| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Region |
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|
| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| noncomputable section |
| open Set unitInterval |
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| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology unitInterval |
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| namespace CannonPlanarCorner |
| open Set Topology Metric |
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|
| def positiveHalfAxis : Set (ℝ × ℝ) := {p | 0≤p.1 ∧ p.2=0} |
|
|
| lemma slit_box_connected {r : ℝ} (hr : 0<r) : |
| IsConnected (CannonFaceIncidence.openBox r\positiveHalfAxis) := by |
| let L : Set (ℝ × ℝ) := Ioo (-r) 0 ×ˢ Ioo (-r) r |
| have hL : IsConnected L := (isConnected_Ioo (neg_neg_of_pos hr)).prod |
| (isConnected_Ioo (by linarith : -r<r)) |
| have hm (b : Bool) : (L∩CannonFaceIncidence.halfBox r b).Nonempty := by |
| cases b |
| · refine ⟨(-r/2,-r/2),?_,?_⟩ |
| · change (-r< -r/2 ∧ -r/2<0) ∧ (-r< -r/2 ∧ -r/2<r) |
| constructor <;> constructor <;> linarith |
| · change (-r< -r/2 ∧ -r/2<r) ∧ (-r< -r/2 ∧ -r/2<0) |
| constructor <;> constructor <;> linarith |
| · refine ⟨(-r/2,r/2),?_,?_⟩ |
| · change (-r< -r/2 ∧ -r/2<0) ∧ (-r<r/2 ∧ r/2<r) |
| constructor <;> constructor <;> linarith |
| · change (-r< -r/2 ∧ -r/2<r) ∧ (0<r/2 ∧ r/2<r) |
| constructor <;> constructor <;> linarith |
| have he : CannonFaceIncidence.openBox r\positiveHalfAxis= |
| (L∪CannonFaceIncidence.halfBox r false)∪CannonFaceIncidence.halfBox r true := by |
| ext p |
| simp only [CannonFaceIncidence.openBox,CannonFaceIncidence.halfBox,L,positiveHalfAxis, |
| Bool.false_eq_true,↓reduceIte,mem_sdiff,mem_ofPred_eq,mem_prod,mem_Ioo,mem_union] |
| constructor |
| · rintro ⟨⟨hx,hy⟩,hn⟩ |
| by_cases hpx : p.1<0 |
| · exact Or.inl (Or.inl ⟨⟨hx.1,hpx⟩,hy⟩) |
| · have hpy : p.2≠0 := fun h => hn ⟨le_of_not_gt hpx,h⟩ |
| rcases lt_or_gt_of_ne hpy with hp|hp |
| · exact Or.inl (Or.inr ⟨hx,hy.1,hp⟩) |
| · exact Or.inr ⟨hx,hp,hy.2⟩ |
| · rintro ((⟨hx,hy⟩|⟨hx,hy⟩)|⟨hx,hy⟩) |
| · exact ⟨⟨⟨hx.1,hx.2.trans hr⟩,hy⟩,fun h => hx.2.not_ge h.1⟩ |
| · exact ⟨⟨hx,hy.1,hy.2.trans hr⟩,fun h => hy.2.ne h.2⟩ |
| · exact ⟨⟨hx,(neg_neg_of_pos hr).trans hy.1,hy.2⟩,fun h => hy.1.ne' h.2⟩ |
| rw [he] |
| apply IsConnected.union _ (IsConnected.union (hm false) hL (CannonFaceIncidence.halfBox_connected hr false)) |
| (CannonFaceIncidence.halfBox_connected hr true) |
| obtain ⟨p,hpL,hpT⟩ := hm true |
| exact ⟨p,Or.inl hpL,hpT⟩ |
| |
| lemma straight_ray_chart (x u : ℂ) (hu : u≠0) : |
| ∃ H : (ℝ × ℝ) ≃ₜ ℂ, H (0,0)=x ∧ |
| H '' positiveHalfAxis={z | ∃ s : ℝ, 0 ≤ s ∧ z=x+s•u} := by |
| let H := (Complex.equivRealProdCLM.toHomeomorph.symm.trans (Homeomorph.mulLeft₀ u hu)).trans |
| (Homeomorph.addLeft x) |
| have ht (t : ℝ) : H (t,0)=x+t•u := by |
| change x+u*Complex.equivRealProdCLM.symm (t,0)=x+t•u |
| congr 1 |
| rw [Complex.real_smul] |
| have he : Complex.equivRealProdCLM.symm (t,0)=(t:ℂ) := by apply Complex.ext <;> rfl |
| rw [he,mul_comm] |
| refine ⟨H,by simpa using ht 0,?_⟩ |
| ext z |
| constructor |
| · rintro ⟨⟨s,t⟩,⟨hs,ht₀⟩,rfl⟩ |
| change t=0 at ht₀ |
| subst t |
| exact ⟨s,hs,ht s⟩ |
| · rintro ⟨s,hs,hz⟩ |
| exact ⟨(s,0),⟨hs,rfl⟩,(ht s).trans hz.symm⟩ |
| end CannonPlanarCorner |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
