import OAI.Combinatorics.PlanarL1.Growth namespace OAI section namespace PlanarL1.HeightTree open Set PlanarL1.Topology attribute [local instance] Classical.propDecidable variable {V : Type} [Fintype V] [LinearOrder V] (T : HeightTree V) {G : SimpleGraph V} (P : PLDrawing G) lemma growth_exists (hG : T.graph ≤ G) (S : Finset V) (hroot : T.root∈S) (hS : T.ParentClosed S) : Nonempty (T.Growth P S) := by classical induction S using Finset.strongInductionOn with | _ S ih => by_cases hs : S={T.root} · subst S; exact T.initial_growth P hG have hne : (S.erase T.root).Nonempty := by by_contra hh have hempty := Finset.not_nonempty_iff_eq_empty.mp hh apply hs exact (Finset.insert_erase hroot).symm.trans (by rw [hempty]; rfl) obtain ⟨v,hv,hmax⟩ := Finset.exists_max_image (S.erase T.root) T.depth hne have hvr : v≠T.root := (Finset.mem_erase.mp hv).1 have hvS : v∈S := (Finset.mem_erase.mp hv).2 have hroot' : T.root∈S.erase v := Finset.mem_erase.mpr ⟨hvr.symm,hroot⟩ have hS' : T.ParentClosed (S.erase v) := by intro w hw have hwv := (Finset.mem_erase.mp hw).1 have hwS := (Finset.mem_erase.mp hw).2 refine Finset.mem_erase.mpr ⟨?_,hS w hwS⟩ intro he by_cases hwr : w=T.root · subst w; exact hvr (he.symm.trans T.at_root) · have hd := T.step_depth (show T.Step v w from ⟨hwr,he.symm⟩) exact (not_lt_of_ge (hmax w (Finset.mem_erase.mpr ⟨hwr,hwS⟩))) hd have hfront : T.Frontier (S.erase v) v := by refine ⟨(by simp),Finset.mem_erase.mpr ⟨T.parent_ne hvr,hS v hvS⟩⟩ obtain ⟨H⟩ := ih (S.erase v) (Finset.erase_ssubset hvS) hroot' hS' simpa only [Finset.insert_erase hvS] using H.advance hS' hG hfront noncomputable def internal : Finset V := Finset.univ.filter (fun v => ¬T.IsLeaf v) omit [LinearOrder V] in lemma mem_internal (v : V) : v∈T.internal ↔ ¬T.IsLeaf v := by simp [internal] lemma internal_closed : T.ParentClosed T.internal := by intro v hv by_cases he : v=T.root · subst v; rwa [T.at_root] · apply (T.mem_internal _).mpr exact fun h => h v ⟨he,rfl⟩ omit [LinearOrder V] in lemma internal_scope (hroot : ¬T.IsLeaf T.root) (v : V) : T.Scope T.internal v := by by_cases h : T.IsLeaf v · refine Or.inr ⟨fun hv => (T.mem_internal v).mp hv h,(T.mem_internal _).mpr ?_⟩ have hvr : v≠T.root := fun he => hroot (he ▸ h) exact fun hh => hh v ⟨hvr,rfl⟩ · exact Or.inl ((T.mem_internal _).mpr h) omit [LinearOrder V] in lemma internal_frontier (hroot : ¬T.IsLeaf T.root) (v : V) : T.Frontier T.internal v ↔ T.IsLeaf v := by constructor · intro h by_contra hh exact h.1 ((T.mem_internal _).mpr hh) · intro h rcases T.internal_scope hroot v with hv|hv · exact False.elim ((T.mem_internal _).mp hv h) · exact hv /-- Every actual polygonally drawn finite tree can be enclosed in a convex disk, with the leaf stubs left as disjoint exterior polygonal arcs. All untouched raw leaf-to-leaf drawing pieces remain outside the closed disk. -/ lemma tree_core (hG : T.graph ≤ G) (hroot : ¬T.IsLeaf T.root) : ∃ C : DiskCore T.graph {v | T.IsLeaf v}, ∃ E : ExteriorPorts C.drawing.point P.point {v | T.IsLeaf v} C.disk, Disjoint (closure C.disk) (P.carrier {v | T.IsLeaf v}) ∧ ∀ v, T.IsLeaf v → ∀ z∈E.region v, z∈P.carrier {v | T.IsLeaf v} → z=P.point v := by classical obtain ⟨H⟩ := T.growth_exists P hG T.internal ((T.mem_internal _).mpr hroot) T.internal_closed let i : V → {v // T.Scope T.internal v} := fun v => ⟨v,T.internal_scope hroot v⟩ have hi : Function.Injective i := fun _ _ h => congrArg Subtype.val h have hgraph : ∀ u v, T.graph.Adj u v → (SimpleGraph.comap Subtype.val T.graph).Adj (i u) (i v) := fun _ _ h => h have hports : ∀ v, T.IsLeaf v → T.Frontier T.internal (i v).val := fun v h => (T.internal_frontier hroot v).mpr h obtain ⟨C,E,hD,hp,hE⟩ := H.core.pullback H.stems i hi hgraph hports have hout : outside P T.internal=P.carrier {v | T.IsLeaf v} := by unfold outside congr 1 ext v simp only [mem_ofPred_eq,T.mem_internal,not_not] refine ⟨C,E,?_,?_⟩ · rw [hD,← hout]; exact H.avoid · intro v hv z hz hzK apply H.contact (i v) (hports v hv) z · exact (hE v) ▸ hz · rwa [hout] end PlanarL1.HeightTree namespace PlanarL1.Topology open Set lemma PolygonallyJoined.exists_path {S : Set ℂ} {x y : ℂ} (h : PolygonallyJoined S x y) (hx : x∈S) : ∃ p : Path x y, range p ⊆ S := by induction h with | refl => exact ⟨Path.refl x,by rintro z ⟨t,rfl⟩; exact hx⟩ | @tail y z h hz ih => obtain ⟨p,hp⟩ := ih exact ⟨p.trans (Path.segment y z),by rw [Path.trans_range,Path.range_segment]; exact union_subset hp hz⟩ namespace DiskCore variable {V : Type} [LinearOrder V] {G : SimpleGraph V} {B : Set V} (C : DiskCore G B) lemma carrier_subset (S : Set V) : C.drawing.carrier S ⊆ closure C.disk := by intro z hz rcases hz with hz|hz · obtain ⟨v,hv⟩ := mem_iUnion.mp hz exact mem_singleton_iff.mp hv ▸ C.point_mem v.val · obtain ⟨u,hu⟩ := mem_iUnion.mp hz obtain ⟨v,hv⟩ := mem_iUnion.mp hu exact C.region_mem _ _ (C.drawing.region_adj hv) hv end DiskCore end PlanarL1.Topology namespace PlanarL1.HeightTree open Set PlanarL1.Topology Real attribute [local instance] Classical.propDecidable variable {V : Type} [LinearOrder V] (T : HeightTree V) lemma ancestor_plpath (P : PLDrawing T.graph) {u v : V} (h : T.Ancestor u v) {S : Set V} (hS : ∀ w, T.Ancestor u w → T.Ancestor w v → w∈S) : ∃ p : Path (P.point u) (P.point v), range p ⊆ P.carrier S := by induction h with | refl => exact ⟨Path.refl _,by rintro z ⟨t,rfl⟩; exact P.point_mem_carrier (hS _ .refl .refl)⟩ | @tail v w huv hvw ih => obtain ⟨p,hp⟩ := ih (fun a ha hb => hS a ha (hb.tail hvw)) obtain ⟨q,hq⟩ := (P.region_joined v w (Or.inl hvw)).exists_path (P.region_start v w (Or.inl hvw)) refine ⟨p.trans q,?_⟩ rw [Path.trans_range] exact union_subset hp (hq.trans (P.region_mem_carrier (hS v huv (.single hvw)) (hS w (huv.tail hvw) .refl))) lemma descendants_pljoined (P : PLDrawing T.graph) (a : V) {u v : V} (hu : T.Ancestor a u) (hv : T.Ancestor a v) : ∃ p : Path (P.point u) (P.point v), range p ⊆ P.carrier {w | T.Ancestor a w} := by obtain ⟨p,hp⟩ := T.ancestor_plpath P hu (fun _ h _ => h) obtain ⟨q,hq⟩ := T.ancestor_plpath P hv (fun _ h _ => h) exact ⟨p.symm.trans q,by rw [Path.trans_range,Path.symm_range]; exact union_subset hp hq⟩ lemma nondescendants_pljoined (P : PLDrawing T.graph) (a : V) {u v : V} (hu : ¬T.Ancestor a u) (hv : ¬T.Ancestor a v) : ∃ p : Path (P.point u) (P.point v), range p ⊆ P.carrier {w | ¬T.Ancestor a w} := by obtain ⟨p,hp⟩ := T.ancestor_plpath P (T.root_ancestor u) (S := {w | ¬T.Ancestor a w}) (fun _ _ h ha => hu (ha.trans h)) obtain ⟨q,hq⟩ := T.ancestor_plpath P (T.root_ancestor v) (S := {w | ¬T.Ancestor a w}) (fun _ _ h ha => hv (ha.trans h)) exact ⟨p.symm.trans q,by rw [Path.trans_range,Path.symm_range]; exact union_subset hp hq⟩ lemma core_prefix_connected {B : Set V} {n : ℕ} (C : DiskCore T.graph B) (h : ℂ ≃ₜ ℂ) (hbound : ∀ z∈closure C.disk, ‖h z‖≤1) (leaf : Fin n → V) (angle : Fin n → ℝ) (hend : ∀ i, h (C.drawing.point (leaf i))=(Circle.exp (angle i) : ℂ)) (horder : StrictMono angle) (hspan : ∀ i j, angle j-angle i<2*π) (a : V) {i j : Fin n} (hi : T.Ancestor a (leaf i)) (hj : T.Ancestor a (leaf j)) : (T.prefixGraph leaf a).Reachable i j := by apply Columns.consecutive_connected (T.prefixGraph leaf a) {k | T.Ancestor a (leaf k)} (fun _ _ hne hc ha hb => ⟨hne,hc,ha,hb⟩) _ hi hj intro w x y z hwx hxy hyz hbad suffices impossible : ∀ (S U : Set V), Disjoint S U → (∃ p : Path (C.drawing.point (leaf w)) (C.drawing.point (leaf y)), range p⊆C.drawing.carrier S) → (∃ q : Path (C.drawing.point (leaf x)) (C.drawing.point (leaf z)), range q⊆C.drawing.carrier U) → False by rcases hbad with ⟨hw,hx,hy,hz⟩ | ⟨hw,hx,hy,hz⟩ · exact impossible {v | T.Ancestor a v} {v | ¬T.Ancestor a v} disjoint_compl_right (T.descendants_pljoined C.drawing a hw hy) (T.nondescendants_pljoined C.drawing a hx hz) · exact impossible {v | ¬T.Ancestor a v} {v | T.Ancestor a v} disjoint_compl_left (T.nondescendants_pljoined C.drawing a hw hy) (T.descendants_pljoined C.drawing a hx hz) intro S U hdis ⟨p,hp⟩ ⟨q,hq⟩ let p' := (p.map h.continuous).cast (hend w).symm (hend y).symm let q' := (q.map h.continuous).cast (hend x).symm (hend z).symm obtain ⟨s,t,hst⟩ := Topology.disk_cyclic_arcs_meet (angle w) (angle x) (angle y) (angle z) (horder hwx) (horder hxy) (horder hyz) (by linarith [hspan w z]) p' q' (fun t => hbound _ (C.carrier_subset S (hp (mem_range_self t)))) (fun t => hbound _ (C.carrier_subset U (hq (mem_range_self t)))) have he : p s=q t := h.injective hst exact disjoint_left.mp (C.drawing.carrier_disjoint hdis) (hp (mem_range_self s)) (he ▸ hq (mem_range_self t)) lemma layout_of_core [Fintype V] (C : DiskCore T.graph {v | T.IsLeaf v}) (P : SimpleGraph {v : V // T.IsLeaf v}) (hheight : ∀ i j, P.Adj i j → T.height i.val=T.height j.val) (hmatch : ∀ i j k, P.Adj i j → P.Adj i k → j=k) (arc : ∀ i j, P.Adj i j → Path (C.drawing.point i.val) (C.drawing.point j.val)) (hout : ∀ i j h t, arc i j h t∉C.disk) (hdis : ∀ i j h k l h', i≠k → i≠l → j≠k → j≠l → Disjoint (range (arc i j h)) (range (arc k l h'))) : ∃ (L : T.Layout (Fintype.card {v : V // T.IsLeaf v})), ∀ i j, L.pair.Adj i j ↔ P.Adj ⟨L.leaf i,L.isLeaf i⟩ ⟨L.leaf j,L.isLeaf j⟩ := by classical obtain ⟨h,hi,hcl,hfr⟩ := exists_homeomorph_image_interior_closure_frontier_eq_unitBall C.convex (show (interior C.disk).Nonempty by rw [C.isOpen.interior_eq]; exact ⟨C.center,C.center_mem⟩) C.bounded have hnorm (v : {v : V // T.IsLeaf v}) : ‖h (C.drawing.point v.val)‖=1 := by have hm := mem_image_of_mem h (C.port v.val v.property) rw [hfr] at hm simpa only [Metric.mem_sphere,dist_zero_right] using hm have hbound (z : ℂ) (hz : z∈closure C.disk) : ‖h z‖≤1 := by have hm := mem_image_of_mem h hz rw [hcl] at hm simpa only [Metric.mem_closedBall,dist_zero_right] using hm have hexterior (z : ℂ) (hz : z∉C.disk) : 1≤‖h z‖ := by by_contra! hn have hm : h z∈Metric.ball (0:ℂ) 1 := by simpa only [Metric.mem_ball,dist_zero_right] using hn rw [← hi] at hm obtain ⟨w,hw,he⟩ := hm exact hz (h.injective he ▸ interior_subset hw) obtain ⟨e,he,hspan⟩ := Topology.finite_port_order (fun v : {v : V // T.IsLeaf v} => h (C.drawing.point v.val)) hnorm (h.injective.comp (C.drawing.point_injective.comp Subtype.val_injective)) let angle (i : Fin (Fintype.card {v : V // T.IsLeaf v})) := Complex.arg (h (C.drawing.point (e i).val)) have hend i : h (C.drawing.point (e i).val)=(Circle.exp (angle i) : ℂ) := (Topology.circle_exp_arg (hnorm (e i))).symm let A : ∀ i j, (SimpleGraph.comap e P).Adj i j → Path (Circle.exp (angle i) : ℂ) (Circle.exp (angle j) : ℂ) := fun i j hij => ((arc (e i) (e j) hij).map h.continuous).cast (hend i).symm (hend j).symm have hnc : ColumnOrder.Noncrossing (SimpleGraph.comap e P) := by apply Topology.noncrossing_of_exterior_arcs _ angle he hspan A · intro i j hij t; exact hexterior _ (hout (e i) (e j) hij t) · intro i j hij k l hkl hik hil hjk hjl rw [disjoint_left] rintro z ⟨s,hs⟩ ⟨t,ht⟩ have hh : h (arc (e i) (e j) hij s)=h (arc (e k) (e l) hkl t) := hs.trans ht.symm exact disjoint_left.mp (hdis (e i) (e j) hij (e k) (e l) hkl (e.injective.ne hik) (e.injective.ne hil) (e.injective.ne hjk) (e.injective.ne hjl)) (mem_range_self s) (h.injective hh ▸ mem_range_self t) let L : T.Layout (Fintype.card {v : V // T.IsLeaf v}) := { leaf := fun i => (e i).val isLeaf := fun i => (e i).property injective := Subtype.val_injective.comp e.injective covers := fun v hv => ⟨e.symm ⟨v,hv⟩,by simp⟩ prefix_connected := fun a i j hi hj => T.core_prefix_connected C h hbound _ angle hend he hspan a hi hj pair := SimpleGraph.comap e P noncrossing := hnc paired_height := fun i j hij => hheight (e i) (e j) hij matching := fun i j k hij hik => e.injective (hmatch (e i) (e j) (e k) hij hik) } exact ⟨L,fun _ _ => Iff.rfl⟩ end PlanarL1.HeightTree namespace PlanarL1.Topology open Set variable {V : Type} [LinearOrder V] {G : SimpleGraph V} (P : PLDrawing G) {p : V → ℂ} {B : Set V} {D : Set ℂ} (E : ExteriorPorts p P.point B D) /-- The untouched raw carrier together with the actual exterior stubs. -/ def stemCarrier (S : Set V) : Set ℂ := P.carrier S ∪ ⋃ v : S, E.region v.val lemma stemCarrier_disjoint (hcontact : ∀ v∈B, ∀ z∈E.region v, z∈P.carrier B → z=P.point v) {S U : Set V} (hS : S⊆B) (hU : U⊆B) (hSU : Disjoint S U) : Disjoint (stemCarrier P E S) (stemCarrier P E U) := by have cross {S U : Set V} (hS : S⊆B) (hU : U⊆B) (hSU : Disjoint S U) : Disjoint (P.carrier S) (⋃ v : U, E.region v.val) := by rw [disjoint_left] intro z hz hz' obtain ⟨v,hv⟩ := mem_iUnion.mp hz' have he := hcontact v.val (hU v.property) z hv (P.carrier_mono hS hz) exact disjoint_left.mp hSU (P.point_carrier_iff.mp (he ▸ hz)) v.property apply Disjoint.union_left · exact Disjoint.union_right (P.carrier_disjoint hSU) (cross hS hU hSU) · apply Disjoint.union_right (cross hU hS hSU.symm).symm rw [disjoint_left] intro z hz hz' obtain ⟨v,hv⟩ := mem_iUnion.mp hz obtain ⟨w,hw⟩ := mem_iUnion.mp hz' exact disjoint_left.mp (E.disjoint v.val (hS v.property) w.val (hU w.property) (fun he => disjoint_left.mp hSU v.property (he ▸ w.property))) hv hw end PlanarL1.Topology namespace PlanarL1.HeightTree open Set PlanarL1.Topology attribute [local instance] Classical.propDecidable variable {V : Type} [Fintype V] [LinearOrder V] (T : HeightTree V) {G : SimpleGraph V} (P : PLDrawing G) /-- The literal planar leaf layout is constructed from a real raw drawing of its tree and terminal matching. Neither a contour order nor a disk is assumed. -/ lemma layout_of_drawing (P : PLDrawing G) (hG : T.graph ≤ G) (hroot : ¬T.IsLeaf T.root) (M : SimpleGraph {v : V // T.IsLeaf v}) (hMG : ∀ i j, M.Adj i j → G.Adj i.val j.val) (hheight : ∀ i j, M.Adj i j → T.height i.val=T.height j.val) (hmatch : ∀ i j k, M.Adj i j → M.Adj i k → j=k) : ∃ (L : T.Layout (Fintype.card {v : V // T.IsLeaf v})), ∀ i j, L.pair.Adj i j ↔ M.Adj ⟨L.leaf i,L.isLeaf i⟩ ⟨L.leaf j,L.isLeaf j⟩ := by classical obtain ⟨C,E,havoid,hcontact⟩ := T.tree_core P hG hroot choose s hs using fun v : {v // T.IsLeaf v} => (E.joined v.val v.property).exists_path (E.start v.val v.property) choose r hr using fun i j (h : M.Adj i j) => (P.region_joined i.val j.val (hMG i j h)).exists_path (P.region_start i.val j.val (hMG i j h)) let A : ∀ i j, M.Adj i j → Path (C.drawing.point i.val) (C.drawing.point j.val) := fun i j h => ((s i).trans (r i j h)).trans (s j).symm have hS (i j : {v // T.IsLeaf v}) : ({i.val,j.val} : Set V) ⊆ {v | T.IsLeaf v} := by intro v hv rcases hv with rfl|hv · exact i.property · rw [mem_singleton_iff] at hv; exact hv ▸ j.property have hArange (i j : {v // T.IsLeaf v}) (h : M.Adj i j) : range (A i j h) ⊆ stemCarrier P E {i.val,j.val} := by change range (((s i).trans (r i j h)).trans (s j).symm) ⊆ _ rw [Path.trans_range,Path.trans_range,Path.symm_range] refine union_subset (union_subset ?_ ?_) ?_ · intro z hz; exact Or.inr (mem_iUnion.mpr ⟨⟨i.val,by simp⟩,hs i hz⟩) · intro z hz; exact Or.inl (P.region_mem_carrier (by simp) (by simp) (hr i j h hz)) · intro z hz; exact Or.inr (mem_iUnion.mpr ⟨⟨j.val,by simp⟩,hs j hz⟩) have houtside (S : Set V) (hS : S⊆{v | T.IsLeaf v}) (z : ℂ) (hz : z∈stemCarrier P E S) : z∉C.disk := by intro hzD rcases hz with hz|hz · exact disjoint_left.mp havoid (subset_closure hzD) (P.carrier_mono hS hz) · obtain ⟨v,hv⟩ := mem_iUnion.mp hz have he := E.inside v.val (hS v.property) z hv (subset_closure hzD) have hf := C.port v.val (hS v.property) have hn : C.drawing.point v.val∉interior C.disk := hf.2 exact hn (by rw [C.isOpen.interior_eq]; exact he ▸ hzD) apply T.layout_of_core C M hheight hmatch A · intro i j h t; exact houtside _ (hS i j) _ (hArange i j h (mem_range_self t)) · intro i j h k l h' hik hil hjk hjl have hSU : Disjoint ({i.val,j.val} : Set V) {k.val,l.val} := by simp only [disjoint_left,mem_insert_iff,mem_singleton_iff] intro v hv hv' rcases hv with rfl|rfl <;> rcases hv' with he|he · exact hik (Subtype.ext he) · exact hil (Subtype.ext he) · exact hjk (Subtype.ext he) · exact hjl (Subtype.ext he) exact (stemCarrier_disjoint P E hcontact (hS i j) (hS k l) hSU).mono (hArange i j h) (hArange k l h') end PlanarL1.HeightTree namespace PlanarL1.Topology.Drawing open Set variable {V : Type} [LinearOrder V] {G : SimpleGraph V} (D : Drawing G) lemma forward_interior_meet (d e : ForwardDart G) {s t : unitInterval} (hs : 0<(s:ℝ)) (hs' : (s:ℝ)<1) (ht : 0<(t:ℝ)) (ht' : (t:ℝ)<1) (he : D.edge d.val.fst d.val.snd d.val.adj s=D.edge e.val.fst e.val.snd e.val.adj t) : d=e ∧ s=t := by have hd := ForwardDart.eq_of_endpoints d e (D.interior_disjoint _ _ _ _ _ _ s t hs hs' ht ht' he) subst e exact ⟨rfl,D.edge_injective _ _ _ he⟩ namespace Split attribute [local instance] Classical.propDecidable variable (S : Set (ForwardDart G)) abbrev Tip := S × Fin 2 abbrev Node := V ⊕ Tip S abbrev Piece := {d : ForwardDart G // d∉S} ⊕ (S × Fin 3) noncomputable def tipTime (k : Fin 2) : unitInterval := ⟨((k.val:ℝ)+1)/3,by constructor · positivity · have hk : (k.val:ℝ)<2 := by exact_mod_cast k.isLt linarith⟩ lemma tipTime_pos (k : Fin 2) : 0<(tipTime k:ℝ) := by dsimp [tipTime]; positivity lemma tipTime_lt_one (k : Fin 2) : (tipTime k:ℝ)<1 := by have hk : (k.val:ℝ)<2 := by exact_mod_cast k.isLt dsimp [tipTime]; linarith noncomputable def point : Node S → ℂ | .inl v => D.point v | .inr (d,k) => D.edge d.val.val.fst d.val.val.snd d.val.val.adj (tipTime k) lemma point_injective : Function.Injective (point D S) := by intro a b he cases a with | inl a => cases b with | inl b => exact congrArg Sum.inl (D.point_injective he) | inr b => exact False.elim (D.interior_avoid _ _ _ _ a (tipTime_pos b.2) (tipTime_lt_one b.2) he.symm) | inr a => cases b with | inl b => exact False.elim (D.interior_avoid _ _ _ _ b (tipTime_pos a.2) (tipTime_lt_one a.2) he) | inr b => obtain ⟨hd,ht⟩ := D.forward_interior_meet a.1.val b.1.val (tipTime_pos a.2) (tipTime_lt_one a.2) (tipTime_pos b.2) (tipTime_lt_one b.2) he have hk : a.2=b.2 := by apply Fin.ext have hh := congrArg Subtype.val ht change ((a.2.val:ℝ)+1)/3=((b.2.val:ℝ)+1)/3 at hh exact_mod_cast (show (a.2.val:ℝ)=(b.2.val:ℝ) by linarith) exact congrArg Sum.inr (Prod.ext (Subtype.ext hd) hk) def meshNode (d : S) (r : Fin 4) : Node S := if h0 : r.val=0 then .inl d.val.val.fst else if h3 : r.val=3 then .inl d.val.val.snd else .inr (d,⟨r.val-1,by omega⟩) noncomputable def meshTime (r : Fin 4) : unitInterval := ⟨(r.val:ℝ)/3,by