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0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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