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3277efa | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.ExternalCopyDescent
namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
open Attachment CosetGeometry
universe u
variable {A J R : Type u} [Group J] {P : MeetOrder A}
variable (f : Family P R) (a : OrderAction J P)
noncomputable def fromFlatQuotient : f.FlatQuotient a → f.Node a :=
Quotient.lift (fun x => (f.copy a x.1).toFun x.2)
(fun _ _ path => FlatPath.copy_eq f a path)
theorem fromFlatQuotient_class (w : External J f.Letters) (x : A) :
f.fromFlatQuotient a (f.flatClass a w x) = (f.copy a w).toFun x := rfl
noncomputable def flatQuotientEquiv : f.FlatQuotient a ≃ f.Node a where
toFun := f.fromFlatQuotient a
invFun := f.toFlatQuotient a
left_inv q := by
induction q using Quotient.ind with
| _ x => exact f.toFlatQuotient_copy a x.1 x.2
right_inv z := by
obtain ⟨w, x, rfl⟩ := f.copy_cover a z
rw [f.toFlatQuotient_copy, f.fromFlatQuotient_class]
theorem flatQuotientEquiv_class (w : External J f.Letters) (x : A) :
f.flatQuotientEquiv a (f.flatClass a w x) = (f.copy a w).toFun x := rfl
theorem FlatPath.translate (g : External J f.Letters) {x y : f.FlatLabel (J := J)}
(path : f.FlatPath a x y) :
f.FlatPath a (g * x.1, x.2) (g * y.1, y.2) := by
apply (f.flatPath_iff_copy_eq a _ _).2
have equal := congrArg (f.translate a g) ((f.flatPath_iff_copy_eq a x y).1 path)
simpa only [f.translate_copy] using equal
theorem flatPath_translate_iff (g : External J f.Letters) (x y : f.FlatLabel (J := J)) :
f.FlatPath a (g * x.1, x.2) (g * y.1, y.2) ↔ f.FlatPath a x y := by
constructor
· intro path
have inverse := FlatPath.translate f a g⁻¹ path
simpa only [Prod.fst, Prod.snd, inv_mul_cancel_left] using inverse
· exact FlatPath.translate f a g
end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
end
end OAI
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