| import OAI.SetTheory.PartitionPrinciple.Models.IncidenceQuotient |
|
|
| namespace OAI |
|
|
|
|
| namespace PartitionPilot.Compression.RecursiveMaps |
|
|
| namespace 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) |
|
|
| def fullIdeal : MeetIso (idealOrder P f.top) P where |
| toFun := Subtype.val |
| le_iff _ _ := Iff.rfl |
| map_meet _ _ := rfl |
| inverse x := ⟨x, f.greatest x⟩ |
| left_inverse _ := rfl |
| right_inverse _ := rfl |
|
|
| noncomputable def vertexIdealIso {B : Type u} {Q : MeetOrder B} (i : IdealEmbedding P Q) : |
| MeetIso P (idealOrder Q (i.toFun f.top)) := |
| RootedGluing.chainIso (inverseIso f.fullIdeal) (i.principalIso f.top) |
|
|
| def Space : TileStars.Vertex J f.Letters R → Type u |
| | .inl _ => A |
| | .inr c => (f.tile c.2).Space |
|
|
| noncomputable def localOrder : (v : TileStars.Vertex J f.Letters R) → MeetOrder (f.Space v) |
| | .inl _ => P |
| | .inr c => (f.tile c.2).order |
|
|
| noncomputable def leftAnchor (e : TileStars.Edge J f.Letters R) : |
| f.Space (TileStars.source J f.Letters R Sigma.fst (e, true)) := by |
| cases e with |
| | inl c => exact f.top |
| | inr e => exact ((f.tile e.2.1).child e.2.2).toFun f.top |
|
|
| noncomputable def rightAnchor (e : TileStars.Edge J f.Letters R) : |
| f.Space (TileStars.target J f.Letters R Sigma.fst (e, true)) := by |
| cases e with |
| | inl c => exact (f.tile c.2).root.toFun f.top |
| | inr e => exact f.top |
|
|
| noncomputable def edgeOverlap (e : TileStars.Edge J f.Letters R) : MeetIso |
| (idealOrder (f.localOrder (TileStars.source J f.Letters R Sigma.fst (e, true))) (f.leftAnchor e)) |
| (idealOrder (f.localOrder (TileStars.target J f.Letters R Sigma.fst (e, true))) (f.rightAnchor e)) := by |
| cases e with |
| | inl c => |
| exact RootedGluing.chainIso f.fullIdeal (RootedGluing.chainIso |
| (inverseIso (a.chart c.1)) (f.vertexIdealIso (f.tile c.2).root)) |
| | inr e => |
| exact RootedGluing.chainIso (inverseIso (f.vertexIdealIso ((f.tile e.2.1).child e.2.2))) |
| (RootedGluing.chainIso (a.chart (e.1 * generator J f.Letters e.2)) (inverseIso f.fullIdeal)) |
|
|
| noncomputable def diagram : TileStars.Spaces J f.Letters R Sigma.fst where |
| Space := f.Space |
| order := f.localOrder |
| leftAnchor := f.leftAnchor |
| rightAnchor := f.rightAnchor |
| overlap := f.edgeOverlap a |
|
|
| def Node := (f.diagram a).Node |
| noncomputable def gluedOrder : MeetOrder (f.Node a) := (f.diagram a).gluedOrder |
|
|
| noncomputable def copy (w : External J f.Letters) : IdealEmbedding P (f.gluedOrder a) := |
| (a.ideal (coordinate J f.Letters w)).comp ((f.diagram a).vertex (.inl (point J f.Letters w))) |
|
|
| noncomputable def placedTile (w : External J f.Letters) (r : R) : |
| IdealEmbedding (f.tile r).order (f.gluedOrder a) := (f.diagram a).vertex (.inr (w, r)) |
|
|
| theorem copy_coordinate (w : External J f.Letters) (j : J) (x : A) : |
| (f.copy a (w * old J f.Letters j)).toFun x = (f.copy a w).toFun ((a.iso j).toFun x) := by |
| change ((f.diagram a).vertex (.inl (point J f.Letters (w * old J f.Letters j)))).toFun |
| ((a.chart (w * old J f.Letters j)).toFun x) = _ |
| rw [point_mul_old, a.chart_mul_old] |
| rfl |
|
|
| theorem copy_one (x : A) : (f.copy a 1).toFun x = |
| ((f.diagram a).vertex (.inl (root J f.Letters))).toFun x := by |
| change ((f.diagram a).vertex (.inl (root J f.Letters))).toFun ((a.chart 1).toFun x) = _ |
| rw [a.chart_one] |
|
|
| theorem stem_formula (w : External J f.Letters) (r : R) (x : A) : |
| ((f.edgeOverlap a (.inl (w, r))).toFun ⟨(a.chart w).toFun x, f.greatest _⟩).val = |
| (f.tile r).root.toFun x := by |
| change (f.tile r).root.toFun ((a.chart w).inverse ((a.chart w).toFun x)) = _ |
| rw [(a.chart w).left_inverse] |
|
|
| theorem branch_formula (w : External J f.Letters) (r : R) (j : f.Child r) (x : A) : |
| ((f.edgeOverlap a (.inr (w, ⟨r, j⟩))).toFun |
| ⟨((f.tile r).child j).toFun x, (((f.tile r).child j).le_iff _ _).mpr (f.greatest x)⟩).val = |
| (a.chart (w * generator J f.Letters ⟨r, j⟩)).toFun x := by |
| have inv := (f.vertexIdealIso ((f.tile r).child j)).left_inverse x |
| change (a.chart (w * generator J f.Letters ⟨r, j⟩)).toFun |
| ((f.vertexIdealIso ((f.tile r).child j)).inverse |
| ((f.vertexIdealIso ((f.tile r).child j)).toFun x)) = _ |
| rw [inv] |
|
|
| theorem root_identification (w : External J f.Letters) (r : R) (x : A) : |
| (f.placedTile a w r).toFun ((f.tile r).root.toFun x) = (f.copy a w).toFun x := by |
| have eq := (f.diagram a).edge_identification (.inl (w, r), true) |
| ⟨(a.chart w).toFun x, f.greatest _⟩ |
| change (f.copy a w).toFun x = (f.placedTile a w r).toFun |
| ((f.edgeOverlap a (.inl (w, r))).toFun ⟨(a.chart w).toFun x, f.greatest _⟩).val at eq |
| rw [f.stem_formula] at eq |
| exact eq.symm |
|
|
| theorem child_identification (w : External J f.Letters) (r : R) (j : f.Child r) (x : A) : |
| (f.placedTile a w r).toFun (((f.tile r).child j).toFun x) = |
| (f.copy a (w * generator J f.Letters ⟨r, j⟩)).toFun x := by |
| have eq := (f.diagram a).edge_identification (.inr (w, ⟨r, j⟩), true) |
| ⟨((f.tile r).child j).toFun x, (((f.tile r).child j).le_iff _ _).mpr (f.greatest x)⟩ |
| change (f.placedTile a w r).toFun (((f.tile r).child j).toFun x) = |
| ((f.diagram a).vertex (.inl (point J f.Letters (w * generator J f.Letters ⟨r, j⟩)))).toFun |
| ((f.edgeOverlap a (.inr (w, ⟨r, j⟩))).toFun |
| ⟨((f.tile r).child j).toFun x, (((f.tile r).child j).le_iff _ _).mpr (f.greatest x)⟩).val at eq |
| rw [f.branch_formula] at eq |
| exact eq |
|
|
| theorem copy_principal (w : External J f.Letters) (z : f.Node a) : |
| (f.gluedOrder a).le z ((f.copy a w).toFun f.top) ↔ ∃ x, (f.copy a w).toFun x = z := |
| (f.copy a w).principal f.top f.greatest z |
|
|
| theorem copy_greatest_part (w : External J f.Letters) (z : f.Node a) (x : A) : |
| (f.gluedOrder a).le ((f.copy a w).toFun x) z ↔ |
| P.le x ((f.copy a w).projection f.top z) := |
| (f.copy a w).greatest_part f.top f.greatest z x |
|
|
| theorem copy_cover (z : f.Node a) : ∃ w x, (f.copy a w).toFun x = z := by |
| rcases (f.diagram a).cover z with ⟨v, x, rfl⟩ |
| cases v with |
| | inl v => |
| refine ⟨representative J f.Letters v, (a.chart (representative J f.Letters v)).inverse x, ?_⟩ |
