| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.SymbolMulDerivativeBound | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.PolicyAngular | |
| 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 Filter | |
| open scoped ENNReal RealInnerProductSpace ContDiff Topology Classical | |
| variable {d : ℕ} | |
| noncomputable def actualSourceSymbol | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (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*(actualPolicyAngular A N q a t L ρ m sig C J i x).1 w), | |
| contDiff_const.mul ((show ContDiff ℝ ∞ Q.1 from Q.2).mul (show ContDiff ℝ ∞ (actualPolicyAngular A N q a t L ρ m sig C J i x).1 from (actualPolicyAngular A N q a t L ρ m sig C J i x).2))⟩ | |
| lemma actualSourceSymbol_measurable | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (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) => | |
| (actualSourceSymbol A N q a t L ρ m sig C J i f Q z.1).1 z.2) := by | |
| exact (hf.comp measurable_fst).mul (((show ContDiff ℝ ∞ Q.1 from Q.2).continuous.measurable.comp measurable_snd).mul | |
| (actualPolicyAngular_measurable A N q a t L ρ m sig C J i)) | |
| lemma actualSourceSymbol_support | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))) | |
| (k : ℕ) (x : EuclideanSpace ℝ (Fin d)) : | |
| tsupport (iteratedFDeriv ℝ k (actualSourceSymbol A N q a t L ρ m sig C J i f Q x).1) ⊆ | |
| tsupport (actualPolicyAngular A N q a t L ρ m sig C J i x).1 := | |
| (tsupport_iteratedFDeriv_subset k).trans (tsupport_mul_subset_right.trans tsupport_mul_subset_right) | |
| noncomputable def actualPolicyAmbientGood | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (x : EuclideanSpace ℝ (Fin d)) : Set (EuclideanSpace ℝ (Fin d)) := | |
| {w | ‖w‖≠1 ∨ ∀ b∈policyDescriptorList A N q a t, (x,w)∈endpointGood ρ m N sig C J i b} | |
| lemma actualPolicyAmbientGood_even | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (x : EuclideanSpace ℝ (Fin d)) {w : EuclideanSpace ℝ (Fin d)} | |
| (hw : w∈actualPolicyAmbientGood A N q a t ρ m sig C J i x) : | |
| -w∈actualPolicyAmbientGood A N q a t ρ m sig C J i x := by | |
| rcases hw with hn | hg | |
| · exact Or.inl (by simpa only [norm_neg] using hn) | |
| · right | |
| intro b hb P hP hx | |
| exact descriptorGood_neg ρ m N sig C J i P b (hg b hb P hP hx) | |
| lemma actualPolicyAmbientGood_measurable | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] | |
| (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) : | |
| MeasurableSet {z : EuclideanSpace ℝ (Fin d) × EuclideanSpace ℝ (Fin d) | | |
| z.2∈actualPolicyAmbientGood A N q a t ρ m sig C J i z.1} := by | |
| change MeasurableSet ({z | ‖z.2‖≠1} ∪ {z | ∀ b∈policyDescriptorList A N q a t, | |
| z∈endpointGood ρ m N sig C J i b}) | |
| refine ((measurable_snd.norm (measurableSet_singleton 1)).compl).union ?_ | |
| simp only [ofPred_forall] | |
| exact Finset.measurableSet_biInter _ (fun b hb => measurable_endpointGood ρ m N sig C J i b) | |
| lemma uniform_actualSourceSymbol_support {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : | |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))), IsFiniteMeasure ρ → | |
| ∀ (A : ℕ → ℝ) (q a t : ℕ), 0<q → t≤N → | |
| ∀ (m : ℝ → ℝ) (C : ℝ) (i : ℕ) | |
| (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))) | |
| (k : ℕ) (x : EuclideanSpace ℝ (Fin d)), | |
| tsupport (iteratedFDeriv ℝ k (actualSourceSymbol A N q a t L ρ m sig C J i f Q x).1) ⊆ | |
| actualPolicyAmbientGood A N q a t ρ m sig C J (i+1) x := by | |
| obtain ⟨N₀,h₀⟩ := uniform_actualPolicyAngular_support (d:=d) hsig hJ | |
| refine ⟨N₀,?_⟩ | |
| intro N hN ρ hf A q a t hq ht m C i L f Q k x w hw | |
| by_cases hn : ‖w‖=1 | |
| · right | |
| have hs := actualSourceSymbol_support A N q a t L ρ m sig C J i f Q k x hw | |
| have hts : tsupport (iteratedFDeriv ℝ 0 (actualPolicyAngular A N q a t L ρ m sig C J i x).1)= | |
| tsupport (actualPolicyAngular A N q a t L ρ m sig C J i x).1 := by | |
| apply congrArg closure | |
| ext z | |
| simp only [Function.mem_support] | |
| rw [← norm_ne_zero_iff,norm_iteratedFDeriv_zero,norm_ne_zero_iff] | |
| apply h₀ N hN ρ hf A q a t hq ht m C i L 0 x w hn | |
| rwa [hts] | |
| · exact Or.inl hn | |
| lemma uniform_actualSourceSymbol_derivatives | |
| (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, ∀ (A : ℕ → ℝ) (q a t : ℕ), 0<q → | |
| ∀ (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 : ℕ) | |
| (f : EuclideanSpace ℝ (Fin d) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin d))), | |
| (∀ b ∈ policyDescriptorList A N q a t, L b ∈ W) → | |
| (∑ b ∈ policyDescriptorList A N q a t, (L b).length ≤ total) → | |
| (∀ x, ‖f x‖≤1) → (∀ j≤M, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j Q.1 w‖≤Cq) → | |
| ∀ (x : EuclideanSpace ℝ (Fin d)), ∀ j ≤ M, ∀ w, ‖w‖=1 → | |
| ‖iteratedFDeriv ℝ j (actualSourceSymbol A N q a t L ρ m sig C J i f Q x).1 w‖ ≤ | |
| E*(((2:ℝ)^N)^sig* | |
| sourceMaskScaleND ((policyDescriptorList A N q a t).sup descriptorLength) ((2:ℝ)^N) sig)^j := by | |
| obtain ⟨D,hD,hDb⟩ := uniform_actualPolicyAngular_derivatives W M B total hW sig hsig | |
| refine ⟨Cq*D^total*2^M,by positivity,?_⟩ | |
| filter_upwards [hDb] with N hDb | |
| intro A q a t hq L ρ m C J i f Q hL htotal hf hQ x j hj w hw | |
| let Z := actualPolicyAngular A N q a t L ρ m sig C J i x | |
| let scale := ((2:ℝ)^N)^sig*sourceMaskScaleND | |
| ((policyDescriptorList A N q a t).sup descriptorLength) ((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 A q a t hq L ρ m C J i hL htotal 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)) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| 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} | |
| lemma actualEndpointSmooth_reached (ρ : Measure (EuclideanSpace ℝ (Fin d))) | |
| (m : ℝ → ℝ) (N : ℕ) (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) | |
| {P : CellPath d} (ha : descriptorAnchor b ≤ P.length) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈treeCell P) (hy : y∈treeCell P) : | |
| actualEndpointSmooth ρ m N sig C J i b x = actualEndpointSmooth ρ m N sig C J i b y := by | |
| have hP : P.take (descriptorAnchor b) ∈ dyadicPaths (descriptorAnchor b) := by | |
| simp only [mem_dyadicPaths,List.length_take,Nat.min_eq_left ha] | |
| rw [actualEndpointSmooth_of_mem ρ m N sig C J i b hP (treeCell_subset_take P _ hx), | |
| actualEndpointSmooth_of_mem ρ m N sig C J i b hP (treeCell_subset_take P _ hy)] | |
| lemma actualEndpointAngular_reached | |