constructor · positivity · have hr : (r.val:ℝ)<4 := by exact_mod_cast r.isLt have hr' : r.val≤3 := by omega have hr'' : (r.val:ℝ)≤3 := by exact_mod_cast hr' linarith⟩ lemma mesh_point (d : S) (r : Fin 4) : point D S (meshNode S d r)=D.edge d.val.val.fst d.val.val.snd d.val.val.adj (meshTime r) := by fin_cases r <;> norm_num [meshNode,point,meshTime,tipTime] noncomputable def label : Piece S → ForwardDart G | .inl d => d.val | .inr (d,_) => d.val def src : Piece S → Node S | .inl d => .inl d.val.val.fst | .inr (d,k) => meshNode S d k.castSucc def dst : Piece S → Node S | .inl d => .inl d.val.val.snd | .inr (d,k) => meshNode S d k.succ noncomputable def first : Piece S → unitInterval | .inl _ => 0 | .inr (_,k) => meshTime k.castSucc noncomputable def last : Piece S → unitInterval | .inl _ => 1 | .inr (_,k) => meshTime k.succ lemma first_lt_last (p : Piece S) : (first S p:ℝ)<(last S p:ℝ) := by cases p with | inl p => exact zero_lt_one | inr p => dsimp [first,last,meshTime]; push_cast; linarith lemma src_point (p : Piece S) : point D S (src S p)=D.edge (label S p).val.fst (label S p).val.snd (label S p).val.adj (first S p) := by cases p with | inl p => exact (Path.source _).symm | inr p => exact mesh_point D S p.1 p.2.castSucc lemma dst_point (p : Piece S) : point D S (dst S p)=D.edge (label S p).val.fst (label S p).val.snd (label S p).val.adj (last S p) := by cases p with | inl p => exact (Path.target _).symm | inr p => exact mesh_point D S p.1 p.2.succ lemma src_ne_dst (D : Drawing G) (p : Piece S) : src S p≠dst S p := by intro he have hp := congrArg (point D S) he rw [src_point,dst_point] at hp have ht := congrArg Subtype.val (D.edge_injective _ _ _ hp) exact (ne_of_lt (first_lt_last S p)) ht lemma interior_point_avoid (p : Piece S) {t : unitInterval} (ht₀ : (first S p:ℝ)<(t:ℝ)) (ht₁ : (t:ℝ)<(last S p:ℝ)) (v : Node S) : D.edge (label S p).val.fst (label S p).val.snd (label S p).val.adj t≠point D S v := by have ht : 0<(t:ℝ) := lt_of_le_of_lt (first S p).property.1 ht₀ have ht' : (t:ℝ)<1 := lt_of_lt_of_le ht₁ (last S p).property.2 intro he cases v with | inl v => exact D.interior_avoid _ _ _ t v ht ht' he | inr v => obtain ⟨hd,htt⟩ := D.forward_interior_meet (label S p) v.1.val ht ht' (tipTime_pos v.2) (tipTime_lt_one v.2) he cases p with | inl p => change p.val=v.1.val at hd exact p.property (hd.symm ▸ v.1.property) | inr p => rw [htt] at ht₀ ht₁ change (p.2.val:ℝ)/3 < ((v.2.val:ℝ)+1)/3 at ht₀ change ((v.2.val:ℝ)+1)/3 < ((p.2.val+1:ℕ):ℝ)/3 at ht₁ have hlo : p.2.val cases q with | inl q => exact congrArg Sum.inl (Subtype.ext hd) | inr q => change p.val=q.1.val at hd exact False.elim (p.property (hd.symm ▸ q.1.property)) | inr p => cases q with | inl q => change p.1.val=q.val at hd exact False.elim (q.property (hd ▸ p.1.property)) | inr q => have hk : p.2=q.2 := by apply Fin.ext change (p.2.val:ℝ)/3 < (s:ℝ) at hs₀ change (s:ℝ)<((p.2.val+1:ℕ):ℝ)/3 at hs₁ change (q.2.val:ℝ)/3 < (s:ℝ) at ht₀ change (s:ℝ)<((q.2.val+1:ℕ):ℝ)/3 at ht₁ push_cast at hs₁ ht₁ have hpq : p.2.val