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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 | import OAI.Geometry.IsometricImmersion.Immersions.BoundedHeightClasses
import Mathlib.Topology.Baire.Lemmas
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry
def patchObstructionSet (g0 : MetricField) (kappa : ℝ) : Set (metricPatchSet g0 kappa) :=
⋂ index : ℕ × ℚ, (closure (boundedHeightClass g0 kappa index.1 index.2))ᶜ
theorem patchObstructionSet_isGdelta (g0 : MetricField) (kappa : ℝ) :
IsGδ (patchObstructionSet g0 kappa) :=
IsGδ.iInter_of_isOpen (fun _ => isClosed_closure.isOpen_compl)
theorem mem_patchObstructionSet_iff {g0 : MetricField} {kappa : ℝ}
{eta : metricPatchSet g0 kappa} :
eta ∈ patchObstructionSet g0 kappa ↔
∀ M : ℕ, ∀ q0 : ℚ, eta ∉ closure (boundedHeightClass g0 kappa M q0) := by
simp only [patchObstructionSet, mem_iInter, mem_compl_iff, Prod.forall]
theorem patchObstructionSet_has_no_admissible_height
{g0 : MetricField} {kappa : ℝ} {eta : metricPatchSet g0 kappa}
(heta : eta ∈ patchObstructionSet g0 kappa) :
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z := by
intro z hz
obtain ⟨M,hM,q0,hq0⟩ := every_admissible_height_has_class hz kappa
have hclass : eta ∈ boundedHeightClass g0 kappa M q0 := ⟨z,hM,hq0⟩
exact (mem_patchObstructionSet_iff.mp heta M q0) (subset_closure hclass)
theorem patchObstructionSet_subset_complement_admissible (g0 : MetricField) (kappa : ℝ) :
patchObstructionSet g0 kappa ⊆ (admissiblePatchMetricSet g0 kappa)ᶜ := by
intro eta heta hbad
obtain ⟨z,hz⟩ := hbad
exact patchObstructionSet_has_no_admissible_height heta z hz
end SmoothLocal.Perturbation
end
end OAI
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