openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Finiteness /BooleanStepProjective.lean
| import OAI.GroupTheory.PolycyclicRecognition.Algebra.BooleanPartitionMaps | |
| import Mathlib.Data.Set.Countable | |
| namespace OAI | |
| /-! # Countable Boolean step rings are projective over the integers | |
| Enumerating the Boolean coordinates gives a sequence of finite partitions. | |
| Their modules are free, the refinement maps split, and their colimit is the | |
| step ring itself. No freeness theorem about all bounded functions is needed. | |
| -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open CategoryTheory CategoryTheory.Limits | |
| variable {X I : Type} (a : I → X → Bool) (b : ℕ → I) | |
| noncomputable def booleanPrefix (n : ℕ) : Finset I := by | |
| classical | |
| exact (Finset.range n).image b | |
| theorem booleanPrefix_mono {i j : ℕ} (h : i ≤ j) : booleanPrefix b i ⊆ booleanPrefix b j := by | |
| classical | |
| exact Finset.image_subset_image (Finset.range_mono h) | |
| theorem exists_booleanPrefix_contains (hb : Function.Surjective b) (s : Finset I) : | |
| ∃ n, s ⊆ booleanPrefix b n := by | |
| classical | |
| induction s using Finset.induction_on with | |
| | empty => exact ⟨0, Finset.empty_subset _⟩ | |
| | @insert i s _ ih => | |
| obtain ⟨n, hn⟩ := ih | |
| obtain ⟨m, hm⟩ := hb i | |
| refine ⟨max n (m + 1), Finset.insert_subset_iff.mpr ⟨?_, ?_⟩⟩ | |
| · apply booleanPrefix_mono b (Nat.le_max_right n (m + 1)) | |
| exact Finset.mem_image.mpr ⟨m, Finset.mem_range.mpr (Nat.lt_succ_self m), hm⟩ | |
| · exact hn.trans (booleanPrefix_mono b (Nat.le_max_left n (m + 1))) | |
| def booleanPartitionDiagram : SequentialRepresentationDiagram PUnit.{1} where | |
| obj n := booleanPartitionStage a (booleanPrefix b n) | |
| map _ _ h := booleanPartitionMap a (booleanPrefix_mono b h) | |
| map_id n := by | |
| apply Rep.hom_ext | |
| apply Representation.IntertwiningMap.ext | |
| apply LinearMap.ext | |
| intro v | |
| funext c | |
| rfl | |
| map_comp i j k hij hjk := by | |
| apply Rep.hom_ext | |
| apply Representation.IntertwiningMap.ext | |
| apply LinearMap.ext | |
| intro v | |
| funext c | |
| rfl | |
| def booleanStepColimit (hb : Function.Surjective b) : | |
| (booleanPartitionDiagram a b).ColimitCocone (booleanStepRepresentation a) where | |
| ι n := booleanPartitionEvaluation a (booleanPrefix b n) | |
| comm _ _ h := booleanPartitionMap_evaluation a (booleanPrefix_mono b h) | |
| surj f := by | |
| obtain ⟨s, hs⟩ := f.property | |
| obtain ⟨n, hn⟩ := exists_booleanPrefix_contains b hb s | |
| obtain ⟨v, hv⟩ := dependsOnBoolean_factorization (hs.mono hn) | |
| refine ⟨n, v, ?_⟩ | |
| apply Subtype.ext | |
| exact hv.symm | |
| zero_exact i v hv := by | |
| refine ⟨i, le_rfl, ?_⟩ | |
| have h0 : v = 0 := booleanPartitionEvaluation_injective a (booleanPrefix b i) | |
| (hv.trans (map_zero _).symm) | |
| rw [h0, map_zero] | |
| theorem booleanStepRepresentation_projective (hb : Function.Surjective b) : | |
| Projective (booleanStepRepresentation a) := by | |
| let D := booleanPartitionDiagram a b | |
| let r : ∀ i, D.obj (i + 1) ⟶ D.obj i := fun i => | |
| booleanPartitionRetraction a (booleanPrefix_mono b (Nat.le_succ i)) | |
| have hr : ∀ i, D.map i (i + 1) (Nat.le_succ i) ≫ r i = 𝟙 _ := | |
| fun i => booleanPartitionMap_retraction a (booleanPrefix_mono b (Nat.le_succ i)) | |
| let : ∀ i, Projective (D.obj i) := fun i => booleanPartitionStage_projective a _ | |
| exact D.split_colimit_projective r hr (booleanStepColimit a b hb) | |
| /-- The small step-function algebra required in the countable group application. -/ | |
| theorem booleanStepRing_integer_projective [Countable I] [Nonempty I] : | |
| Module.Projective ℤ (booleanStepRing a) := by | |
| obtain ⟨b, hb⟩ := exists_surjective_nat I | |
| let := booleanStepRepresentation_projective a b hb | |
| exact integer_projective_of_projective_rep (booleanStepRepresentation a) | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |