| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonTriangleCuts |
| open Finset |
|
|
| variable {V F : Type*} [Fintype V] [DecidableEq V] [Fintype F] [DecidableEq F] |
|
|
| def demand (out : Finset V) (v : V) : ℕ := if v∈out then 1 else 6 |
| omit [Fintype V] in |
| lemma demand_pos (out : Finset V) (v : V) : 0<demand out v := by |
| unfold demand |
| split_ifs <;> omega |
| omit [Fintype V] in |
| lemma demand_sum (out T : Finset V) : |
| (∑ v∈T,demand out v)+5*(T.filter fun v=>v∈out).card=6*T.card := by |
| have h (v : V) : demand out v+5*(if v∈out then 1 else 0)=6 := by |
| unfold demand |
| split_ifs <;> omega |
| calc |
| _=∑ v∈T,(demand out v+5*(if v∈out then 1 else 0)) := by |
| rw [sum_add_distrib,←mul_sum] |
| simp [Finset.filter_mem_eq_inter] |
| _=6*T.card := by simp_rw [h]; simp [mul_comm] |
|
|
| theorem disk_demands (corners : F → Finset V) (out : F) |
| (hout : (corners out).card=3) |
| (hEuler : Fintype.card F+4=2*Fintype.card V) |
| (hbound : ∀ T : Finset V,T≠Finset.univ → |
| (∃ f,corners f⊆T) → (Finset.univ.filter fun f=>corners f⊆T).card+5≤2*T.card) : |
| (∑ v,demand (corners out) v)=3*Fintype.card {f : F |
| ∀ S : Finset V,S.Nonempty → S≠Finset.univ → |
| (∑ v∈S,demand (corners out) v)< |
| 3*(Finset.univ.filter fun f : {f : F // f≠out}=>∃ v∈S,v∈corners f.val).card := by |
| classical |
| let A := {f : F |
| have hAc : Fintype.card A+1=Fintype.card F := by |
| have he : Fintype.card A=(Finset.univ.erase out : Finset F).card := |
| Fintype.card_of_subtype _ (fun _=>by simp) |
| rw [he,card_erase_add_one (mem_univ out),card_univ] |
| have hsum := demand_sum (corners out) (Finset.univ : Finset V) |
| have hfilter : (Finset.univ.filter fun v=>v∈corners out)=corners out := by ext v; simp |
| rw [hfilter,hout,card_univ] at hsum |
| have htotal : (∑ v,demand (corners out) v)=3*Fintype.card A := by omega |
| refine ⟨htotal,?_⟩ |
| intro S hS hSp |
| let T := Sᶜ |
| let B : Finset F := Finset.univ.filter (fun f=>corners f⊆T) |
| let B' : Finset A := Finset.univ.filter (fun f=>corners f.val⊆T) |
| let N : Finset A := Finset.univ.filter (fun f=>∃ v∈S,v∈corners f.val) |
| have hTp : T≠Finset.univ := by |
| obtain ⟨v,hv⟩ := hS |
| intro he |
| have ht : v∈T := he ▸ mem_univ v |
| exact (mem_compl.mp ht) hv |
| have hT : T.Nonempty := by |
| by_contra hn |
| have he : T=∅ := not_nonempty_iff_eq_empty.mp hn |
| apply hSp |
| ext v |
| simp only [mem_univ,iff_true] |
| by_contra hv |
| have ht : v∈T := mem_compl.mpr hv |
| simp only [he,notMem_empty] at ht |
| have hBN : B'.card+N.card=Fintype.card A := by |
| have he : N=Finset.univ.filter (fun f : A=>¬corners f.val⊆T) := by |
| ext f |
| simp only [N,mem_filter,mem_univ,true_and,not_subset] |
| constructor |
| · rintro ⟨v,hvS,hvf⟩ |
| exact ⟨v,hvf,fun hvT=>(mem_compl.mp hvT) hvS⟩ |
| · rintro ⟨v,hvf,hvT⟩ |
| exact ⟨v,by simpa only [T,mem_compl,not_not] using hvT,hvf⟩ |
| rw [he] |
| exact (card_filter_add_card_filter_not (s:=Finset.univ) (fun f : A=>corners f.val⊆T)).trans (card_univ) |
| have himage : B'.image (fun f : A=>f.val)=B.erase out := by |
| ext f |
| simp only [mem_image,mem_filter,mem_univ,true_and,mem_erase,B',B] |
| constructor |
| · rintro ⟨g,hg,rfl⟩ |
| exact ⟨g.property,hg⟩ |
| · rintro ⟨hf,hB⟩ |
| exact ⟨⟨f,hf⟩,hB,rfl⟩ |
| have hBcard : B'.card=(B.erase out).card := by |
| rw [←himage,card_image_of_injective _ Subtype.val_injective] |
| have hsplit : (∑ v∈S,demand (corners out) v)+(∑ v∈T,demand (corners out) v)= |
| ∑ v,demand (corners out) v := sum_add_sum_compl S _ |
| have hstrict : 3*B'.card<(∑ v∈T,demand (corners out) v) := by |
| by_cases hb : B.Nonempty |
| · obtain ⟨f,hf⟩ := hb |
| have hpl := hbound T hTp ⟨f,(mem_filter.mp hf).2⟩ |
| change B.card+5≤2*T.card at hpl |
| have hd := demand_sum (corners out) T |
| by_cases ho : corners out⊆T |
| · have hB : out∈B := mem_filter.mpr ⟨mem_univ _,ho⟩ |
| have hbc : B'.card+1=B.card := by rw [hBcard,card_erase_add_one hB] |
| have hfi : T.filter (fun v=>v∈corners out)=corners out := by |
| ext v |
| simp only [mem_filter] |
| exact ⟨fun h=>h.2,fun h=>⟨ho h,h⟩⟩ |
| rw [hfi,hout] at hd |
| omega |
| · have hB : out∉B := by simpa only [B,mem_filter,mem_univ,true_and] using ho |
| have hbc : B'.card=B.card := by rw [hBcard,erase_eq_of_notMem hB] |
| have hsmall : (T.filter fun v=>v∈corners out).card<3 := by |
| rw [←hout] |
| apply card_lt_card |
| refine Finset.ssubset_iff_subset_ne.mpr ⟨fun v hv=>(mem_filter.mp hv).2,?_⟩ |
| intro he |
| apply ho |
| intro v hv |
| exact (mem_filter.mp (he.symm ▸ hv)).1 |
| omega |
| · have hz : B=∅ := not_nonempty_iff_eq_empty.mp hb |
| have hz' : B'.card=0 := by rw [hBcard,hz,erase_empty,card_empty] |
| rw [hz',mul_zero] |
| exact sum_pos (fun v _=>demand_pos _ v) hT |
| change (∑ v∈S,demand (corners out) v)<3*N.card |
| omega |
| end CannonTriangleCuts |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
