| import OAI.SetTheory.PartitionPrinciple.Geometry.ExternalCopyDescent |
|
|
| namespace OAI |
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|
| section |
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|
| 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 |
|
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| end |
|
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| end OAI |
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|
