| 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 | |