| import OAI.Combinatorics.CompleteCrossing.DiskChords |
|
|
| namespace OAI |
|
|
| /-! Continuous assembled edges, their incidences, crossings, and absence of triple points. -/ |
|
|
| section |
|
|
| /-! Actual simple replacement edges, including both graph-vertex spokes. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : FiniteIntersectionDrawing graph} {C : DrawingPassages D} |
|
|
| def startSpoke (C : DrawingPassages D) (v : Vertex → Plane) (e : Edge) : |
| Path (v (graph.src e)) (C.arcStart (C.firstArc e)) := Path.segment _ _ |
|
|
| def endSpoke (C : DrawingPassages D) (v : Vertex → Plane) (e : Edge) : |
| Path (C.arcEnd (C.lastArc e)) (v (graph.dst e)) := (Path.segment _ _).symm |
|
|
| def assembledEdge (E : PolygonalExterior C) (v : Vertex → Plane) |
| (e : Edge) : Path (v (graph.src e)) (v (graph.dst e)) := |
| ((startSpoke C v e).trans (E.anchoredMiddle e)).trans (endSpoke C v e) |
|
|
| theorem startSpoke_injective (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : Edge) : |
| Function.Injective (startSpoke C v e) := C.vertex_spoke_injective v hv _ (C.startPortAt e) |
|
|
| theorem endSpoke_range (v : Vertex → Plane) (e : Edge) : |
| Set.range (endSpoke C v e)=Set.range (Path.segment (v (graph.dst e)) (C.portPoint (C.endPortAt e).val)) := by |
| exact Path.symm_range _ |
|
|
| theorem endSpoke_injective (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : Edge) : |
| Function.Injective (endSpoke C v e) := by |
| intro s t he |
| have hh := C.vertex_spoke_injective v hv _ (C.endPortAt e) he |
| have hval := congrArg Subtype.val hh |
| apply Subtype.ext |
| change 1-s.val=1-t.val at hval |
| linarith |
|
|
| theorem assembledEdge_injective (E : PolygonalExterior C) (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : Edge) : |
| Function.Injective (E.assembledEdge v e) := by |
| have hfirst : Function.Injective ((startSpoke C v e).trans (E.anchoredMiddle e)) := by |
| apply path_trans_injective _ _ (startSpoke_injective v hv e) (E.anchoredMiddle_injective e) |
| intro x hx hy |
| exact E.middle_spoke_meeting v hv e _ (C.startPortAt e) x hy hx |
| apply path_trans_injective _ _ hfirst (endSpoke_injective v hv e) |
| intro x hx hy |
| rw [Path.trans_range] at hx |
| rw [endSpoke_range] at hy |
| rcases hx with hx | hx |
| · have hd := C.vertex_spokes_disjoint_of_vertices_ne v hv (graph.src e) (graph.dst e) |
| (graph.loopless e) (C.startPortAt e) (C.endPortAt e) |
| exact False.elim (Set.disjoint_left.mp hd hx hy) |
| · exact E.middle_spoke_meeting v hv e _ (C.endPortAt e) x hx hy |
|
|
| theorem assembledEdge_range_iff (E : PolygonalExterior C) (v : Vertex → Plane) |
| (e : Edge) (x : Plane) : |
| x ∈ Set.range (E.assembledEdge v e) ↔ |
| (x ∈ Set.range (startSpoke C v e) ∨ |
| (∃ i : C.ArcIndex, i.1=e ∧ x ∈ Set.range (E.chain i).path) ∨ |
| (∃ p : C.InnerIndex, p.1=e ∧ x ∈ Set.range (C.crossingChord p))) ∨ |
| x ∈ Set.range (endSpoke C v e) := by |
| simp only [assembledEdge,Path.trans_range,Set.mem_union,anchoredMiddle_range,E.middlePath_range_iff] |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The assembled replacement is an actual continuous drawing with simple, vertex-avoiding edges. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| theorem vertex_spoke_vertex_eq (C : DrawingPassages D) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (j k : Vertex) (p : C.PortsAt (C.vertexMark j)) |