| change ((f.diagram a).vertex (.inl (point J f.Letters (representative J f.Letters v)))).toFun |
| ((a.chart (representative J f.Letters v)).toFun |
| ((a.chart (representative J f.Letters v)).inverse x)) = _ |
| erw [(a.chart (representative J f.Letters v)).right_inverse, representative_spec] |
| | inr c => |
| rcases (f.tile c.2).cover x with ⟨y, rfl⟩ | ⟨j, y, rfl⟩ |
| · exact ⟨c.1, y, (f.root_identification a c.1 c.2 y).symm⟩ |
| · exact ⟨c.1 * generator J f.Letters ⟨c.2, j⟩, y, |
| (f.child_identification a c.1 c.2 j y).symm⟩ |
|
|
| theorem comparison_in_copy (x y : f.Node a) (h : (f.gluedOrder a).le x y) : |
| ∃ w b c, (f.copy a w).toFun b = x ∧ (f.copy a w).toFun c = y ∧ P.le b c := by |
| rcases f.copy_cover a y with ⟨w, c, rfl⟩ |
| rcases (f.copy a w).ideal c x h with ⟨b, rfl⟩ |
| exact ⟨w, b, c, rfl, rfl, ((f.copy a w).le_iff b c).mp h⟩ |
|
|
| theorem copy_injective (w : External J f.Letters) (x y : A) : |
| (f.copy a w).toFun x = (f.copy a w).toFun y ↔ x = y := by |
| constructor |
| · exact embedding_injective (f.copy a w).toMeetEmbedding |
| · exact congrArg (f.copy a w).toFun |
|
|
| end TileRequests.Family |
|
|
|
|
|
|
| namespace TileRequests |
|
|
| open Attachment CosetGeometry |
| universe u |
| variable {A J : Type u} [Group J] {P : MeetOrder A} |
|
|
| namespace Family |
| variable {R : Type u} (f : Family P R) (a : OrderAction J P) |
|
|
| theorem transfer_child_root (w : External J f.Letters) (r : R) (j : f.Child r) (x y : A) |
| (h : ((f.tile r).child j).toFun x = (f.tile r).root.toFun y) : |
| (f.copy a (w * generator J f.Letters ⟨r, j⟩)).toFun x = (f.copy a w).toFun y := by |
| rw [← f.child_identification, ← f.root_identification, h] |
|
|
| theorem transfer_child_root_meet (w : External J f.Letters) (r : R) (j : f.Child r) (x y z : A) |
| (h : (f.tile r).order.meet (((f.tile r).child j).toFun x) ((f.tile r).root.toFun y) = |
| (f.tile r).root.toFun z) : |
| (f.gluedOrder a).meet ((f.copy a (w * generator J f.Letters ⟨r, j⟩)).toFun x) |
| ((f.copy a w).toFun y) = (f.copy a w).toFun z := by |
| rw [← f.child_identification, ← f.root_identification, ← f.root_identification, |
| ← (f.placedTile a w r).map_meet, h] |
|
|
| theorem transfer_child_child_meet (w : External J f.Letters) (r : R) |
| (j k : f.Child r) (x y z : A) |
| (h : (f.tile r).order.meet (((f.tile r).child j).toFun x) (((f.tile r).child k).toFun y) = |
| ((f.tile r).child j).toFun z) : |
| (f.gluedOrder a).meet ((f.copy a (w * generator J f.Letters ⟨r, j⟩)).toFun x) |
| ((f.copy a (w * generator J f.Letters ⟨r, k⟩)).toFun y) = |
| (f.copy a (w * generator J f.Letters ⟨r, j⟩)).toFun z := by |
| rw [← f.child_identification, ← f.child_identification, ← f.child_identification, |
| ← (f.placedTile a w r).map_meet, h] |
|
|
| end Family |
|
|
| namespace AllGeometry |
| variable (T : A) (greatest : ∀ x, P.le x T) (a : OrderAction J P) |
|
|
| noncomputable abbrev family : Family P (Request P) := Family.all T greatest |
| abbrev Letters := (family T greatest).Letters |
|
|
