openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /DescriptorGoodNeg.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.TensorCapTsupportCoord | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.NominalRadius | |
| 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 | |
| variable {d : ℕ} | |
| lemma descriptorGood_neg (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) {w : EuclideanSpace ℝ (Fin d)} | |
| (hw : w ∈ descriptorGood ρ m N sig C J i P b) : | |
| -w ∈ descriptorGood ρ m N sig C J i P b := by | |
| classical | |
| cases b with | |
| | parallel g p => | |
| have he : fullParallelCount ρ g p P (testWidth N J i p sig) (-w)= | |
| fullParallelCount ρ g p P (testWidth N J i p sig) w := by | |
| unfold fullParallelCount | |
| congr 1 | |
| ext Q | |
| simp only [Finset.mem_filter,graphPerpND,inner_neg_right,neg_smul,smul_neg,neg_neg] | |
| constructor | |
| · intro h | |
| apply Finset.mem_filter.mpr | |
| exact h | |
| · intro h | |
| exact Finset.mem_filter.mp h | |
| exact ⟨by simpa only [norm_neg] using hw.1,by rw [he]; exact hw.2⟩ | |
| | projection p t => | |
| simpa only [descriptorGood,Set.mem_ofPred_eq,norm_neg,fullProjectionCollision, | |
| inner_neg_right,abs_neg] using hw | |
| noncomputable def actualDescriptorMask (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) : EuclideanSpace ℝ (Fin d) → ℝ := | |
| evenBufferedMaskND (sourceMaskRadiusND d (descriptorLength b) ((2:ℝ)^N) sig) | |
| (descriptorGood ρ m N sig C J i P b) | |
| lemma actualDescriptorMask_smooth (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) : ContDiff ℝ ∞ (actualDescriptorMask ρ m N sig C J i P b) := | |
| evenBufferedMaskND_contDiff _ _ | |
| lemma actualDescriptorMask_even (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) : Function.Even (actualDescriptorMask ρ m N sig C J i P b) := | |
| evenBufferedMaskND_even _ _ | |
| lemma actualDescriptorMask_bounds (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) {w : EuclideanSpace ℝ (Fin d)} (hw : ‖w‖ ≤ 1) : | |
| 0 ≤ actualDescriptorMask ρ m N sig C J i P b w ∧ | |
| actualDescriptorMask ρ m N sig C J i P b w ≤ 1 := | |
| evenBufferedMaskND_bounds (sourceMaskRadiusND_pos _ _ _ _) _ hw | |
| lemma actualDescriptorMask_one (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) {w : EuclideanSpace ℝ (Fin d)} | |
| (hw : w ∈ descriptorGood ρ m N sig C J i P b) : | |
| actualDescriptorMask ρ m N sig C J i P b w = 1 := by | |
| have hnorm : ‖w‖=1 := by cases b <;> exact hw.1 | |
| exact evenBufferedMaskND_eq_one (sourceMaskRadiusND_pos _ _ _ _) | |
| (fun _ hv => descriptorGood_neg ρ m N sig C J i P b hv) hnorm.le hw | |
| lemma sourceMaskRadiusND_near_nominal (N l : ℕ) (sig : ℝ) : | |
| 2*(d+1:ℝ)*sourceMaskRadiusND d l ((2:ℝ)^N) sig ≤ nominalRadius N sig l := by | |
| have hD : 0 < sourceMaskScaleND l ((2:ℝ)^N) sig := sourceMaskScaleND_pos _ _ _ | |
| have h₁ : 2*(d+1:ℝ)*sourceMaskRadiusND d l ((2:ℝ)^N) sig ≤ | |
| (sourceMaskScaleND l ((2:ℝ)^N) sig)⁻¹ := by | |
| unfold sourceMaskRadiusND | |
| have hd : 2*(d+1:ℝ) ≤ 100*(d+1:ℝ)^2 := by nlinarith [Nat.cast_nonneg (α := ℝ) d] | |
| exact (mul_le_mul_of_nonneg_right hd (by positivity : 0 ≤ | |
| (sourceMaskScaleND l ((2:ℝ)^N) sig)⁻¹/(100*(d+1:ℝ)^2))).trans_eq (by | |
| field_simp) | |
| apply h₁.trans | |
| apply le_min | |
| · exact (inv_le_one₀ hD).mpr (sourceMaskScaleND_one_le _ _ _) | |
| · simpa only [Nat.sub_zero,pow_zero,inv_one,one_mul,one_div] using | |
| (sourceMaskScaleND_inv_depth (g := 0) (p := l) (Nat.zero_le l) | |
| (R := (2:ℝ)^N) (σ := sig) (by positivity)) | |
| lemma uniform_actualDescriptorMask_support {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))), IsFiniteMeasure ρ → | |
| ∀ (m : ℝ → ℝ) (C : ℝ) (i : ℕ) (b : PolicyDescriptor), ValidDescriptor N b → | |
| ∀ P : CellPath d, P.length=descriptorAnchor b → ∀ k : ℕ, | |
| ∀ w : EuclideanSpace ℝ (Fin d), ‖w‖=1 → | |
| w ∈ tsupport (iteratedFDeriv ℝ k (actualDescriptorMask ρ m N sig C J i P b)) → | |
| w ∈ descriptorGood ρ m N sig C J (i+1) P b := by | |
| obtain ⟨N₀,h₀⟩ := uniform_descriptorGood_transfer (d := d) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ρ hf m C i b hb P hP k w hw hsupport | |
| have hs : w ∈ tsupport (actualDescriptorMask ρ m N sig C J i P b) := | |
| tsupport_iteratedFDeriv_subset k hsupport | |
| obtain ⟨v,hv,hd⟩ := evenBufferedMaskND_support_near (sourceMaskRadiusND_pos _ _ _ _) | |
| (fun _ hv => descriptorGood_neg ρ m N sig C J i P b hv) hs | |
| apply h₀ N hN ρ hf m C i b hb P hP v w hw _ hv | |
| rw [dist_comm] | |
| exact hd.trans (sourceMaskRadiusND_near_nominal N (descriptorLength b) sig) | |
| lemma actualDescriptorMask_derivatives (d B : ℕ) : | |
| ∃ C₀ : ℝ, 0 ≤ C₀ ∧ ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (P : CellPath d) | |
| (b : PolicyDescriptor) (k : ℕ), k ≤ B → ∀ w, | |
| ‖iteratedFDeriv ℝ k (actualDescriptorMask ρ m N sig C J i P b) w‖ ≤ | |
| C₀*(sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig)^k := by | |
| obtain ⟨C₀,hC,h₀⟩ := evenBufferedMaskND_derivative_bound d B | |
| let A : ℝ := 100*(d+1)^2 | |
| have hA : 1≤A := by dsimp [A]; nlinarith [sq_nonneg (d:ℝ),Nat.cast_nonneg (α := ℝ) d] | |
| refine ⟨C₀*A^B,by positivity,?_⟩ | |
| intro ρ m N sig C J i P b k hk w | |
| have he : (sourceMaskRadiusND d (descriptorLength b) ((2:ℝ)^N) sig)⁻¹= | |
| A*sourceMaskScaleND (descriptorLength b) ((2:ℝ)^N) sig := by | |
| rw [sourceMaskRadiusND,inv_div,div_inv_eq_mul] | |
| apply (h₀ _ _ (sourceMaskRadiusND_pos _ _ _ _) k hk w).trans | |
| rw [he,mul_pow,← mul_assoc] | |
| exact mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (pow_le_pow_right₀ hA hk) hC) | |
| (pow_nonneg (sourceMaskScaleND_pos _ _ _).le k) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |