File size: 4,003 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Region
namespace OAI
section
section
section
noncomputable section
open Set unitInterval Topology
noncomputable section
noncomputable section
noncomputable section
open Set Topology unitInterval
noncomputable section
open Set Metric Topology unitInterval
noncomputable section
open Set Metric Topology
noncomputable section
noncomputable section
open Set unitInterval
noncomputable section
noncomputable section
open Set Metric Topology
noncomputable section
open Set Metric Topology
noncomputable section
open Set Metric Topology
namespace CannonFaceIncidence
open Set Topology
variable {X D : Type*} [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D]
variable {Y : Type*} [TopologicalSpace Y]
def chartLabel (e : Y → X) {V : Set X} (π : V → D) : (e ⁻¹' V) → D :=
fun x => π ⟨e x.val,x.property⟩
omit [DiscreteTopology D] in
lemma chartLabel_continuous (e : Y → X) (he : Continuous e) {V : Set X}
{π : V → D} (hπ : Continuous π) : Continuous (chartLabel e π) :=
hπ.comp ((he.comp continuous_subtype_val).subtype_mk _)
omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] [TopologicalSpace Y] in
lemma region_chartLabel (e : Y → X) {V : Set X} (π : V → D) (d : D) :
region _ (chartLabel e π) d=e ⁻¹' region V π d := by
ext x
rfl
omit [TopologicalSpace D] [DiscreteTopology D] in
lemma incident_chartLabel (e : Y → X) (he : IsOpenMap e) (hc : Continuous e)
{V : Set X} (π : V → D) (x : Y) :
incident _ (chartLabel e π) x=incident V π (e x) := by
ext d
change x∈closure (region _ (chartLabel e π) d) ↔ e x∈closure (region V π d)
rw [region_chartLabel]
rw [← he.preimage_closure_eq_closure_preimage hc (region V π d)]
rfl
omit [TopologicalSpace Y] in
lemma chartLabel_surjective (e : Y → X) (he : DenseRange e) {V : Set X}
(hV : IsOpen V) {π : V → D} (hπ : Continuous π) (hsur : Function.Surjective π) :
Function.Surjective (chartLabel e π) := by
intro d
obtain ⟨x,hx⟩ := hsur d
obtain ⟨y,hy,hyd⟩ := he.exists_mem_open (region_isOpen hV hπ d) ⟨x.val,x.property,hx⟩
exact ⟨⟨y,hy⟩,hyd⟩
end CannonFaceIncidence
namespace CannonFaceIncidence
open Set Topology
variable {X Y D : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[TopologicalSpace D] [DiscreteTopology D]
def labelPull (e : Y → X) {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V)
(π : V → D) : W → D := chartLabel e π ∘ (Homeomorph.setCongr hW)
omit [DiscreteTopology D] in
lemma labelPull_continuous (e : Y → X) (he : Continuous e)
{V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) {π : V → D} (hπ : Continuous π) :
Continuous (labelPull e hW π) :=
(chartLabel_continuous e he hπ).comp (Homeomorph.setCongr hW).continuous
omit [TopologicalSpace X] [TopologicalSpace D] [DiscreteTopology D] in
lemma region_labelPull (e : Y → X) {V : Set X} {W : Set Y} (hW : W=e ⁻¹' V)
(π : V → D) (d : D) : region W (labelPull e hW π) d=e ⁻¹' region V π d := by
subst W
exact region_chartLabel e π d
omit [TopologicalSpace D] [DiscreteTopology D] in
lemma incident_labelPull (e : Y → X) (he : IsOpenMap e) (hc : Continuous e)
{V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) (π : V → D) (x : Y) :
incident W (labelPull e hW π) x=incident V π (e x) := by
ext d
change x∈closure (region W (labelPull e hW π) d) ↔ e x∈closure (region V π d)
rw [region_labelPull, ← he.preimage_closure_eq_closure_preimage hc (region V π d)]
rfl
lemma labelPull_surjective (e : Y → X) (he : DenseRange e)
{V : Set X} {W : Set Y} (hW : W=e ⁻¹' V) (hV : IsOpen V)
{π : V → D} (hπ : Continuous π) (hsur : Function.Surjective π) :
Function.Surjective (labelPull e hW π) :=
(chartLabel_surjective e he hV hπ hsur).comp (Homeomorph.setCongr hW).surjective
end CannonFaceIncidence
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
|