| theorem separation_fixes (w : External J (Letters T greatest)) (W x : A) (below : P.le x W) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.separation W, ⟨()⟩⟩)).toFun x = |
| ((family T greatest).copy a w).toFun x := |
| (family T greatest).transfer_child_root a w (.separation W) ⟨()⟩ x x |
| (Request.separation_agrees T greatest W x below) |
|
|
| theorem separation_meet (w : External J (Letters T greatest)) (W D X : A) |
| (belowD : P.le W D) (belowX : P.le W X) : |
| ((family T greatest).gluedOrder a).meet |
| (((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.separation W, ⟨()⟩⟩)).toFun D) |
| (((family T greatest).copy a w).toFun X) = ((family T greatest).copy a w).toFun W := |
| (family T greatest).transfer_child_root_meet a w (.separation W) ⟨()⟩ D X W |
| (BinaryTiles.separation_cross_meet P W D X belowD belowX) |
|
|
| theorem compression_restriction (w : External J (Letters T greatest)) (K : A) |
| (S : MeetIso P (idealOrder P K)) (x : A) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.compression K S, ⟨()⟩⟩)).toFun x = |
| ((family T greatest).copy a w).toFun (S.toFun x).val := |
| (family T greatest).transfer_child_root a w (.compression K S) ⟨()⟩ x (S.toFun x).val |
| (Request.compression_agrees T greatest K S x) |
|
|
| theorem compression_top (w : External J (Letters T greatest)) (K : A) |
| (S : MeetIso P (idealOrder P K)) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.compression K S, ⟨()⟩⟩)).toFun T = |
| ((family T greatest).copy a w).toFun K := |
| (family T greatest).transfer_child_root a w (.compression K S) ⟨()⟩ T K |
| (BinaryTiles.compression_top P T K greatest S) |
|
|
| theorem rectangle_first_agrees (w : External J (Letters T greatest)) (r : Rectangle.Input P) |
| (x : A) (below : P.le x r.C) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨false⟩⟩)).toFun x = |
| ((family T greatest).copy a w).toFun (r.g1.toFun x) := |
| (family T greatest).transfer_child_root a w (.rectangle r) ⟨false⟩ x (r.g1.toFun x) |
| (r.first_agrees x below) |
|
|
| theorem rectangle_second_agrees (w : External J (Letters T greatest)) (r : Rectangle.Input P) |
| (x : A) (below : P.le x r.C) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨true⟩⟩)).toFun x = |
| ((family T greatest).copy a w).toFun (r.g2.toFun x) := |
| (family T greatest).transfer_child_root a w (.rectangle r) ⟨true⟩ x (r.g2.toFun x) |
| (r.second_agrees x below) |
|
|
| theorem rectangle_children_intersection (w : External J (Letters T greatest)) |
| (r : Rectangle.Input P) (x y : A) : |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨false⟩⟩)).toFun x = |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨true⟩⟩)).toFun y ↔ |
| P.le x r.B ∧ x = y := by |
| erw [← (family T greatest).child_identification, ← (family T greatest).child_identification] |
| constructor |
| · intro h |
| exact (r.children_intersection x y).mp |
| (embedding_injective ((family T greatest).placedTile a w (.rectangle r)).toMeetEmbedding h) |
| · intro h |
| exact congrArg ((family T greatest).placedTile a w (.rectangle r)).toFun |
| ((r.children_intersection x y).mpr h) |
|
|
| theorem rectangle_first_meet_root (w : External J (Letters T greatest)) |
| (r : Rectangle.Input P) (x : A) : |
| ((family T greatest).gluedOrder a).meet |
| (((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨false⟩⟩)).toFun r.D) |
| (((family T greatest).copy a w).toFun x) = |
| ((family T greatest).copy a w).toFun (P.meet (r.g1.toFun r.C) x) := |
| (family T greatest).transfer_child_root_meet a w (.rectangle r) ⟨false⟩ r.D x _ |
| (r.first_meet_root x) |
|
|
| theorem rectangle_second_meet_root (w : External J (Letters T greatest)) |
| (r : Rectangle.Input P) (x : A) : |
| ((family T greatest).gluedOrder a).meet |
| (((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨true⟩⟩)).toFun r.D) |
| (((family T greatest).copy a w).toFun x) = |
| ((family T greatest).copy a w).toFun (P.meet (r.g2.toFun r.C) x) := |
| (family T greatest).transfer_child_root_meet a w (.rectangle r) ⟨true⟩ r.D x _ |
| (r.second_meet_root x) |
|
|
| theorem rectangle_children_meet (w : External J (Letters T greatest)) (r : Rectangle.Input P) : |
| ((family T greatest).gluedOrder a).meet |
| (((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨false⟩⟩)).toFun r.D) |
| (((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨true⟩⟩)).toFun r.D) = |
| ((family T greatest).copy a |
| (w * generator J (Letters T greatest) ⟨.rectangle r, ⟨false⟩⟩)).toFun r.B := |
| (family T greatest).transfer_child_child_meet a w (.rectangle r) ⟨false⟩ ⟨true⟩ r.D r.D r.B |
| r.children_meet |
|
|
| end AllGeometry |
|
|
| end TileRequests |
|
|
|
|
|
|
| namespace CosetGeometry.TileStars.Spaces |
|
|
| universe u |
| variable {J S R : Type u} [Group J] {request : S → R} (g : Spaces J S R request) |
| variable {B : Type u} (value : ∀ v, g.Space v → B) |
| variable (edges : ∀ d, g.Equation value d) |
|
|
| include edges in |
| theorem values_respect_parents {x y : g.scheme.Label} (h : g.scheme.Related x y) : |
| value x.1 x.2 = value y.1 y.2 := by |
| induction h with |
| | parent v hv x => exact g.parents_of_equations value edges v hv x |
| | refl => rfl |
| | symm h ih => exact ih.symm |
| | trans h1 h2 ih1 ih2 => exact ih1.trans ih2 |
|
|
| noncomputable def descend (z : g.Node) : B := |
| Quotient.lift (fun x => value x.1 x.2) |
| (fun _ _ h => g.values_respect_parents value edges h) (g.scheme.toGenerated z) |
|
|
| theorem descend_vertex (v : Vertex J S R) (x : g.Space v) : |
| g.descend value edges ((g.vertex v).toFun x) = value v x := by |
| change Quotient.lift _ _ (g.scheme.toGenerated ((g.scheme.vertex v).toFun x)) = _ |
| erw [g.scheme.toGenerated_vertex] |
| rfl |
|
|
| theorem descend_unique (h : g.Node → B) (vertices : ∀ v x, h ((g.vertex v).toFun x) = value v x) : |
| h = g.descend value edges := by |
| funext z |
| rcases g.cover z with ⟨v, x, rfl⟩ |
| exact (vertices v x).trans (g.descend_vertex value edges v x).symm |
|
|
| end CosetGeometry.TileStars.Spaces |
|
|
|
|
|
|
| namespace 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 translationValue (g : External J f.Letters) : |
| (v : TileStars.Vertex J f.Letters R) → f.Space v → f.Node a |
| | .inl v, x => (f.copy a (g * representative J f.Letters v)).toFun x |
| | .inr c, x => (f.placedTile a (g * c.1) c.2).toFun x |
|
|
| theorem translationValue_copy (g w : External J f.Letters) (x : A) : |
| f.translationValue a g (.inl (point J f.Letters w)) ((a.chart w).toFun x) = |
| (f.copy a (g * w)).toFun x := by |
| change (f.copy a (g * representative J f.Letters (point J f.Letters w))).toFun |
| ((a.iso (coordinate J f.Letters w)).toFun x) = _ |
| rw [← f.copy_coordinate, mul_assoc, coordinate_spec] |
|
|
| theorem translation_stem (g w : External J f.Letters) (r : R) : |
| (f.diagram a).Equation (f.translationValue a g) (.inl (w, r), true) := by |
| intro x |
| change f.translationValue a g (.inl (point J f.Letters w)) x.val = |
| (f.placedTile a (g * w) r).toFun ((f.tile r).root.toFun ((a.chart w).inverse x.val)) |
| rw [f.root_identification] |
| have eq := f.translationValue_copy a g w ((a.chart w).inverse x.val) |
| erw [(a.chart w).right_inverse] at eq |
| exact eq |
|
|
| theorem translation_branch (g w : External J f.Letters) (r : R) (j : f.Child r) : |
| (f.diagram a).Equation (f.translationValue a g) (.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.placedTile a (g * w) r).toFun (((f.tile r).child j).toFun y) = |
| f.translationValue a g (.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.translationValue_copy, ← mul_assoc] |
| exact f.child_identification a (g * w) r j y |
|
|
| theorem translation_edges (g : External J f.Letters) (d : TileStars.Dart J f.Letters R) : |
| (f.diagram a).Equation (f.translationValue a g) d := by |
| rcases d with ⟨e, b⟩ |
| have forward : (f.diagram a).Equation (f.translationValue a g) (e, true) := by |
| cases e with |
| | inl c => exact f.translation_stem a g c.1 c.2 |
| | inr c => exact f.translation_branch a g c.1 c.2.1 c.2.2 |
| cases b |
| · exact (f.diagram a).equation_of_reverse (f.translationValue a g) (e, false) forward |
| · exact forward |
|
|
| noncomputable def translate (g : External J f.Letters) : f.Node a → f.Node a := |
| (f.diagram a).descend (f.translationValue a g) (f.translation_edges a g) |
|
|
| theorem translate_vertex (g : External J f.Letters) (v : TileStars.Vertex J f.Letters R) |
| (x : f.Space v) : |
| f.translate a g (((f.diagram a).vertex v).toFun x) = f.translationValue a g v x := |
| (f.diagram a).descend_vertex (f.translationValue a g) (f.translation_edges a g) v x |
|
|
| theorem translate_copy (g w : External J f.Letters) (x : A) : |
| f.translate a g ((f.copy a w).toFun x) = (f.copy a (g * w)).toFun x := by |
| change f.translate a g (((f.diagram a).vertex (.inl (point J f.Letters w))).toFun |
| ((a.chart w).toFun x)) = _ |
| erw [f.translate_vertex, f.translationValue_copy] |
|
|
| theorem translate_tile (g w : External J f.Letters) (r : R) (x : (f.tile r).Space) : |
| f.translate a g ((f.placedTile a w r).toFun x) = (f.placedTile a (g * w) r).toFun x := |
| f.translate_vertex a g (.inr (w, r)) x |
|
|
| end TileRequests.Family |
|
|
| end PartitionPilot.Compression.RecursiveMaps |
|
|
| end OAI |
|
|
