File size: 2,037 Bytes
8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import Mathlib
import OAI.Geometry.Cannon.Energy.SquareMap
namespace OAI
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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 {z : CannonGrid.Square}
(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 {X : Type*} [TopologicalSpace X] (e : Plane → X)
(he : IsOpenEmbedding e) {z : CannonGrid.Square}
(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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