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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 78 79 80 81 82 83 84 85 86 87 | 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
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