| import OAI.SetTheory.PartitionPrinciple.Definability.ExternalCopyRelations |
|
|
| 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 flatTileValue (w : External J f.Letters) (r : R) |
| (z : (f.tile r).Space) : f.FlatQuotient a := |
| f.flatClass a (f.flatWord w r (Classical.choose (f.flatVertex_cover r z))) |
| (Classical.choose (Classical.choose_spec (f.flatVertex_cover r z))) |
|
|
| theorem flatTileValue_vertex (w : External J f.Letters) (r : R) |
| (i : Option (f.Child r)) (x : A) : |
| f.flatTileValue a w r ((f.flatVertex r i).toFun x) = |
| f.flatClass a (f.flatWord w r i) x := by |
| unfold flatTileValue |
| apply f.flatClass_tile a |
| exact Classical.choose_spec (Classical.choose_spec (f.flatVertex_cover r _)) |
|
|
| noncomputable def flatIncidenceValue : |
| (v : TileStars.Vertex J f.Letters R) → f.Space v → f.FlatQuotient a |
| | .inl v, x => f.flatClass a (representative J f.Letters v) x |
| | .inr c, x => f.flatTileValue a c.1 c.2 x |
|
|
| theorem flatIncidenceValue_copy (w : External J f.Letters) (x : A) : |
| f.flatIncidenceValue a (.inl (point J f.Letters w)) ((a.chart w).toFun x) = |
| f.flatClass a w x := by |
| change f.flatClass a (representative J f.Letters (point J f.Letters w)) |
| ((a.iso (coordinate J f.Letters w)).toFun x) = _ |
| rw [← f.flatClass_coordinate, coordinate_spec] |
|
|
| theorem flatIncidence_stem (w : External J f.Letters) (r : R) : |
| (f.diagram a).Equation (f.flatIncidenceValue a) (.inl (w, r), true) := by |
| intro x |
| change f.flatIncidenceValue a (.inl (point J f.Letters w)) x.val = |
| f.flatTileValue a w r ((f.tile r).root.toFun ((a.chart w).inverse x.val)) |
| have vertex := f.flatTileValue_vertex a w r none ((a.chart w).inverse x.val) |
| change f.flatTileValue a w r ((f.tile r).root.toFun ((a.chart w).inverse x.val)) = |
| f.flatClass a w ((a.chart w).inverse x.val) at vertex |
| rw [vertex] |
| have eq := f.flatIncidenceValue_copy a w ((a.chart w).inverse x.val) |
| erw [(a.chart w).right_inverse] at eq |
| exact eq |
|
|
| theorem flatIncidence_branch (w : External J f.Letters) (r : R) (j : f.Child r) : |
| (f.diagram a).Equation (f.flatIncidenceValue a) (.inr (w, ⟨r, j⟩), true) := by |
| intro x |
| let i := f.vertexIdealIso ((f.tile r).child j) |
| obtain ⟨y, rfl⟩ : ∃ y, i.toFun y = x := ⟨i.inverse x, i.right_inverse x⟩ |
| change f.flatTileValue a w r (((f.tile r).child j).toFun y) = |
| f.flatIncidenceValue a (.inl (point J f.Letters (w * generator J f.Letters ⟨r, j⟩))) |
| ((f.edgeOverlap a (.inr (w, ⟨r, j⟩))).toFun (i.toFun y)).val |
| have formula : ((f.edgeOverlap a (.inr (w, ⟨r, j⟩))).toFun (i.toFun y)).val = |
| (a.chart (w * generator J f.Letters ⟨r, j⟩)).toFun y := f.branch_formula a w r j y |
| rw [formula, f.flatIncidenceValue_copy] |
| exact f.flatTileValue_vertex a w r (some j) y |
|
|
| theorem flatIncidence_edges (d : TileStars.Dart J f.Letters R) : |
| (f.diagram a).Equation (f.flatIncidenceValue a) d := by |
| rcases d with ⟨e, b⟩ |
| have forward : (f.diagram a).Equation (f.flatIncidenceValue a) (e, true) := by |
| cases e with |
| | inl c => exact f.flatIncidence_stem a c.1 c.2 |
| | inr c => exact f.flatIncidence_branch a c.1 c.2.1 c.2.2 |
| cases b |
| · exact (f.diagram a).equation_of_reverse (f.flatIncidenceValue a) (e, false) forward |
| · exact forward |
|
|
| noncomputable def toFlatQuotient (z : f.Node a) : f.FlatQuotient a := |
| (f.diagram a).descend (f.flatIncidenceValue a) (f.flatIncidence_edges a) z |
|
|
| theorem toFlatQuotient_copy (w : External J f.Letters) (x : A) : |
| f.toFlatQuotient a ((f.copy a w).toFun x) = f.flatClass a w x := by |
| change (f.diagram a).descend (f.flatIncidenceValue a) (f.flatIncidence_edges a) |
| (((f.diagram a).vertex (.inl (point J f.Letters w))).toFun ((a.chart w).toFun x)) = _ |
| erw [(f.diagram a).descend_vertex, f.flatIncidenceValue_copy] |
|
|
| theorem flatPath_iff_copy_eq (x y : f.FlatLabel (J := J)) : |
| f.FlatPath a x y ↔ (f.copy a x.1).toFun x.2 = (f.copy a y.1).toFun y.2 := by |
| constructor |
| · exact FlatPath.copy_eq f a |
| · intro equal |
| have classes := congrArg (f.toFlatQuotient a) equal |
| rw [f.toFlatQuotient_copy, f.toFlatQuotient_copy] at classes |
| exact Quotient.exact classes |
|
|
| end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family |
|
|
| end |
|
|
| end OAI |
|
|