| (hk : v k ∈ Set.range (Path.segment (v j) (C.portPoint p.val))) : k=j := by |
| by_contra hkj |
| have hd := C.disks_disjoint _ _ (C.vertexMark_injective.ne (Ne.symm hkj)) |
| exact Set.disjoint_left.mp hd (C.vertex_spoke_range_in_disk v hv j p hk) (hv k).le |
|
|
| end DrawingPassages |
| namespace PolygonalExterior |
| variable {D : FiniteIntersectionDrawing graph} {C : DrawingPassages D} |
|
|
| theorem middlePath_avoids_vertices (E : PolygonalExterior C) (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : Edge) |
| (j : Vertex) : v j ∉ Set.range (E.middlePath e) := by |
| intro hx |
| rcases (E.middlePath_range_iff e (v j)).mp hx with ⟨i,hi,⟨t,ht⟩⟩ | ⟨p,hp,hx⟩ |
| · exact E.avoids_new_vertices v hv i t j ht |
| · exact Set.disjoint_left.mp (C.crossingChord_avoids_vertex_disks p j) hx (hv j).le |
|
|
| theorem assembledEdge_vertex_eq (E : PolygonalExterior C) (v : Vertex → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : Edge) |
| (j : Vertex) (hj : v j ∈ Set.range (E.assembledEdge v e)) : |
| j=graph.src e ∨ j=graph.dst e := by |
| rw [assembledEdge,Path.trans_range,Path.trans_range] at hj |
| rcases hj with (hj | hj) | hj |
| · exact Or.inl (C.vertex_spoke_vertex_eq v hv _ j (C.startPortAt e) hj) |
| · exact False.elim (E.middlePath_avoids_vertices v hv e j hj) |
| · rw [endSpoke_range] at hj |
| exact Or.inr (C.vertex_spoke_vertex_eq v hv _ j (C.endPortAt e) hj) |
|
|
| theorem assembledEdge_interior_avoids_vertices (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (e : Edge) (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) |
| (j : Vertex) : E.assembledEdge v e t ≠ v j := by |
| intro he |
| rcases E.assembledEdge_vertex_eq v hv e j ⟨t,he⟩ with rfl | rfl |
| · have hh := E.assembledEdge_injective v hv e (he.trans (E.assembledEdge v e).source.symm) |
| exact ht.1.ne hh.symm |
| · have hh := E.assembledEdge_injective v hv e (he.trans (E.assembledEdge v e).target.symm) |
| exact ht.2.ne hh |
|
|
| def assembledDrawing (E : PolygonalExterior C) (v : Vertex → Plane) |
| (hvinj : Function.Injective v) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) : |
| Paper170.ContinuousDrawing graph where |
| vertex := v |
| vertex_injective := hvinj |
| edge := E.assembledEdge v |
| edge_injective := E.assembledEdge_injective v hv |
| interior_avoids_vertices := E.assembledEdge_interior_avoids_vertices v hv |
|
|
| theorem exists_assembledDrawing [Fintype Vertex] [Fintype Edge] (D : FiniteIntersectionDrawing graph) : |
| ∃ (C : DrawingPassages D) (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2), |
| ∀ e, Function.Injective ((E.assembledDrawing v hvinj hv).edge e) := by |
| obtain ⟨C⟩ := nonempty_drawingPassages D |
| obtain ⟨E⟩ := nonempty_polygonalExterior C |
| obtain ⟨v,hvinj,hv,hne,hdet⟩ := C.exists_drawing_vertex_placement |
| exact ⟨C,E,v,hvinj,hv,E.assembledEdge_injective v hv⟩ |
|
|
| end PolygonalExterior |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Cross-edge incidence of the actual middle paths and exterior pieces. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : FiniteIntersectionDrawing graph} {C : DrawingPassages D} |
