import OAI.Combinatorics.PolynomialVacuum.Geometry.PolygonLaminar namespace OAI namespace PolynomialVacuumBridge.Corridor @[simp] theorem mem_polygonSystems (D : Finset Vertex) (f : FaceIndex) (S : Finset Polygon) : S ∈ polygonSystems D f ↔ S ⊆ polygons D ∧ (S : Set Polygon).Pairwise (fun P Q => Disjoint P.vertices Q.vertices) ∧ ∀ P ∈ S, f ∈ P.interiorFaces := by classical simp only [polygonSystems,Finset.mem_filter,Finset.mem_powerset] @[simp] theorem empty_mem_polygonSystems (D : Finset Vertex) (f : FaceIndex) : ∅ ∈ polygonSystems D f := by simp [mem_polygonSystems,Set.Pairwise] theorem Polygon.not_vertices_subset_interior (P : Polygon) : ¬ P.vertices ⊆ P.interior := by intro h obtain ⟨v,hv⟩ := P.vertices_nonempty exact ((P.mem_interior v).mp (h hv)).2 hv theorem Polygon.not_mutually_nested (P Q : Polygon) (h : P.vertices ⊆ Q.interior) : ¬ Q.vertices ⊆ P.interior := by intro h' have hl := P.interior_card_lt Q.interior Q.interior_filled ((mem_polygons _ _).mpr h) have hl' := Q.interior_card_lt P.interior P.interior_filled ((mem_polygons _ _).mpr h') omega theorem polygonSystem_outermost (D : Finset Vertex) (f : FaceIndex) (S : Finset Polygon) (hS : S ∈ polygonSystems D f) (hne : S.Nonempty) : ∃ P ∈ S, ∀ Q ∈ S, Q ≠ P → Q.vertices ⊆ P.interior := by obtain ⟨P,hP,hmax⟩ := S.exists_max_image (fun P => P.interior.card) hne refine ⟨P,hP,?_⟩ intro Q hQ hneq have hs := (mem_polygonSystems D f S).mp hS rcases Q.disjoint_common_face_nested P (hs.2.1 hQ hP hneq) f (hs.2.2 Q hQ) (hs.2.2 P hP) with h | h · exact h · have hl := P.interior_card_lt Q.interior Q.interior_filled ((mem_polygons _ _).mpr h) have hm := hmax Q hQ omega theorem polygonSystem_erase_outermost (D : Finset Vertex) (f : FaceIndex) (S : Finset Polygon) (hS : S ∈ polygonSystems D f) (P : Polygon) (hout : ∀ Q ∈ S, Q ≠ P → Q.vertices ⊆ P.interior) : S.erase P ∈ polygonSystems P.interior f := by classical have hs := (mem_polygonSystems D f S).mp hS apply (mem_polygonSystems _ _ _).mpr refine ⟨?_,?_,?_⟩ · intro Q hQ exact (mem_polygons _ _).mpr (hout Q (Finset.mem_of_mem_erase hQ) (Finset.ne_of_mem_erase hQ)) · intro Q hQ R hR hne exact hs.2.1 (Finset.mem_of_mem_erase hQ) (Finset.mem_of_mem_erase hR) hne · intro Q hQ exact hs.2.2 Q (Finset.mem_of_mem_erase hQ) theorem not_mem_interior_system (P : Polygon) (f : FaceIndex) (S : Finset Polygon) (hS : S ∈ polygonSystems P.interior f) : P ∉ S := by intro hP exact P.not_vertices_subset_interior ((mem_polygons _ _).mp (((mem_polygonSystems _ _ _).mp hS).1 hP)) theorem polygonSystem_insert_outer (D : Finset Vertex) (f : FaceIndex) (P : Polygon) (hD : Filled D) (hP : P ∈ polygons D) (hf : f ∈ P.interiorFaces) (S : Finset Polygon) (hS : S ∈ polygonSystems P.interior f) : insert P S ∈ polygonSystems D f := by classical have hs := (mem_polygonSystems _ _ _).mp hS have hQ (Q : Polygon) (hQ : Q ∈ S) : Q.vertices ⊆ P.interior := (mem_polygons _ _).mp (hs.1 hQ) have hdis (Q : Polygon) (hq : Q ∈ S) : Disjoint P.vertices Q.vertices := by apply Finset.disjoint_left.mpr intro v hvP hvQ exact ((P.mem_interior v).mp (hQ Q hq hvQ)).2 hvP apply (mem_polygonSystems _ _ _).mpr refine ⟨?_,?_,?_⟩ · intro Q hq rcases Finset.mem_insert.mp hq with rfl | hq · exact hP · exact (mem_polygons _ _).mpr ((hQ Q hq).trans (hD P hP)) · intro Q hq R hr hne rcases Finset.mem_insert.mp hq with hQP | hQS · subst Q rcases Finset.mem_insert.mp hr with hRP | hRS · exact (hne hRP.symm).elim · exact hdis R hRS · rcases Finset.mem_insert.mp hr with hRP | hRS · subst R exact (hdis Q hQS).symm · exact hs.2.1 hQS hRS hne · intro Q hq rcases Finset.mem_insert.mp hq with rfl | hq · exact hf · exact hs.2.2 Q hq theorem polygonSystem_insert_unique (f : FaceIndex) (P Q : Polygon) (S T : Finset Polygon) (hS : S ∈ polygonSystems P.interior f) (hT : T ∈ polygonSystems Q.interior f) (he : insert P S = insert Q T) : P = Q := by classical by_contra hn have hp : P ∈ T := by have : P ∈ insert Q T := he ▸ Finset.mem_insert_self P S exact (Finset.mem_insert.mp this).resolve_left hn have hq : Q ∈ S := by have : Q ∈ insert P S := he.symm ▸ Finset.mem_insert_self Q T exact (Finset.mem_insert.mp this).resolve_left (Ne.symm hn) exact P.not_mutually_nested Q ((mem_polygons _ _).mp (((mem_polygonSystems _ _ _).mp hT).1 hp)) ((mem_polygons _ _).mp (((mem_polygonSystems _ _ _).mp hS).1 hq)) noncomputable def outerSystems (P : Polygon) (f : FaceIndex) : Finset (Finset Polygon) := by classical exact if f ∈ P.interiorFaces then (polygonSystems P.interior f).image (insert P) else ∅ theorem polygonSystems_split (D : Finset Vertex) (f : FaceIndex) (hD : Filled D) : polygonSystems D f = insert ∅ ((polygons D).biUnion (fun P => outerSystems P f)) := by classical ext S constructor · intro hS by_cases h0 : S = ∅ · simp [h0] · obtain ⟨P,hP,hout⟩ := polygonSystem_outermost D f S hS (Finset.nonempty_iff_ne_empty.mpr h0) apply Finset.mem_insert_of_mem apply Finset.mem_biUnion.mpr refine ⟨P,((mem_polygonSystems D f S).mp hS).1 hP,?_⟩ rw [outerSystems,ite_eq_left (((mem_polygonSystems D f S).mp hS).2.2 P hP)] exact Finset.mem_image.mpr ⟨S.erase P,polygonSystem_erase_outermost D f S hS P hout, Finset.insert_erase hP⟩ · intro hS rcases Finset.mem_insert.mp hS with rfl | hS · exact empty_mem_polygonSystems D f · obtain ⟨P,hP,hS⟩ := Finset.mem_biUnion.mp hS by_cases hf : f ∈ P.interiorFaces · simp only [outerSystems,ite_eq_left hf,Finset.mem_image] at hS obtain ⟨T,hT,rfl⟩ := hS exact polygonSystem_insert_outer D f P hD hP hf T hT · simp [outerSystems,hf] at hS theorem outerSystems_disjoint (f : FaceIndex) (P Q : Polygon) (hne : P ≠ Q) : Disjoint (outerSystems P f) (outerSystems Q f) := by classical apply Finset.disjoint_left.mpr intro S hP hQ by_cases hf : f ∈ P.interiorFaces · by_cases hf' : f ∈ Q.interiorFaces · simp only [outerSystems,ite_eq_left hf,ite_eq_left hf',Finset.mem_image] at hP hQ obtain ⟨T,hT,hT'⟩ := hP obtain ⟨U,hU,hU'⟩ := hQ exact hne (polygonSystem_insert_unique f P Q T U hT hU (hT'.trans hU'.symm)) · simp [outerSystems,hf'] at hQ · simp [outerSystems,hf] at hP theorem empty_not_mem_outerSystems (P : Polygon) (f : FaceIndex) : ∅ ∉ outerSystems P f := by classical unfold outerSystems split_ifs <;> simp theorem faceNestingMass_recursion (D : Finset Vertex) (f : FaceIndex) (hD : Filled D) : faceNestingMass D f = 1 + ∑ P ∈ polygons D, if f ∈ P.interiorFaces then 2 * criticalActivity ^ P.length * faceNestingMass P.interior f else 0 := by classical unfold faceNestingMass rw [polygonSystems_split D f hD,Finset.sum_insert] · simp only [Finset.prod_empty] congr 1 rw [Finset.sum_biUnion] · apply Finset.sum_congr rfl intro P hP rw [outerSystems] split_ifs with hf · rw [Finset.sum_image] · rw [Finset.mul_sum] apply Finset.sum_congr rfl intro S hS exact Finset.prod_insert (not_mem_interior_system P f S hS) · intro S hS T hT he have hs := not_mem_interior_system P f S hS have ht := not_mem_interior_system P f T hT have h := congrArg (fun U : Finset Polygon => U.erase P) he simpa [hs,ht] using h · simp · intro P hP Q hQ hne exact outerSystems_disjoint f P Q hne · intro hn obtain ⟨P,hP,h0⟩ := Finset.mem_biUnion.mp hn exact empty_not_mem_outerSystems P f h0 end PolynomialVacuumBridge.Corridor end OAI