openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /DescriptorSmooth.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularWordOperatorBound | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer.TreeND | |
| open Set Metric MeasureTheory PlanarFalconer | |
| open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical | |
| variable {d : ℕ} | |
| noncomputable def actualDescriptorSmooth (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) (b : PolicyDescriptor) : | |
| SmoothFunctionND (EuclideanSpace ℝ (Fin d)) := | |
| ⟨fun w => (actualDescriptorMask ρ m N sig C J i P b w : ℂ), | |
| Complex.ofRealCLM.contDiff.comp (actualDescriptorMask_smooth ρ m N sig C J i P b)⟩ | |
| lemma actualEndpointSmooth_of_mem (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {P : CellPath d} (hP : P ∈ dyadicPaths (descriptorAnchor b)) | |
| {x : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) : | |
| actualEndpointSmooth ρ m N sig C J i b x=actualDescriptorSmooth ρ m N sig C J i P b := by | |
| apply Subtype.ext | |
| funext w | |
| exact congrArg Complex.ofReal (congrFun (actualEndpointMask_of_mem ρ m N sig C J i b hP hx) w) | |
| lemma actualEndpointSmooth_of_absent (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {x : EuclideanSpace ℝ (Fin d)} | |
| (hx : ∀ P ∈ dyadicPaths (descriptorAnchor b), x ∉ treeCell P) : | |
| actualEndpointSmooth ρ m N sig C J i b x=0 := by | |
| apply Subtype.ext | |
| funext w | |
| change (actualEndpointMask ρ m N sig C J i b x w : ℂ)=0 | |
| rw [actualEndpointMask_of_absent ρ m N sig C J i b hx] | |
| rfl | |
| lemma actualEndpointAngular_partition | |
| (L : List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (N : ℕ) | |
| (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) (x : EuclideanSpace ℝ (Fin d)) : | |
| actualEndpointAngular L ρ m N sig C J i b x= | |
| ∑ P ∈ dyadicPaths (descriptorAnchor b), | |
| if x ∈ treeCell P then | |
| normalizedAngularWordND (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig) L | |
| (actualDescriptorSmooth ρ m N sig C J i P b) else 0 := by | |
| classical | |
| by_cases hx : ∃ P ∈ dyadicPaths (descriptorAnchor b), x ∈ treeCell P | |
| · obtain ⟨P,hP,hx⟩ := hx | |
| unfold actualEndpointAngular | |
| rw [actualEndpointSmooth_of_mem ρ m N sig C J i b hP hx,Finset.sum_eq_single P] | |
| · simp only [hx,ite_true] | |
| · intro Q hQ hQP | |
| have hnot : x ∉ treeCell Q := fun hxQ => | |
| Set.disjoint_left.mp (treeCell_disjoint | |
| (((mem_dyadicPaths _ _).mp hQ).trans ((mem_dyadicPaths _ _).mp hP).symm) hQP) hxQ hx | |
| simp only [hnot,ite_false] | |
| · exact fun h => (h hP).elim | |
| · have hx' : ∀ P ∈ dyadicPaths (descriptorAnchor b), x ∉ treeCell P := by | |
| simpa only [not_exists,not_and] using hx | |
| unfold actualEndpointAngular normalizedAngularWordND | |
| rw [actualEndpointSmooth_of_absent ρ m N sig C J i b hx',map_zero,smul_zero] | |
| exact (Finset.sum_eq_zero fun P hP => ite_eq_right (hx' P hP)).symm | |
| lemma actualEndpointAngular_measurable | |
| (L : List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (N : ℕ) | |
| (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) : | |
| Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) => | |
| (actualEndpointAngular L ρ m N sig C J i b z.1).1 z.2) := by | |
| classical | |
| have he : (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) => | |
| (actualEndpointAngular L ρ m N sig C J i b z.1).1 z.2)= | |
| fun z => ∑ P ∈ dyadicPaths (descriptorAnchor b), if z.1 ∈ treeCell P then | |
| (normalizedAngularWordND (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig) L | |
| (actualDescriptorSmooth ρ m N sig C J i P b)).1 z.2 else 0 := by | |
| funext z | |
| rw [actualEndpointAngular_partition] | |
| simp only [Submodule.coe_sum,Finset.sum_apply] | |
| apply Finset.sum_congr rfl | |
| intro P _hP | |
| split_ifs <;> rfl | |
| rw [he] | |
| apply Finset.measurable_sum | |
| intro P _hP | |
| exact Measurable.ite ((measurableSet_treeCell P).preimage measurable_fst) | |
| ((smoothFunctionND_contDiff _).continuous.measurable.comp measurable_snd) measurable_const | |
| lemma actualEndpointSmooth_even (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| (x : EuclideanSpace ℝ (Fin d)) : Function.Even (actualEndpointSmooth ρ m N sig C J i b x).1 := by | |
| classical | |
| intro w | |
| change (actualEndpointMask ρ m N sig C J i b x (-w):ℂ)=_ | |
| unfold actualEndpointMask | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro P _hP | |
| split_ifs | |
| · exact actualDescriptorMask_even ρ m N sig C J i P b w | |
| · rfl | |
| lemma actualEndpointAngular_even | |
| (L : List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (N : ℕ) | |
| (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) (x : EuclideanSpace ℝ (Fin d)) : | |
| Function.Even (actualEndpointAngular L ρ m N sig C J i b x).1 := | |
| normalizedAngularWordND_even _ L _ (actualEndpointSmooth_even ρ m N sig C J i b x) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |