openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /AssemblyListsCard.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.AssemblyEndpointSmooth | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphSymbolJetConj | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth 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 Topology ComplexConjugate | |
| variable {n : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| lemma assemblyLists_card (e₁ e₂ : RecursiveEnvironmentND n) (hN : 1≤e₁.N) | |
| (heN : e₂.N=e₁.N) (hq₁ : 0<e₁.q) (hq₂ : 0<e₂.q) {c : ℕ} (hc : c≤e₁.N) : | |
| max 1 (((assemblyDescriptorList e₁.profile e₁.N e₁.q c).card+ | |
| (assemblyDescriptorList e₂.profile e₂.N e₂.q c).card:ℕ):ℝ) ≤ 16*(e₁.N:ℝ)^2 := by | |
| have h₁ := validDescriptorList_card hN (assemblyDescriptorList e₁.profile e₁.N e₁.q c) | |
| (fun _ hb => assemblyDescriptorList_valid e₁.profile hq₁ hc hb) | |
| have h₂ := validDescriptorList_card (by omega : 1≤e₂.N) (assemblyDescriptorList e₂.profile e₂.N e₂.q c) | |
| (fun _ hb => assemblyDescriptorList_valid e₂.profile hq₂ (by omega) hb) | |
| rw [heN] at h₂ | |
| have hN' : (1:ℝ)≤e₁.N := by exact_mod_cast hN | |
| apply max_le | |
| · nlinarith only [hN',sq_nonneg ((e₁.N:ℝ)-1)] | |
| · push_cast | |
| rw [heN] | |
| linarith only [h₁,h₂] | |
| lemma assemblyDescriptor_scale (e : RecursiveEnvironmentND n) (hq : 0<e.q) | |
| {c : ℕ} (hc : 2*c≤e.N) {b : PolicyDescriptor} | |
| (hb : b∈assemblyDescriptorList e.profile e.N e.q c) : | |
| 1 ≤ sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ∧ | |
| sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ≤ | |
| Attempt4PrefixShell.sourceMaskScaleND (e.N/2) ((2:ℝ)^e.N) e.sig := by | |
| refine ⟨sourceMaskScaleND_one_le _ _ _,?_⟩ | |
| change sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig ≤ | |
| sourceMaskScaleND (e.N/2) ((2:ℝ)^e.N) e.sig | |
| apply sourceMaskScaleND_mono _ (by positivity) | |
| have h := assemblyDescriptorList_length e.profile hq hc | |
| have hb' := Finset.le_sup (f:=descriptorLength) hb | |
| omega | |
| lemma assembly_pairMask_eq (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ) (z : V×V) (w : V) : | |
| (Attempt4PrefixShell.pairMaskProductND | |
| (assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c) | |
| (assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1 w = | |
| ((∏ b∈assemblyDescriptorList e₁.profile e₁.N e₁.q c, | |
| actualEndpointMask e₁.ρ e₁.mass e₁.N e₁.sig e₁.C e₁.J 0 b z.1 w)* | |
| (∏ b∈assemblyDescriptorList e₂.profile e₂.N e₂.q c, | |
| actualEndpointMask e₂.ρ e₂.mass e₂.N e₂.sig e₂.C e₂.J 0 b z.2 w):ℝ) := by | |
| simp only [Attempt4PrefixShell.pairMaskProductND,Attempt4PrefixShell.smoothFinsetProductND, | |
| Finset.prod_disjSum,Sum.elim_inl,Sum.elim_inr,Attempt4PrefixShell.smoothStarND, | |
| assemblyEndpointSmooth,Complex.star_def,Complex.conj_ofReal,Complex.ofReal_mul,Complex.ofReal_prod] | |
| lemma assembly_endpoint_radial_left (e : RecursiveEnvironmentND n) (b : PolicyDescriptor) : | |
| Measurable (fun z : V×V => actualEndpointMask e.ρ e.mass e.N e.sig e.C e.J 0 b z.1 | |
| ((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by | |
| simpa only [radialVector,dist_eq_norm,Function.comp_def] using | |
| (actualEndpointMask_measurable e.ρ e.mass e.N e.sig e.C e.J 0 b).comp | |
| (measurable_fst.prodMk measurable_radialVector) | |
| lemma assembly_endpoint_radial_right (e : RecursiveEnvironmentND n) (b : PolicyDescriptor) : | |
| Measurable (fun z : V×V => actualEndpointMask e.ρ e.mass e.N e.sig e.C e.J 0 b z.2 | |
| ((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by | |
| simpa only [radialVector,dist_eq_norm,Function.comp_def] using | |
| (actualEndpointMask_measurable e.ρ e.mass e.N e.sig e.C e.J 0 b).comp | |
| (measurable_snd.prodMk measurable_radialVector) | |
| lemma assembly_pairMask_radial_measurable (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ) : | |
| Measurable (fun z : V×V => | |
| (Attempt4PrefixShell.pairMaskProductND | |
| (assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c) | |
| (assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1 | |
| ((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by | |
| simp only [assembly_pairMask_eq] | |
| apply Complex.continuous_ofReal.measurable.comp | |
| exact (Finset.measurable_prod _ (fun b _ => assembly_endpoint_radial_left e₁ b)).mul | |
| (Finset.measurable_prod _ (fun b _ => assembly_endpoint_radial_right e₂ b)) | |
| lemma assembly_pairMask_measurable (e₁ e₂ : RecursiveEnvironmentND n) (c : ℕ) | |
| {f g : V → ℂ} (hf : Measurable f) (hg : Measurable g) : | |
| Measurable (fun z : V×V => (f z.1*conj (g z.2))* | |
| (Attempt4PrefixShell.pairMaskProductND | |
| (assemblyDescriptorList e₁.profile e₁.N e₁.q c) (assemblyDescriptorList e₂.profile e₂.N e₂.q c) | |
| (assemblyEndpointSmooth e₁) (assemblyEndpointSmooth e₂) z).1 | |
| ((dist z.1 z.2)⁻¹ • (z.1-z.2))) := by | |
| exact ((hf.comp measurable_fst).mul (Complex.continuous_conj.measurable.comp (hg.comp measurable_snd))).mul | |
| (assembly_pairMask_radial_measurable e₁ e₂ c) | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |