openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonAlternatingStabilization.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonPlacementAlternating | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonAlternatingStabilization | |
| open Classical Set | |
| namespace PolygonTracks | |
| variable {a k n : ℕ} | |
| theorem stabilize_threeSlot {g : polygonFullGroup a n} (hg : IsThreeSlotCycle g) : | |
| IsThreeSlotCycle (stabilize a k n g) := by | |
| obtain ⟨U,s,hU,hs,hg,hfix⟩ := hg | |
| let t : Fin 3 → Fin (k+n) × (CutRing × CutRing) := fun i => ((s i).1.natAdd k,(s i).2) | |
| have ht (x : Fin 3 × U.val) : SlotMap a (k+n) U t x=rightIncl a k n (SlotMap a n U s x) := rfl | |
| refine ⟨U,t,hU,?_,?_,?_⟩ | |
| · intro x y hxy | |
| apply hs | |
| exact rightIncl_injective (by simpa only [ht] using hxy) | |
| · intro i x | |
| rw [ht,ht] | |
| change (stabilize a k n g).val ((SlotMap a n U s (i,x)).1.natAdd k, | |
| (SlotMap a n U s (i,x)).2)=_ | |
| rw [stabilize_right,hg] | |
| rfl | |
| · intro p hp | |
| obtain ⟨q,rfl⟩ := (join a k n).surjective p | |
| cases q with | |
| | inl q => exact stabilize_left g q | |
| | inr q => | |
| rw [join_inr,stabilize_right] | |
| have hq : q∉Set.range (SlotMap a n U s) := by | |
| rintro ⟨x,rfl⟩ | |
| exact hp ⟨x,rfl⟩ | |
| rw [hfix q hq] | |
| theorem stabilize_mem_alternating (g : polygonAlternatingGroup a n) : | |
| stabilize a k n g.val∈polygonAlternatingGroup a (k+n) := by | |
| have hl : polygonAlternatingGroup a n ≤ | |
| (polygonAlternatingGroup a (k+n)).comap (stabilize a k n) := by | |
| apply (Subgroup.closure_le _).mpr | |
| intro f hf | |
| exact Subgroup.subset_closure (stabilize_threeSlot hf) | |
| exact hl g.property | |
| noncomputable def stabilizeAlternating (a k n : ℕ) : | |
| polygonAlternatingGroup a n →* polygonAlternatingGroup a (k+n) := | |
| (stabilize a k n).restrict (fun g hg => stabilize_mem_alternating ⟨g,hg⟩) | |
| @[simp] theorem stabilizeAlternating_val (g : polygonAlternatingGroup a n) : | |
| (stabilizeAlternating a k n g).val=stabilize a k n g.val := rfl | |
| theorem stabilizeAlternating_injective : Function.Injective (stabilizeAlternating a k n) := by | |
| intro g h he | |
| apply Subtype.ext | |
| exact stabilize_injective (congrArg Subtype.val he) | |
| theorem exists_alternating_agrees_right (f : polygonFullGroup a (k+n)) (hm : 6*n+7<k+n) : | |
| ∃c : polygonAlternatingGroup a (k+n),∀x : TrackPoint a n, | |
| c.val.val (rightIncl a k n x)=f.val (rightIncl a k n x) := by | |
| let e : Fin n → PolygonPlacement a (k+n) := fun i => PolygonPlacement.standard (i.natAdd k) | |
| have he : Pairwise (fun i j => PolygonPlacement.Apart (e i) (e j)) := by | |
| intro i j hij | |
| exact PolygonPlacement.standard_apart (fun h => hij (Fin.ext (Nat.add_left_cancel | |
| (congrArg Fin.val h)))) | |
| have hf : Pairwise (fun i j => PolygonPlacement.Apart (f • e i) (f • e j)) := by | |
| intro i j hij | |
| exact (PolygonPlacement.apart_smul_iff f _ _).mpr (he hij) | |
| obtain ⟨c,hc⟩ := PolygonPlacement.alternating_configuration_transitive e (fun i => f • e i) he hf hm | |
| refine ⟨c,fun x => ?_⟩ | |
| exact congrArg (fun z : PolygonPlacement a (k+n) => z x.2) (hc x.1) | |
| end PolygonTracks | |
| end PolygonAlternatingStabilization | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |