openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /ListSourceSymbol.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularOperator2 | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.SymbolMulDerivativeBound | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.AllMaskProductAdmissible | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.MaskScale2 | |
| 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 Metric MeasureTheory PlanarFalconer Filter | |
| open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical ComplexConjugate | |
| variable {d : ℕ} | |
| noncomputable def actualListSourceSymbol | |
| (S : Finset PolicyDescriptor) | |
| (L : PolicyDescriptor → 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) (f : EuclideanSpace ℝ (Fin d) → ℂ) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) (x : EuclideanSpace ℝ (Fin d)) : | |
| SmoothFunctionND (EuclideanSpace ℝ (Fin d)) := | |
| ⟨fun w => f x*(Q.1 w*(actualAngularProduct S id L ρ m N sig C J i x).1 w), | |
| contDiff_const.mul ((show ContDiff ℝ ∞ Q.1 from Q.2).mul | |
| (actualAngularProduct S id L ρ m N sig C J i x).2)⟩ | |
| lemma actualListSourceSymbol_measurable | |
| (S : Finset PolicyDescriptor) | |
| (L : PolicyDescriptor → 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) (f : EuclideanSpace ℝ (Fin d) → ℂ) (hf : Measurable f) | |
| (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) : | |
| Measurable (fun z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) => | |
| (actualListSourceSymbol S L ρ m N sig C J i f Q z.1).1 z.2) := | |
| (hf.comp measurable_fst).mul (((show ContDiff ℝ ∞ Q.1 from Q.2).continuous.measurable.comp measurable_snd).mul | |
| (actualAngularProduct_measurable S id L ρ m N sig C J i)) | |
| lemma validDescriptorList_card {N : ℕ} (hN : 1 ≤ N) (S : Finset PolicyDescriptor) | |
| (hS : ∀ b ∈ S, ValidDescriptor N b) : (S.card:ℝ) ≤ 8*(N:ℝ)^2 := by | |
| let K := (Finset.range (N+1)).product (Finset.range (N+1)) | |
| let U := K.image (fun z => PolicyDescriptor.parallel z.1 z.2) ∪ | |
| K.image (fun z => PolicyDescriptor.projection z.1 z.2) | |
| have hsub : S ⊆ U := by | |
| intro b hb | |
| have hv := hS b hb | |
| cases b with | |
| | parallel g p => | |
| apply Finset.mem_union_left | |
| exact Finset.mem_image.mpr ⟨(g,p),Finset.mem_product.mpr | |
| ⟨Finset.mem_range.mpr (by dsimp [ValidDescriptor] at hv; omega), | |
| Finset.mem_range.mpr (by exact Nat.lt_succ_of_le hv.2)⟩,rfl⟩ | |
| | projection p t => | |
| apply Finset.mem_union_right | |
| exact Finset.mem_image.mpr ⟨(p,t),Finset.mem_product.mpr | |
| ⟨Finset.mem_range.mpr (by dsimp [ValidDescriptor] at hv; omega), | |
| Finset.mem_range.mpr (by exact Nat.lt_succ_of_le hv.2)⟩,rfl⟩ | |
| have hc : S.card ≤ 2*(N+1)^2 := by | |
| calc | |
| _ ≤ U.card := Finset.card_le_card hsub | |
| _ ≤ (K.image (fun z => PolicyDescriptor.parallel z.1 z.2)).card+ | |
| (K.image (fun z => PolicyDescriptor.projection z.1 z.2)).card := Finset.card_union_le _ _ | |
| _ ≤ K.card+K.card := add_le_add Finset.card_image_le Finset.card_image_le | |
| _ = _ := by simp [K,Finset.product_eq_sprod,Finset.card_product]; ring | |
| have hNr : (1:ℝ) ≤ N := by exact_mod_cast hN | |
| calc | |
| _ ≤ 2*((N:ℝ)+1)^2 := by exact_mod_cast hc | |
| _ ≤ 8*(N:ℝ)^2 := by nlinarith | |
| lemma uniform_actualListSourceSymbol_jets | |
| (W : Finset (List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))) | |
| (M B total : ℕ) (hW : ∀ L ∈ W, L.length ≤ B) (sig : ℝ) (hsig : 0<sig) | |
| (Cq : ℝ) (hCq : 0≤Cq) : | |
| ∃ E : ℝ, 0≤E ∧ ∀ᶠ N : ℕ in atTop, ∀ (S : Finset PolicyDescriptor) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (C : ℝ) (J i l : ℕ) | |
| (f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))), | |
| (∀ e ∈ S, ValidDescriptor N e) → (∀ e ∈ S, descriptorLength e ≤ l) → | |
| (∀ e ∈ S, L e ∈ W) → (∑ e ∈ S, (L e).length ≤ total) → | |
| Measurable f → (∀ x, ‖f x‖≤1) → | |
| (∀ j≤M, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → | |
| _root_.OAI.CurrentFalconer.GraphSymbolJet | |
| (fun x w => (actualListSourceSymbol S L ρ m N sig C J i f Q x).1 w) M | |
| (((2:ℝ)^N)^sig*sourceMaskScaleND l ((2:ℝ)^N) sig) E := by | |
| obtain ⟨D,hD,hDb⟩ := uniform_actualAngularProduct_derivatives (ι:=PolicyDescriptor) | |
| W M B total 2 hW 8 sig hsig | |
| refine ⟨Cq*D^total*2^M,by positivity,?_⟩ | |
| filter_upwards [hDb,Filter.eventually_ge_atTop 1] with N hDb hN | |
| intro S L ρ m C J i l f Q hValid hl hL htotal hfm hf hQ | |
| refine ⟨actualListSourceSymbol_measurable S L ρ m N sig C J i f hfm Q, | |
| fun x => (actualListSourceSymbol S L ρ m N sig C J i f Q x).2,by positivity,?_⟩ | |
| intro x j hj w hw | |
| let Z := actualAngularProduct S id L ρ m N sig C J i x | |
| let scale := ((2:ℝ)^N)^sig*sourceMaskScaleND l ((2:ℝ)^N) sig | |
| have hscale : 1 ≤ scale := one_le_mul_of_one_le_of_one_le | |
| (Real.one_le_rpow (one_le_pow₀ (by norm_num : (1:ℝ)≤2)) hsig.le) | |
| (sourceMaskScaleND_one_le _ _ _) | |
| have hb := symbol_mul_derivative_boundND Q Z j hCq (pow_nonneg (zero_le_one.trans hD) total) | |
| (zero_le_one.trans hscale) w (fun k hk => | |
| (hQ k (hk.trans hj) w hw).trans (le_mul_of_one_le_right hCq (one_le_pow₀ hscale))) | |
| (fun k hk => hDb S id L ρ m C J i l (validDescriptorList_card hN S hValid) | |
| hL htotal hl x k (hk.trans hj) w hw) | |
| have hh := symbol_const_mul_derivative_boundND (smoothMulND Q Z) (f x) (hf x) j w | |
| apply hh.trans (hb.trans ?_) | |
| rw [mul_pow] | |
| change Cq*D^total*(2^j*scale^j) ≤ Cq*D^total*2^M*scale^j | |
| rw [mul_assoc (Cq*D^total) (2^M)] | |
| exact mul_le_mul_of_nonneg_left | |
| (mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hj) | |
| (pow_nonneg (zero_le_one.trans hscale) j)) (mul_nonneg hCq (pow_nonneg (zero_le_one.trans hD) total)) | |
| lemma uniform_actualListSourceSymbol_test {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) (S : Finset PolicyDescriptor), | |
| (∀ e ∈ S, ValidDescriptor N e) → | |
| ∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) | |
| (x w : EuclideanSpace ℝ (Fin d)), ‖w‖=1 → | |
| (actualListSourceSymbol S L ρ m N sig C J i f Q x).1 w ≠ 0 → | |
| ∀ e ∈ S, (x,w) ∈ endpointGood ρ m N sig C J (i+1) e := by | |
| obtain ⟨N₀,h₀⟩ := uniform_actualAngularProduct_support (d:=d) (ι:=PolicyDescriptor) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ρ hρ m C i S hValid L f Q x w hw hn e he | |
| have hp : (actualAngularProduct S id L ρ m N sig C J i x).1 w ≠ 0 := by | |
| intro hz | |
| apply hn | |
| simp [actualListSourceSymbol,hz] | |
| apply h₀ N hN ρ hρ m C i S id hValid L 0 x w hw _ e he | |
| apply subset_closure | |
| rw [Function.mem_support,←norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] | |
| exact hp | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |