openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /AssemblyInnerSymbol.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.InheritedPolicyState | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.AssemblyDescriptorList | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveUnit | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.TensorCapCompact | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.TreeND | |
| open Set MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff Classical | |
| variable {n : ℕ} | |
| noncomputable def assemblyInnerSymbol (e : RecursiveEnvironmentND n) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) : | |
| EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ := | |
| fun x w => f x*(actualAngularProduct (policyDescriptorList e.profile e.N e.q c e.N) | |
| id L e.ρ e.mass e.N e.sig e.C e.J 0 x).1 w | |
| noncomputable def assemblyOuterFactor (e : RecursiveEnvironmentND n) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (P : CellPath (n+1)) : EuclideanSpace ℝ (Fin (n+1)) → ℂ := | |
| fun w => Q.1 w*(actualEndpointAngular (L (.parallel 0 c)) e.ρ e.mass e.N e.sig e.C e.J 0 | |
| (.parallel 0 c) (treeCenter P)).1 w | |
| lemma assemblySymbol_factor (e : RecursiveEnvironmentND n) (hq : 0<e.q) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (P : CellPath (n+1)) (hPc : c≤P.length) {x : EuclideanSpace ℝ (Fin (n+1))} (hx : x∈treeCell P) | |
| (w : EuclideanSpace ℝ (Fin (n+1))) : | |
| assemblySymbol e c L f Q x w=assemblyOuterFactor e c L Q P w*assemblyInnerSymbol e c L f x w := by | |
| have hn : PolicyDescriptor.parallel 0 c∉policyDescriptorList e.profile e.N e.q c e.N := by | |
| intro h | |
| have hh := policyDescriptor_anchor_gt_start e.profile hq h | |
| simp only [descriptorAnchor] at hh | |
| omega | |
| have hc := actualEndpointAngular_reached (L (.parallel 0 c)) e.ρ e.mass e.N e.sig e.C e.J 0 | |
| (.parallel 0 c) (show descriptorAnchor (.parallel 0 c)≤P.length by simpa only [descriptorAnchor] using hPc) hx (treeCenter_mem P) | |
| change Q.1 w*f x*(∏ b∈insert (.parallel 0 c) (policyDescriptorList e.profile e.N e.q c e.N), | |
| (actualEndpointAngular (L b) e.ρ e.mass e.N e.sig e.C e.J 0 b x).1 w)=_ | |
| rw [Finset.prod_insert hn,hc] | |
| simp only [assemblyOuterFactor,assemblyInnerSymbol,actualAngularProduct,smoothFinsetProductND,id_eq] | |
| ring | |
| lemma assemblyTransform_factor (e : RecursiveEnvironmentND n) (hq : 0<e.q) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (P : CellPath (n+1)) (hPc : c≤P.length) (M r : ℝ) (w : EuclideanSpace ℝ (Fin (n+1))) : | |
| _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell P) M (assemblySymbol e c L f Q) r w= | |
| assemblyOuterFactor e c L Q P w* | |
| _root_.OAI.CurrentFalconer.sourceAngularTransformND e.ρ (treeCell P) M (assemblyInnerSymbol e c L f) r w := by | |
| unfold _root_.OAI.CurrentFalconer.sourceAngularTransformND | |
| rw [show (∫ x in treeCell P, _root_.OAI.CurrentFalconer.spherePlaneWaveND r x w*assemblySymbol e c L f Q x w ∂e.ρ)= | |
| ∫ x in treeCell P, assemblyOuterFactor e c L Q P w* | |
| (_root_.OAI.CurrentFalconer.spherePlaneWaveND r x w*assemblyInnerSymbol e c L f x w) ∂e.ρ by | |
| apply setIntegral_congr_fun (measurableSet_treeCell P) | |
| intro x hx | |
| dsimp only | |
| rw [assemblySymbol_factor e hq c L f Q P hPc hx w] | |
| ring] | |
| rw [integral_const_mul] | |
| exact (mul_smul_comm _ _ _).symm | |
| lemma assemblyInnerSymbol_initial (e : RecursiveEnvironmentND n) (hq : 0<e.q) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (P Q : CellPath (n+1)) : | |
| assemblyInnerSymbol e c L f=(e.angularData c e.N c e.N 0 P Q).symbol L f recursiveUnit c := by | |
| ext x w | |
| simp only [RecursiveAngularND.symbol,RecursiveEnvironmentND.angularData,RecursiveAngularND.profile, | |
| actualResidualSourceSymbol,unreachedDescriptorList_initial _ _ _ _ _ hq, | |
| recursiveUnit,one_mul,assemblyInnerSymbol] | |
| lemma assemblyInnerSymbol_measurable (e : RecursiveEnvironmentND n) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (hf : Measurable f) : | |
| Measurable (Function.uncurry (assemblyInnerSymbol e c L f)) := | |
| (hf.comp measurable_fst).mul (actualAngularProduct_measurable _ _ _ _ _ _ _ _ _ _) | |
| lemma assemblyInnerSymbol_bounded (e : RecursiveEnvironmentND n) (c : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (hf : ∀ x,‖f x‖≤1) : | |
| ∃ H : ℝ,0≤H ∧ ∀ x w,‖assemblyInnerSymbol e c L f x w‖≤H := by | |
| obtain ⟨H,hH,h⟩ := actualAngularProduct_global_bound | |
| (policyDescriptorList e.profile e.N e.q c e.N) id L e.ρ e.mass e.N e.sig e.C e.J 0 | |
| refine ⟨H,hH,?_⟩ | |
| intro x w | |
| change ‖f x*(actualAngularProduct _ _ _ _ _ _ _ _ _ _ x).1 w‖≤H | |
| rw [norm_mul] | |
| exact (mul_le_mul_of_nonneg_right (hf x) (norm_nonneg _)).trans (by simpa only [one_mul] using h x w) | |
| lemma uniform_assemblyOuterFactor_bound | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (B : ℕ) (hW : ∀ L∈W,L.length≤B) {sig : ℝ} (hsig : 0<sig) : | |
| ∃ D : ℝ,1≤D ∧ ∀ᶠ N : ℕ in atTop,∀ e : RecursiveEnvironmentND n,e.N=N → e.sig=sig → | |
| ∀ c L (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) (Cq : ℝ), | |
| L (.parallel 0 c)∈W → (∀ w,‖w‖=1 → ‖Q.1 w‖≤Cq) → | |
| ∀ P w,‖w‖=1 → ‖assemblyOuterFactor e c L Q P w‖≤Cq*D^B := by | |
| obtain ⟨D,hD,h⟩ := uniform_actualAngularProduct_derivatives (ι:=PolicyDescriptor) W 0 B B 0 hW 1 sig hsig | |
| refine ⟨D,hD,?_⟩ | |
| filter_upwards [h] with N hN | |
| intro e he hs c L Q Cq hL hQ P w hw | |
| subst N | |
| have hh := hN {PolicyDescriptor.parallel 0 c} id L e.ρ e.mass e.C e.J 0 c | |
| (by simp) (by simpa using hL) (by simpa using hW _ hL) | |
| (by simp [descriptorLength]) (treeCenter P) 0 (by omega) w hw | |
| simp only [norm_iteratedFDeriv_zero,pow_zero,mul_one] at hh | |
| have hb : ‖(actualEndpointAngular (L (.parallel 0 c)) e.ρ e.mass e.N e.sig e.C e.J 0 | |
| (.parallel 0 c) (treeCenter P)).1 w‖≤D^B := by | |
| simpa only [actualAngularProduct,smoothFinsetProductND,Finset.prod_singleton,id_eq,hs] using hh | |
| exact (norm_mul _ _).trans_le (mul_le_mul (hQ w hw) hb (norm_nonneg _) ((norm_nonneg _).trans (hQ w hw))) | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace CurrentFalconer.AssemblyEstimate | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate | |
| variable {V : Type*} [MeasurableSpace V] | |
| open Attempt4Near | |
| lemma sourceAngularHND_two_profile_power (d : ℕ) (μ : Measure V) [IsFiniteMeasure μ] | |
| (a v : ℝ) {U W : ℝ × V → ℂ} (hU : Measurable U) (hW : Measurable W) | |
| {C E : ℝ} (hbU : ∀ r,∀ᵐ w ∂μ,‖U (r,w)‖≤C) | |
| (hbW : ∀ r,∀ᵐ w ∂μ,‖W (r,w)‖≤E) | |
| {Ψ : ℝ → ℝ} (hiΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) | |
| (hΨ : ∀ r,0≤Ψ r) {R g₁ g₂ : ℝ} (hR : 0<R) | |
| (h₁ : sourceAngularHND d μ a v Ψ U U≤R^g₁) | |
| (h₂ : sourceAngularHND d μ a v Ψ W W≤R^g₂) : | |
| sourceAngularHND d μ a v Ψ U W≤R^((g₁+g₂)/2) := by | |
| let t := R^((g₂-g₁)/2) | |
| have ht : 0<t := Real.rpow_pos_of_pos hR _ | |
| have hi₁ := graph_weighted_moments_integrableND μ hU hU hbU hbU hiΨ | |
| have hi₂ := graph_weighted_moments_integrableND μ hW hW hbW hbW hiΨ | |
| have hpoint (A B : ℝ) : A*B≤(t*A*A+t⁻¹*B*B)/2 := by | |
| have hs := sq_nonneg (t*A-B) | |
| have hti : t*t⁻¹=1 := mul_inv_cancel₀ ht.ne' | |
| have he : t*(t*A*A+t⁻¹*B*B-2*(A*B))=(t*A-B)^2 := by | |
| calc | |
| _ = t^2*A^2+(t*t⁻¹)*B^2-2*t*A*B := by ring | |
| _ = _ := by rw [hti]; ring | |
| have hh : 0≤t*A*A+t⁻¹*B*B-2*(A*B) := | |
| (mul_nonneg_iff_of_pos_left ht).mp (by rw [he]; exact hs) | |
| linarith | |
| have hh : (∫ r, Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r) ≤ | |
| (t*(∫ r, Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ U r*sourceAngularMomentND μ U r)+ | |
| t⁻¹*(∫ r, Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ W r*sourceAngularMomentND μ W r))/2 := by | |
| rw [← integral_const_mul,← integral_const_mul,← integral_add (hi₁.const_mul t) (hi₂.const_mul t⁻¹), | |
| ← integral_div] | |
| apply integral_mono_of_nonneg | |
| (Filter.Eventually.of_forall fun r => mul_nonneg | |
| (mul_nonneg (hΨ _) (sourceAngularMomentND_nonneg _ _ _)) (sourceAngularMomentND_nonneg _ _ _)) | |
| (((hi₁.const_mul t).add (hi₂.const_mul t⁻¹)).div_const 2) | |
| filter_upwards with r | |
| have h := mul_le_mul_of_nonneg_left | |
| (hpoint (sourceAngularMomentND μ U r) (sourceAngularMomentND μ W r)) (hΨ (r/(2:ℝ)^v)) | |
| simp only [Pi.add_apply] | |
| convert h using 1 <;> ring | |
| have hen : sourceAngularHND d μ a v Ψ U W ≤ | |
| (t*sourceAngularHND d μ a v Ψ U U+t⁻¹*sourceAngularHND d μ a v Ψ W W)/2 := by | |
| have h := mul_le_mul_of_nonneg_left hh | |
| (show 0≤(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹ by positivity) | |
| unfold sourceAngularHND | |
| nlinarith only [h] | |
| have hr₁ : t*R^g₁=R^((g₁+g₂)/2) := by | |
| rw [← Real.rpow_add hR]; congr 1; ring | |
| have hr₂ : t⁻¹*R^g₂=R^((g₁+g₂)/2) := by | |
| dsimp only [t] | |
| rw [← Real.rpow_neg hR.le,← Real.rpow_add hR]; congr 1; ring | |
| calc | |
| _ ≤ (t*sourceAngularHND d μ a v Ψ U U+t⁻¹*sourceAngularHND d μ a v Ψ W W)/2 := hen | |
| _ ≤ (t*R^g₁+t⁻¹*R^g₂)/2 := by gcongr | |
| _ = _ := by rw [hr₁,hr₂]; ring | |
| end CurrentFalconer.AssemblyEstimate | |
| end | |
| end | |
| end OAI | |