|
|
| theorem middle_exterior_edge_eq (E : PolygonalExterior C) (e : Edge) |
| (i : C.ArcIndex) (x : Plane) (hx : x ∈ Set.range (E.middlePath e)) |
| (hy : x ∈ Set.range (E.chain i).path) : e=i.1 := by |
| rcases (E.middlePath_range_iff e x).mp hx with ⟨j,hj,hx⟩ | ⟨p,hp,hx⟩ |
| · have he : j=i := by |
| by_contra hji |
| exact Set.disjoint_left.mp (E.disjoint j i hji) hx hy |
| exact hj.symm.trans (congrArg Sigma.fst he) |
| · obtain ⟨s,hs⟩ := hy |
| obtain ⟨t,ht⟩ := hx |
| rcases E.exterior_chord_meeting_parameters i p s t (hs.trans ht.symm) with |
| ⟨hi,_,_⟩ | ⟨hi,_,_⟩ |
| · exact hp.symm.trans (congrArg Sigma.fst hi).symm |
| · exact hp.symm.trans (congrArg Sigma.fst hi).symm |
|
|
| theorem middle_spoke_edge_eq (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (e : Edge) (j : Vertex) (p : C.PortsAt (C.vertexMark j)) |
| (x : Plane) (hx : x ∈ Set.range (E.middlePath e)) |
| (hy : x ∈ Set.range (Path.segment (v j) (C.portPoint p.val))) : e=p.val.1.1 := by |
| rcases (E.middlePath_range_iff e x).mp hx with ⟨i,hi,⟨s,hs⟩⟩ | ⟨q,hq,hx⟩ |
| · obtain ⟨t,ht⟩ := hy |
| have hm := E.exterior_spoke_meeting_parameters v hv i j p s t (hs.trans ht.symm) |
| rcases hm.2 with ⟨_,he⟩ | ⟨_,he⟩ |
| · exact hi.symm.trans (congrArg (fun r : C.Port => r.1.1) he) |
| · exact hi.symm.trans (congrArg (fun r : C.Port => r.1.1) he) |
| · exact False.elim (Set.disjoint_left.mp (C.crossingChord_avoids_vertex_disks q j) |
| hx (C.vertex_spoke_range_in_disk v hv j p hy)) |
|
|
| theorem middle_middle_meeting_chords (E : PolygonalExterior C) (e f : Edge) |
| (hef : e ≠ f) (x : Plane) (hx : x ∈ Set.range (E.middlePath e)) |
| (hy : x ∈ Set.range (E.middlePath f)) : |
| ∃ p q : C.InnerIndex, p.1=e ∧ q.1=f ∧ |
| x ∈ Set.range (C.crossingChord p) ∧ x ∈ Set.range (C.crossingChord q) := by |
| have he : ∃ p : C.InnerIndex, p.1=e ∧ x ∈ Set.range (C.crossingChord p) := by |
| rcases (E.middlePath_range_iff e x).mp hx with ⟨i,hi,hh⟩ | hh |
| · exact False.elim (hef ((E.middle_exterior_edge_eq f i x hy hh).trans hi).symm) |
| · exact hh |
| have hf : ∃ q : C.InnerIndex, q.1=f ∧ x ∈ Set.range (C.crossingChord q) := by |
| rcases (E.middlePath_range_iff f x).mp hy with ⟨i,hi,hh⟩ | hh |
| · exact False.elim (hef ((E.middle_exterior_edge_eq e i x hx hh).trans hi)) |
| · exact hh |
| obtain ⟨p,hp,hxp⟩ := he |
| obtain ⟨q,hq,hxq⟩ := hf |
| exact ⟨p,q,hp,hq,hxp,hxq⟩ |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Every nonvertex intersection of assembled edges is an actual crossing-disk chord intersection. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| theorem vertex_spokes_meeting (C : DrawingPassages D) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (j k : Vertex) (p : C.PortsAt (C.vertexMark j)) (q : C.PortsAt (C.vertexMark k)) |
| (hpq : p.val ≠ q.val) (x : Plane) |
| (hx : x ∈ Set.range (Path.segment (v j) (C.portPoint p.val))) |
| (hy : x ∈ Set.range (Path.segment (v k) (C.portPoint q.val))) : x=v j := by |
| by_cases hjk : j=k |
| · subst k |
| exact spokes_meet_only_at_center _ _ _ (hdet j p q (fun he => hpq (congrArg Subtype.val he))) |
