| import Mathlib | |
| import OAI.Geometry.SmoothYau.Estimates.RealPhaseLinear | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open MvPolynomial | |
| open Set Filter | |
| open scoped ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open Set Filter Matrix | |
| open scoped Topology Matrix Matrix.Norms.Elementwise | |
| namespace YauCounterexamples | |
| variable {X : Type*} [TopologicalSpace X] | |
| lemma isClosed_phaseMatrixValid (H : X → RealForm PhaseSpace) (z : X → Fin 3 → ℂ) | |
| (hH : Continuous (fun x => phaseFormMatrix (H x))) (hz : Continuous z) | |
| (κ C : ℝ) : | |
| IsClosed {p : X × ComplexPhaseMatrix | PhaseMatrixValid (H p.1) (z p.1) κ C p.2} := by | |
| have hs : IsClosed {p : X × ComplexPhaseMatrix | p.2ᵀ = p.2} := | |
| isClosed_eq (by fun_prop) continuous_snd | |
| have ha : IsClosed {p : X × ComplexPhaseMatrix | p.2 *ᵥ z p.1 = 0} := | |
| isClosed_eq (by fun_prop) continuous_const | |
| have hb : IsClosed {p : X × ComplexPhaseMatrix | ‖p.2‖ ≤ C} := | |
| isClosed_le continuous_snd.norm continuous_const | |
| have hg : IsClosed {p : X × ComplexPhaseMatrix | ∀ v, | |
| 4*κ*squareSum v ≤ realQuadratic (phaseFormMatrix (H p.1)-p.2.map Complex.re) v} := by | |
| simp only [ofPred_forall] | |
| apply isClosed_iInter | |
| intro v | |
| apply isClosed_le continuous_const | |
| unfold realQuadratic | |
| have hM : Continuous (fun p : X × ComplexPhaseMatrix => phaseFormMatrix (H p.1)) := | |
| hH.comp continuous_fst | |
| fun_prop | |
| exact hs.inter (ha.inter (hb.inter hg)) | |
| def boundedPhaseSet (H : X → RealForm PhaseSpace) (B κ C : ℝ) : | |
| Set ((X × (Fin 3 → ℂ)) × ComplexPhaseMatrix) := | |
| {p | ‖p.1.2‖ ≤ B ∧ (∑ i, p.1.2 i * p.1.2 i = -1) ∧ | |
| PhaseMatrixValid (H p.1.1) p.1.2 κ C p.2} | |
| theorem boundedPhaseSet_isCompact [CompactSpace X] | |
| (H : X → RealForm PhaseSpace) (hH : Continuous (fun x => phaseFormMatrix (H x))) | |
| (B κ C : ℝ) : IsCompact (boundedPhaseSet H B κ C) := by | |
| have hz : IsClosed {p : ((X × (Fin 3 → ℂ)) × ComplexPhaseMatrix) | ‖p.1.2‖ ≤ B} := | |
| isClosed_le (continuous_snd.comp continuous_fst).norm continuous_const | |
| have hn : IsClosed {p : ((X × (Fin 3 → ℂ)) × ComplexPhaseMatrix) | | |
| ∑ i, p.1.2 i * p.1.2 i = -1} := isClosed_eq (by fun_prop) continuous_const | |
| have hv := isClosed_phaseMatrixValid (fun p : X × (Fin 3 → ℂ) => H p.1) | |
| (fun p => p.2) (hH.comp continuous_fst) continuous_snd κ C | |
| have hclosed : IsClosed (boundedPhaseSet H B κ C) := hz.inter (hn.inter hv) | |
| apply ((isCompact_univ.prod (isCompact_closedBall (0 : Fin 3 → ℂ) B)).prod | |
| (isCompact_closedBall (0 : ComplexPhaseMatrix) C)).of_isClosed_subset hclosed | |
| intro p hp | |
| exact ⟨⟨mem_univ _, by simpa only [Metric.mem_closedBall, dist_zero_right] using hp.1⟩, | |
| by simpa only [Metric.mem_closedBall, dist_zero_right] using hp.2.2.2.2.1⟩ | |
| end YauCounterexamples | |
| end | |
| end OAI | |