openAI-math-dataset / lean /OAI /Geometry /NodalSets /SmoothLimit /SphereFiniteWeakJetIteration.lean
| import OAI.Geometry.NodalSets.Charts.SphereWeakJetSuccessor | |
| import OAI.Geometry.NodalSets.Elliptic.RealPositiveNestedRadii | |
| namespace OAI | |
| namespace Yau.Target | |
| open MeasureTheory Set Yau.Geometry | |
| open scoped ContDiff | |
| noncomputable section | |
| theorem sphere_finite_weak_jet_iteration (d : SphereEnergyData) (p : Base) | |
| (B : Yau.Jets.Coord → ℝ) (hB : ContDiff ℝ ∞ B) (m : ℕ) : | |
| ∃ K > 0, ∀ U : List (Fin 4) → Yau.Jets.Coord → ℝ, | |
| let Q := Yau.realCenteredCube 4 (1/8) | |
| (∀ es, es.length ≤ 4 → MemLp (U es) 2 (volume.restrict Q)) → | |
| (∀ es, es.length ≤ 3 → ∀ i psi, | |
| ContDiff ℝ ∞ psi → HasCompactSupport psi → tsupport psi ⊆ Q → | |
| (∫ x in Q, U es x*Yau.coordPartial psi x i)=-(∫ x in Q, U (i::es) x*psi x)) → | |
| (∀ psi, ContDiff ℝ ∞ psi → HasCompactSupport psi → tsupport psi ⊆ Q → | |
| (∑ a, ∑ j, ∫ x in Q, sphereChartPrincipalDensity d p x a j*U [a] x*Yau.coordPartial psi x j) = | |
| ∫ x in Q, B x*U [] x*psi x) → | |
| ∀ E : ℝ, 0 ≤ E → (∀ es, es.length ≤ 4 → (∫ x in Q, (U es x)^2) ≤ E) → | |
| ∃ V : List (Fin 4) → Yau.Jets.Coord → ℝ, | |
| let q := Yau.realCenteredCube 4 (Yau.realWeakJetRadius m) | |
| (∀ es, es.length ≤ 4 → V es=U es) ∧ | |
| (∀ es, es.length ≤ m+4 → MemLp (V es) 2 (volume.restrict q) ∧ | |
| (∫ x in q, (V es x)^2) ≤ K*E) ∧ | |
| (∀ es, es.length ≤ m+3 → ∀ i psi, | |
| ContDiff ℝ ∞ psi → HasCompactSupport psi → tsupport psi ⊆ q → | |
| (∫ x in q, V es x*Yau.coordPartial psi x i)=-(∫ x in q, V (i::es) x*psi x)) ∧ | |
| ∀ psi, ContDiff ℝ ∞ psi → HasCompactSupport psi → tsupport psi ⊆ q → | |
| (∑ a, ∑ j, ∫ x in q, sphereChartPrincipalDensity d p x a j*V [a] x*Yau.coordPartial psi x j) = | |
| ∫ x in q, B x*V [] x*psi x := by | |
| induction m with | |
| | zero => | |
| refine ⟨1,by norm_num,?_⟩ | |
| intro U | |
| dsimp only | |
| intro hU hw he E hE hUE | |
| refine ⟨U,fun _ _ ↦ rfl,?_,?_,?_⟩ | |
| · simpa only [Yau.realWeakJetRadius_zero,Nat.zero_add,one_mul] using | |
| (show ∀ es, es.length ≤ 4 → MemLp (U es) 2 (volume.restrict (Yau.realCenteredCube 4 (1/8))) ∧ | |
| (∫ x in Yau.realCenteredCube 4 (1/8), (U es x)^2) ≤ E from fun es hh ↦ ⟨hU es hh,hUE es hh⟩) | |
| · simpa only [Yau.realWeakJetRadius_zero,Nat.zero_add] using hw | |
| · simpa only [Yau.realWeakJetRadius_zero] using he | |
| | succ m ih => | |
| obtain ⟨C,hC,hbase⟩ := ih | |
| obtain ⟨D,hD,hstep⟩ := sphere_weak_jet_successor d p B hB | |
| (Yau.realWeakJetRadius m) (Yau.realWeakJetRadius (m+1)) (Yau.realWeakJetRadius_succ_lt m) | |
| ((Yau.realWeakJetRadius_upper m).trans (by norm_num)) (m+3) | |
| refine ⟨D*C,mul_pos hD hC,?_⟩ | |
| intro U | |
| dsimp only | |
| intro hU hw he E hE hUE | |
| obtain ⟨V,hlo,hV,hvw,hve⟩ := hbase U hU hw he E hE hUE | |
| obtain ⟨W,hWlo,hW,hww,hwe⟩ := hstep V (fun es hh ↦ (hV es (by omega)).1) | |
| hvw hve (C*E) (mul_nonneg hC.le hE) (fun es hh ↦ (hV es (by omega)).2) | |
| refine ⟨W,fun es hh ↦ (hWlo es (by omega)).trans (hlo es hh),?_,?_,hwe⟩ | |
| · intro es hh | |
| have h := hW es (by omega) | |
| exact ⟨h.1,by simpa only [mul_assoc] using h.2⟩ | |
| · intro es hh i psi hp hc hs | |
| exact (hww es (by omega) i psi hp hc hs).2.2 | |
| end | |
| end Yau.Target | |
| end OAI | |