| (by simpa only [Path.range_segment] using hx) (by simpa only [Path.range_segment] using hy) |
| · exact False.elim (Set.disjoint_left.mp (C.vertex_spokes_disjoint_of_vertices_ne v hv j k hjk p q) hx hy) |
|
|
| end DrawingPassages |
| namespace PolygonalExterior |
| variable {D : FiniteIntersectionDrawing graph} {C : DrawingPassages D} |
|
|
| theorem assembled_spoke_meeting (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (e : Edge) (j : Vertex) (p : C.PortsAt (C.vertexMark j)) |
| (hep : e ≠ p.val.1.1) (x : Plane) (hx : x ∈ Set.range (E.assembledEdge v e)) |
| (hy : x ∈ Set.range (Path.segment (v j) (C.portPoint p.val))) : x=v j := by |
| rw [assembledEdge,Path.trans_range,Path.trans_range] at hx |
| rcases hx with (hx | hx) | hx |
| · exact C.vertex_spokes_meeting v hv hdet j _ p (C.startPortAt e) |
| (fun he => hep (congrArg (fun r : C.Port => r.1.1) he).symm) x hy hx |
| · exact False.elim (hep (E.middle_spoke_edge_eq v hv e j p x hx hy)) |
| · rw [endSpoke_range] at hx |
| exact C.vertex_spokes_meeting v hv hdet j _ p (C.endPortAt e) |
| (fun he => hep (congrArg (fun r : C.Port => r.1.1) he).symm) x hy hx |
|
|
| theorem assembled_meeting_chords (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (e f : Edge) (hef : e ≠ f) (x : Plane) |
| (havoid : ∀ j, x ≠ v j) (hx : x ∈ Set.range (E.assembledEdge v e)) |
| (hy : x ∈ Set.range (E.assembledEdge v f)) : |
| ∃ p q : C.InnerIndex, p.1=e ∧ q.1=f ∧ |
| x ∈ Set.range (C.crossingChord p) ∧ x ∈ Set.range (C.crossingChord q) := by |
| have hm (a b : Edge) (hab : a ≠ b) |
| (ha : x ∈ Set.range (E.assembledEdge v a)) (hb : x ∈ Set.range (E.assembledEdge v b)) : |
| x ∈ Set.range (E.middlePath a) := by |
| rw [assembledEdge,Path.trans_range,Path.trans_range] at ha |
| rcases ha with (ha | ha) | ha |
| · exact False.elim (havoid _ (E.assembled_spoke_meeting v hv hdet b _ (C.startPortAt a) |
| (Ne.symm hab) x hb ha)) |
| · exact ha |
| · rw [endSpoke_range] at ha |
| exact False.elim (havoid _ (E.assembled_spoke_meeting v hv hdet b _ (C.endPortAt a) |
| (Ne.symm hab) x hb ha)) |
| exact E.middle_middle_meeting_chords e f hef x (hm e f hef hx hy) (hm f e (Ne.symm hef) hy hx) |
|
|
| theorem assembled_crossings_subset (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) : |
| Paper170.crossingPoints (E.assembledDrawing v hvinj hv) ⊆ C.chordCrossings := by |
| rintro x ⟨e,f,hef,⟨s,hs,he⟩,⟨t,ht,hf⟩⟩ |
| obtain ⟨p,q,hp,hq,hxp,hxq⟩ := E.assembled_meeting_chords v hv hdet e f hef x |
| (fun j heq => E.assembledEdge_interior_avoids_vertices v hv e s hs j (he.trans heq)) |
| ⟨s,he⟩ ⟨t,hf⟩ |
| exact ⟨p,q,(fun heq => hef (hp.symm.trans ((congrArg Sigma.fst heq).trans hq))),hxp,hxq⟩ |
|
|
| theorem assembled_crossings_finite (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) : |
| (Paper170.crossingPoints (E.assembledDrawing v hvinj hv)).Finite := |
| C.chordCrossings_finite.subset (E.assembled_crossings_subset v hvinj hv hdet) |
|
|
| end PolygonalExterior |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! No triple intersection is introduced by the actual replacement drawing. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : FiniteIntersectionDrawing graph} {C : DrawingPassages D} |
|
|
| theorem assembled_no_triples (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (e f g : Edge) (hef : e ≠ f) (heg : e ≠ g) (hfg : f ≠ g) : |
| Paper170.edgeInterior (E.assembledDrawing v hvinj hv) e ∩ |
| Paper170.edgeInterior (E.assembledDrawing v hvinj hv) f ∩ |
| Paper170.edgeInterior (E.assembledDrawing v hvinj hv) g=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro x ⟨⟨⟨s,hs,he⟩,⟨t,ht,hf⟩⟩,⟨u,hu,hg⟩⟩ |
| have ha : ∀ j, x ≠ v j := fun j heq => |
| E.assembledEdge_interior_avoids_vertices v hv e s hs j (he.trans heq) |
| obtain ⟨p,q,hp,hq,hxp,hxq⟩ := E.assembled_meeting_chords v hv hdet e f hef x ha ⟨s,he⟩ ⟨t,hf⟩ |
| obtain ⟨r,z,hr,hz,hxr,hxz⟩ := E.assembled_meeting_chords v hv hdet e g heg x ha ⟨s,he⟩ ⟨u,hg⟩ |
| apply C.innerMark_no_triples p q z |
| · intro heq |
| exact hef (hp.symm.trans ((congrArg Sigma.fst heq).trans hq)) |
| · intro heq |
| exact heg (hp.symm.trans ((congrArg Sigma.fst heq).trans hz)) |
| · intro heq |
| exact hfg (hq.symm.trans ((congrArg Sigma.fst heq).trans hz)) |
| · exact C.chord_meeting_marks_eq p q hxp hxq |
| · exact C.chord_meeting_marks_eq p z hxp hxz |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Each actual boundary port belongs to exactly its assigned assembled edge. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
|
|
| theorem portPoint_mem_exterior (E : PolygonalExterior C) (p : C.Port) : |
| C.portPoint p ∈ Set.range (E.chain p.1).path := by |
| rcases p with ⟨i,b⟩ |
| cases b |
| · exact ⟨0,(E.chain i).path.source⟩ |
| · exact ⟨1,(E.chain i).path.target⟩ |
|
|
| theorem assembled_port_mem_range (E : PolygonalExterior C) (v : Vertex → Plane) |
| (p : C.Port) : C.portPoint p ∈ Set.range (E.assembledEdge v p.1.1) := by |
| apply (E.assembledEdge_range_iff v p.1.1 (C.portPoint p)).mpr |
| exact Or.inl (Or.inr (Or.inl ⟨p.1,rfl,E.portPoint_mem_exterior p⟩)) |
|
|
| theorem assembled_exterior_edge_eq (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (e : Edge) (i : C.ArcIndex) (x : Plane) |
| (hx : x ∈ Set.range (E.assembledEdge v e)) (hy : x ∈ Set.range (E.chain i).path) : e=i.1 := by |
| rw [assembledEdge,Path.trans_range,Path.trans_range] at hx |
| rcases hx with (hx | hx) | hx |
| · obtain ⟨s,hs⟩ := hy |
| obtain ⟨t,ht⟩ := hx |
| have hh := E.exterior_spoke_meeting_parameters v hv i _ (C.startPortAt e) s t (hs.trans ht.symm) |
| rcases hh.2 with ⟨_,he⟩ | ⟨_,he⟩ |
| · exact (congrArg (fun p : C.Port => p.1.1) he).symm |
| · exact (congrArg (fun p : C.Port => p.1.1) he).symm |
| · exact E.middle_exterior_edge_eq e i x hx hy |
| · rw [endSpoke_range] at hx |
| obtain ⟨s,hs⟩ := hy |
| obtain ⟨t,ht⟩ := hx |
| have hh := E.exterior_spoke_meeting_parameters v hv i _ (C.endPortAt e) s t (hs.trans ht.symm) |
| rcases hh.2 with ⟨_,he⟩ | ⟨_,he⟩ |
| · exact (congrArg (fun p : C.Port => p.1.1) he).symm |
| · exact (congrArg (fun p : C.Port => p.1.1) he).symm |
|
|
| theorem assembled_port_unique_edge (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (e : Edge) (p : C.Port) |
| (hp : C.portPoint p ∈ Set.range (E.assembledEdge v e)) : e=p.1.1 := |
| E.assembled_exterior_edge_eq v hv e p.1 (C.portPoint p) hp (E.portPoint_mem_exterior p) |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| end OAI |
|
|
