝ (Set.range (fun q => (e q).toAdd)) = ⊤) : | |
| IsCompact (badUnitCharacters ρ e) := by | |
| rw [badUnitCharacters_eq_sdiff ρ e hspan] | |
| have hs : IsCompact {v : E | ‖v‖ = 1} := by | |
| simpa only [Metric.sphere, dist_zero_right] using (isCompact_sphere (0 : E) 1) | |
| exact hs.inter_right (positiveIdentityRegion_open ρ e).isClosed_compl | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |