| import OAI.Geometry.Relativity.CKS.ComparatorDefinitions |
| import OAI.Geometry.Relativity.CKS.BoundaryCharts |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Manifold Bundle Filter |
| open scoped ContDiff Topology |
| namespace CKSBoundarySurface |
| attribute [local instance] Classical.propDecidable |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace H3 M] |
| [@IsManifold ℝ _ E3 _ _ H3 |
| (@instTopologicalSpaceEuclideanHalfSpace 3 halfSpaceDimension_neZero) I3 ∞ M _ _] |
| lemma liftHalf_drop_chart (x : M) (y : Boundary M) (hy : y.val ∈ (chartAt H3 x).source) : |
| liftHalf (dropPlane (I3 (chartAt H3 x y.val))) = chartAt H3 x y.val := by |
| apply Subtype.ext |
| exact lift_drop ((boundary_iff_zero x y.val hy).mp y.property) |
|
|
| lemma boundaryInverse_mem (x : Boundary M) {z : E2} |
| (hz : liftHalf z ∈ (chartAt H3 x.val).target) : |
| (chartAt H3 x.val).symm (liftHalf z) ∈ Boundary M := |
| chart_inverse_boundary x.val hz |
|
|
| lemma boundaryInverse_val (x : Boundary M) {z : E2} |
| (hz : liftHalf z ∈ (chartAt H3 x.val).target) : |
| (boundaryInverse x z).val = (chartAt H3 x.val).symm (liftHalf z) := by |
| simp only [boundaryInverse,dite_eq_left hz] |
|
|
| @[simp] lemma boundaryChart_source (x : Boundary M) : |
| (boundaryChart x).source = Subtype.val ⁻¹' (chartAt H3 x.val).source := rfl |
| @[simp] lemma boundaryChart_target (x : Boundary M) : |
| (boundaryChart x).target = liftHalf ⁻¹' (chartAt H3 x.val).target := rfl |
| @[simp] lemma boundaryChart_apply (x y : Boundary M) : |
| boundaryChart x y = dropPlane (I3 (chartAt H3 x.val y.val)) := rfl |
| lemma boundaryChart_symm_val (x : Boundary M) {z : E2} (hz : z ∈ (boundaryChart x).target) : |
| ((boundaryChart x).symm z).val = (chartAt H3 x.val).symm (liftHalf z) := |
| boundaryInverse_val x hz |
|
|
| end CKSBoundarySurface |
|
|
| end |
|
|
| end OAI |
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|
