openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonAlternatingExtension.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.PolygonBankEmbedding | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonAlternatingExtension | |
| open Classical Set | |
| namespace PolygonObject | |
| namespace BankEmbedding | |
| variable {a m : ℕ} | |
| theorem alternating_extension (U B : Fin m → polygonAlgebra a) | |
| (f : BankEmbedding (PolygonObject.mk m U) m) | |
| (hU : ∀p : (PolygonObject.mk m U).Point,p.val.2∉(B p.val.1).val) | |
| (hf : ∀p,(f p).2∉(B (f p).1).val) | |
| (harea : 10*(∑i,PolygonArea.area (U i))<∑i,PolygonArea.area (B i)ᶜ) : | |
| ∃c : polygonAlternatingGroup a m, | |
| (∀p,c.val.val p.val=f p) ∧ | |
| (∀y : TrackPoint a m,y.2∈(B y.1).val → c.val.val y=y) := by | |
| let u := inclusion U | |
| have hB : (∑i,PolygonArea.area (B i))≤(m:ENNReal) := by | |
| calc | |
| _≤∑_i : Fin m,(1:ENNReal) := Finset.sum_le_sum (fun i _ => PolygonArea.area_le_one (B i)) | |
| _=_ := by simp | |
| have hBn : (∑i,PolygonArea.area (B i))≠⊤ := ne_top_of_le_ne_top (by simp) hB | |
| have htotal : (∑i,PolygonArea.area (B i))+(∑i,PolygonArea.area (B i)ᶜ)=m := by | |
| rw [←Finset.sum_add_distrib] | |
| simp only [PolygonArea.area_compl,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] | |
| have guard (r : ENNReal) (hr : r≤10) : | |
| (∑i,PolygonArea.area (B i))+r*(∑i,PolygonArea.area (U i)) < m := by | |
| rw [←htotal] | |
| exact ENNReal.add_lt_add_left hBn ((mul_le_mul hr le_rfl (by positivity) (by positivity)).trans_lt harea) | |
| obtain ⟨w₁,hw₁B,hw₁⟩ := exists_away U B ![u,f] (by apply guard; norm_num) | |
| obtain ⟨w₂,hw₂B,hw₂⟩ := exists_away U B ![u,f,w₁] (by apply guard; norm_num) | |
| obtain ⟨w₃,hw₃B,hw₃⟩ := exists_away U B ![u,f,w₁,w₂] (by apply guard; norm_num) | |
| obtain ⟨c₁,hc₁,hout₁⟩ := cycle_three u w₁ w₂ | |
| (fun p q he => hw₁ 0 q p he.symm) | |
| (fun p q he => hw₂ 0 q p he.symm) | |
| (fun p q he => hw₂ 2 q p he.symm) | |
| obtain ⟨c₂,hc₂,hout₂⟩ := cycle_three w₁ f w₃ | |
| (fun p q he => hw₁ 1 p q he) | |
| (fun p q he => hw₃ 2 q p he.symm) | |
| (fun p q he => hw₃ 1 q p he.symm) | |
| refine ⟨c₂*c₁,?_,?_⟩ | |
| · intro p | |
| change c₂.val.val (c₁.val.val (u p))=f p | |
| rw [hc₁,hc₂] | |
| · intro y hy | |
| have avoid (e : BankEmbedding (PolygonObject.mk m U) m) | |
| (he : ∀p,(e p).2∉(B (e p).1).val) : ∀p,e p≠y := by | |
| intro p hp | |
| exact he p (by simpa only [hp] using hy) | |
| have hu : ∀p,u p≠y := avoid u hU | |
| have hf' : ∀p,f p≠y := avoid f hf | |
| have hw₁' := avoid w₁ hw₁B | |
| have hw₂' := avoid w₂ hw₂B | |
| have hw₃' := avoid w₃ hw₃B | |
| change c₂.val.val (c₁.val.val y)=y | |
| rw [hout₁ y hu hw₁' hw₂',hout₂ y hw₁' hf' hw₃'] | |
| end BankEmbedding | |
| end PolygonObject | |
| end PolygonAlternatingExtension | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |