| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.Amenability.BarrierCells |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section PartitionArrows |
| open Classical Set |
|
|
| abbrev TrackPolygonPartition (a m : ℕ) := Fin m → FinitePolygonPartition a |
| abbrev PartitionCell {a m : ℕ} (P : TrackPolygonPartition a m) := Σi : Fin m,↥(P i).cells |
|
|
| noncomputable def partitionCellSet {a m : ℕ} {P : TrackPolygonPartition a m} |
| (c : PartitionCell P) : Set (TrackPoint a m) := {x | x.1=c.1 ∧ x.2∈c.2.val.val} |
|
|
| theorem partitionCellSet_nonempty {a m : ℕ} {P : TrackPolygonPartition a m} |
| (c : PartitionCell P) : (partitionCellSet c).Nonempty := by |
| obtain ⟨p,hp⟩ := (P c.1).nonempty c.2.val c.2.property |
| exact ⟨(c.1,p),rfl,hp⟩ |
|
|
| theorem partitionCellSet_cover {a m : ℕ} (P : TrackPolygonPartition a m) (x : TrackPoint a m) : |
| ∃!c : PartitionCell P,x∈partitionCellSet c := by |
| obtain ⟨C,⟨hC,hx⟩,hu⟩ := (P x.1).cover x.2 |
| refine ⟨⟨x.1,⟨C,hC⟩⟩,⟨rfl,hx⟩,?_⟩ |
| rintro ⟨i,D,hD⟩ ⟨hi,hxD⟩ |
| dsimp at hi |
| subst i |
| have he := hu D ⟨hD,hxD⟩ |
| subst D |
| rfl |
|
|
| theorem partitionCellSet_eq_of_mem {a m : ℕ} {P : TrackPolygonPartition a m} |
| {c d : PartitionCell P} {x : TrackPoint a m} |
| (hc : x∈partitionCellSet c) (hd : x∈partitionCellSet d) : c=d := |
| (partitionCellSet_cover P x).unique hc hd |
|
|
| def PartitionArrow {a m : ℕ} (P Q : TrackPolygonPartition a m) (g : polygonFullGroup a m) : Prop := |
| ∀c : PartitionCell P,∃d : PartitionCell Q, |
| g.val '' partitionCellSet c=partitionCellSet d ∧ ∃u : CutRing×CutRing, |
| ∀x∈partitionCellSet c,(g.val x).2.val=x.2.val+(ordinary u.1,ordinary u.2) |
|
|
| namespace PartitionArrow |
| variable {a m : ℕ} {P Q R : TrackPolygonPartition a m} {g h : polygonFullGroup a m} |
|
|
| theorem one (P : TrackPolygonPartition a m) : PartitionArrow P P 1 := by |
| intro c |
| refine ⟨c,?_,0,?_⟩ |
| · exact Set.image_id _ |
| · intro x _ |
| change x.2.val=x.2.val+(ordinary (0:CutRing),ordinary (0:CutRing)) |
| simp only [map_zero] |
| exact (add_zero x.2.val).symm |
|
|
| theorem mul (hg : PartitionArrow Q R g) (hh : PartitionArrow P Q h) : PartitionArrow P R (g*h) := by |
| intro c |
| obtain ⟨d,hd,u,hu⟩ := hh c |
| obtain ⟨e,he,v,hv⟩ := hg d |
| refine ⟨e,?_,u+v,?_⟩ |
| · change (g.val ∘ h.val) '' partitionCellSet c=partitionCellSet e |
| exact (Set.image_image g.val h.val (partitionCellSet c)).symm.trans (by rw [hd,he]) |
| · intro x hx |
| have hhx : h.val x∈partitionCellSet d := by rw [←hd]; exact ⟨x,hx,rfl⟩ |
| change (g.val (h.val x)).2.val=x.2.val+_ |
| rw [hv _ hhx,hu _ hx] |
| simp only [Prod.fst_add,Prod.snd_add,map_add,Prod.mk_add_mk,add_assoc] |
|
|
| theorem inv (hg : PartitionArrow P Q g) : PartitionArrow Q P g⁻¹ := by |
| intro d |
| obtain ⟨y,hy⟩ := partitionCellSet_nonempty d |
| obtain ⟨c,hc,-⟩ := partitionCellSet_cover P (g.val.symm y) |
| obtain ⟨e,he,u,hu⟩ := hg c |
| have hye : y∈partitionCellSet e := by rw [←he]; exact ⟨g.val.symm y,hc,g.val.apply_symm_apply y⟩ |
| have hde : d=e := partitionCellSet_eq_of_mem hy hye |
| subst e |
| refine ⟨c,?_, -u,?_⟩ |
| · rw [←he] |
| exact g.val.symm_image_image _ |
| · intro x hx |
| obtain ⟨w,hw,rfl⟩ := (Set.mem_image g.val _ _).mp (he.symm ▸ hx) |
| have h := hu w hw |
| change (g.val.symm (g.val w)).2.val=(g.val w).2.val+_ |
| rw [g.val.symm_apply_apply,h] |
| simp only [Prod.fst_neg,Prod.snd_neg,map_neg] |
| apply Prod.ext <;> simp only [Prod.fst_add,Prod.snd_add] <;> ring |
|
|
| end PartitionArrow |
| end PartitionArrows |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
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|
