File size: 2,830 Bytes
59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Coordinates.PatchAddressModel
import OAI.Geometry.IsometricImmersion.Obstructions.SmallPatchObstruction
import OAI.Geometry.IsometricImmersion.Metrics.PatchTensorPositivity
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Geometry
open SmoothLocal.Perturbation
theorem exists_obstructing_patch_tensor (a : PatchAddress) :
∃ eta : SymmetricPerturbation,
eta ∈ patchTensorBudgetNeighborhood 1 a ∧
eta ∈ metricPatchSet (patchAddressInitialMetric a) (patchAddressAmplitude a) ∧
SmoothPositiveOn (perturbedMetric (patchAddressInitialMetric a) eta) patchAddressModelDomain ∧
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric (patchAddressInitialMetric a) eta) z :=
exists_supported_patch_obstruction_in_open (patchAddressInitialMetric_smoothPositive a)
patchAddressModelDomain_isOpen modelSquare_subset_patchAddressModelDomain
(patchAddressAmplitude_pos a) (fun p hp => patchAddressInitialMetric_curvature a hp)
(patchTensorBudgetNeighborhood_isOpen 1 a)
(zero_mem_patchTensorBudgetNeighborhood (by norm_num : (0 : ℝ) < 1) a)
def selectedPatchTensor (a : PatchAddress) : SymmetricPerturbation :=
Classical.choose (exists_obstructing_patch_tensor a)
theorem selectedPatchTensor_spec (a : PatchAddress) :
selectedPatchTensor a ∈ patchTensorBudgetNeighborhood 1 a ∧
selectedPatchTensor a ∈ metricPatchSet (patchAddressInitialMetric a) (patchAddressAmplitude a) ∧
SmoothPositiveOn (perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a))
patchAddressModelDomain ∧
∀ z : Coord → ℝ,
¬ PatchAdmissibleHeight (perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a)) z :=
Classical.choose_spec (exists_obstructing_patch_tensor a)
theorem selectedPatchTensor_small (a : PatchAddress) :
selectedPatchTensor a ∈ patchTensorBudgetNeighborhood 1 a :=
(selectedPatchTensor_spec a).1
theorem selected_local_metric_eq (a : PatchAddress) :
localPatchMetric initialAssemblyMetric selectedPatchTensor a =
perturbedMetric (patchAddressInitialMetric a) (selectedPatchTensor a) := rfl
theorem selectedPatchTensor_local_positive (a : PatchAddress) (q : Coord) (hq : q ∈ modelSquare) :
(localPatchMetric initialAssemblyMetric selectedPatchTensor a q).PosDef := by
rw [selected_local_metric_eq]
exact (selectedPatchTensor_spec a).2.1.1 q hq
theorem selectedPatchTensor_local_no_height (a : PatchAddress) :
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (localPatchMetric initialAssemblyMetric selectedPatchTensor a) z := by
rw [selected_local_metric_eq]
exact (selectedPatchTensor_spec a).2.2.2
end SmoothLocal.Geometry
end
end OAI
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