openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /OrderedAngularAssignmentsSublist.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.CasimirWords | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open scoped ContDiff | |
| variable {V ι α : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fintype ι] [DecidableEq α] | |
| lemma orderedAngularAssignmentsND_sublist (S : Finset α) (L : List (V →L[ℝ] V)) | |
| {k : α → List (V →L[ℝ] V)} (hk : k ∈ orderedAngularAssignmentsND S L) (i : α) : List.Sublist (k i) L := by | |
| induction L generalizing k with | |
| | nil => simp only [orderedAngularAssignmentsND,List.mem_singleton] at hk; subst k; exact .refl _ | |
| | cons A L ih => | |
| obtain ⟨l,hl,hk⟩ := List.mem_flatMap.mp hk | |
| obtain ⟨j,hj,rfl⟩ := List.mem_map.mp hk | |
| by_cases hij : i=j | |
| · subst i | |
| simpa only [Function.update_self] using (ih hl).cons_cons A | |
| · simpa only [Function.update_of_ne hij] using (ih hl).cons A | |
| noncomputable def polynomialAngularSubwordsND (B : OrthonormalBasis ι ℝ V) (p : Polynomial ℂ) : | |
| Finset (List (V →L[ℝ] V)) := by | |
| classical | |
| exact ((casimirPolynomialTermsND B p).flatMap (fun a => a.2.sublists)).toFinset | |
| lemma polynomialLeibnizTermsND_subword (B : OrthonormalBasis ι ℝ V) (p : Polynomial ℂ) | |
| (S : Finset α) {a : ℂ × (α → List (V →L[ℝ] V))} | |
| (ha : a ∈ polynomialLeibnizTermsND B p S) (i : α) : | |
| a.2 i ∈ polynomialAngularSubwordsND B p := by | |
| classical | |
| obtain ⟨b,hb,ha⟩ := List.mem_flatMap.mp ha | |
| obtain ⟨k,hk,rfl⟩ := List.mem_map.mp ha | |
| apply List.mem_toFinset.mpr | |
| exact List.mem_flatMap.mpr ⟨b,hb,List.mem_sublists.mpr (orderedAngularAssignmentsND_sublist S b.2 hk i)⟩ | |
| lemma polynomialAngularSubwordsND_budget (B : OrthonormalBasis ι ℝ V) (p : Polynomial ℂ) | |
| {L : List (V →L[ℝ] V)} (hL : L ∈ polynomialAngularSubwordsND B p) : L.length ≤ 2*p.natDegree := by | |
| classical | |
| obtain ⟨b,hb,hL⟩ := List.mem_flatMap.mp (List.mem_toFinset.mp hL) | |
| exact (List.mem_sublists.mp hL).length_le.trans (casimirPolynomialTermsND_budget B p hb) | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Set Metric | |
| open scoped ContDiff Topology | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| lemma compact_contDiffOn_derivative_bound {f : E → F} {U K : Set E} | |
| (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) (hf : ContDiffOn ℝ ∞ f U) | |
| (m : ℕ) : ∃ C : ℝ, 1 ≤ C ∧ ∀ j ≤ m, ∀ x ∈ K, | |
| ‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := by | |
| have hbound (j : ℕ) : ∃ C : ℝ, ∀ x ∈ K, | |
| ‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := | |
| hK.exists_bound_of_continuousOn | |
| ((hf.continuousOn_iteratedFDerivWithin (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)) hU.uniqueDiffOn).mono hKU) | |
| induction m with | |
| | zero => | |
| obtain ⟨C,hC⟩ := hbound 0 | |
| refine ⟨max 1 C, le_max_left _ _, ?_⟩ | |
| intro j hj x hx | |
| have : j=0 := Nat.eq_zero_of_le_zero hj | |
| subst j | |
| exact (hC x hx).trans (le_max_right _ _) | |
| | succ m ih => | |
| obtain ⟨D,hD,hboundD⟩ := ih | |
| obtain ⟨C,hC⟩ := hbound (m+1) | |
| refine ⟨max D C, hD.trans (le_max_left _ _), ?_⟩ | |
| intro j hj x hx | |
| by_cases hjm : j ≤ m | |
| · exact (hboundD j hjm x hx).trans (le_max_left _ _) | |
| · have : j=m+1 := by omega | |
| subst j | |
| exact (hC x hx).trans (le_max_right _ _) | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |