| import Mathlib |
| import OAI.Geometry.Cannon.Energy.SquareMap |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set Metric Topology |
| open Set Topology unitInterval |
| open Set Filter Topology |
| open Set MeasureTheory |
| open scoped BigOperators |
| open Set unitInterval |
|
|
| namespace CannonCharts |
|
|
| lemma squareMap_range : range squareMap = Icc (1 : ℝ) 2 ×ˢ Icc (-6 : ℝ) 6 := by |
| ext z |
| constructor |
| · rintro ⟨w,rfl⟩ |
| change 1+(w.1:ℝ) ∈ Icc 1 2 ∧ 12*(w.2:ℝ)-6 ∈ Icc (-6) 6 |
| constructor <;> constructor <;> linarith [w.1.property.1,w.1.property.2,w.2.property.1,w.2.property.2] |
| · rintro ⟨hx,hy⟩ |
| let u : unitInterval := ⟨z.1-1,by constructor <;> linarith [hx.1,hx.2]⟩ |
| let v : unitInterval := ⟨(z.2+6)/12,by constructor <;> linarith [hy.1,hy.2]⟩ |
| refine ⟨(u,v),?_⟩ |
| dsimp [squareMap,u,v] |
| ext <;> dsimp <;> ring |
|
|
| lemma squareMap_interior |
| (hx : (z.1:ℝ) ∈ Ioo (0:ℝ) 1) (hy : (z.2:ℝ) ∈ Ioo (0:ℝ) 1) : |
| squareMap z ∈ interior (range squareMap) := by |
| rw [squareMap_range, interior_prod_eq, interior_Icc, interior_Icc] |
| change 1+(z.1:ℝ) ∈ Ioo 1 2 ∧ 12*(z.2:ℝ)-6 ∈ Ioo (-6) 6 |
| constructor <;> constructor <;> linarith [hx.1,hx.2,hy.1,hy.2] |
|
|
| lemma chart_square_interior [TopologicalSpace X] (e : Plane → X) |
| (he : IsOpenEmbedding e) |
| (hx : (z.1:ℝ) ∈ Ioo (0:ℝ) 1) (hy : (z.2:ℝ) ∈ Ioo (0:ℝ) 1) : |
| e (squareMap z) ∈ interior (range (e ∘ squareMap)) := by |
| apply mem_interior.mpr |
| refine ⟨e '' interior (range squareMap),?_,he.isOpenMap _ isOpen_interior,?_⟩ |
| · rintro y ⟨w,hw,rfl⟩ |
| obtain ⟨v,rfl⟩ := interior_subset hw |
| exact ⟨v,rfl⟩ |
| · exact mem_image_of_mem e (squareMap_interior hx hy) |
|
|
| end CannonCharts |
|
|
| open Set MeasureTheory |
| open scoped NNReal ENNReal |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