| (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) {P : CellPath d} (ha : descriptorAnchor b ≤ P.length) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈treeCell P) (hy : y∈treeCell P) : | |
| actualEndpointAngular L ρ m N sig C J i b x = actualEndpointAngular L ρ m N sig C J i b y := by | |
| unfold actualEndpointAngular | |
| rw [actualEndpointSmooth_reached ρ m N sig C J i b ha hx hy] | |
| lemma actualAngularProduct_reached {ι : Type*} [DecidableEq ι] | |
| (S : Finset ι) (b : ι → PolicyDescriptor) | |
| (L : ι → 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) {P : CellPath d} (ha : ∀ e∈S, descriptorAnchor (b e) ≤ P.length) | |
| {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈treeCell P) (hy : y∈treeCell P) : | |
| actualAngularProduct S b L ρ m N sig C J i x = actualAngularProduct S b L ρ m N sig C J i y := by | |
| apply Subtype.ext | |
| funext w | |
| apply Finset.prod_congr rfl | |
| intro e he | |
| exact congrArg (fun f : SmoothFunctionND (EuclideanSpace ℝ (Fin d)) => f.1 w) | |
| (actualEndpointAngular_reached (L e) ρ m N sig C J i (b e) (ha e he) hx hy) | |
| noncomputable def reachedDescriptorList (A : ℕ → ℝ) (N q a t r : ℕ) : Finset PolicyDescriptor := | |
| (policyDescriptorList A N q a t).filter (fun b => descriptorAnchor b≤r) | |
| noncomputable def unreachedDescriptorList (A : ℕ → ℝ) (N q a t r : ℕ) : Finset PolicyDescriptor := | |
| (policyDescriptorList A N q a t).filter (fun b => r<descriptorAnchor b) | |
| lemma actualPolicyAngular_reached_factorization | |
| (A : ℕ → ℝ) (N q a t r : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (x : EuclideanSpace ℝ (Fin d)) : | |
| actualPolicyAngular A N q a t L ρ m sig C J i x = | |
| smoothMulND | |
| (actualAngularProduct (reachedDescriptorList A N q a t r) id L ρ m N sig C J i x) | |
| (actualAngularProduct (unreachedDescriptorList A N q a t r) id L ρ m N sig C J i x) := by | |
| apply Subtype.ext | |
| funext w | |
| change (∏ b∈policyDescriptorList A N q a t, (actualEndpointAngular (L b) ρ m N sig C J i b x).1 w) = _ | |
| simpa only [reachedDescriptorList,unreachedDescriptorList,actualAngularProduct,smoothMulND,smoothFinsetProductND, | |
| id_eq, not_le] using (Finset.prod_filter_mul_prod_filter_not | |
| (policyDescriptorList A N q a t) (fun b => descriptorAnchor b≤r) | |
| (fun b => (actualEndpointAngular (L b) ρ m N sig C J i b x).1 w)).symm | |
| lemma reachedSymbol_constant_on_cell | |
| (A : ℕ → ℝ) (N q a t : ℕ) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin d))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| {P : CellPath d} {x y : EuclideanSpace ℝ (Fin d)} (hx : x∈treeCell P) (hy : y∈treeCell P) : | |
| actualAngularProduct (reachedDescriptorList A N q a t P.length) id L ρ m N sig C J i x = | |
| actualAngularProduct (reachedDescriptorList A N q a t P.length) id L ρ m N sig C J i y := | |
| actualAngularProduct_reached _ _ _ _ _ _ _ _ _ _ | |
| (fun _ hb => (Finset.mem_filter.mp hb).2) hx hy | |
| lemma uniform_reachedSymbol_bound | |
| (W : Finset (List (EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)))) | |
| (B total : ℕ) (hW : ∀ L∈W, L.length≤B) (sig : ℝ) (hsig : 0<sig) : | |
| ∃ D : ℝ, 1≤D ∧ ∀ᶠ N : ℕ in Filter.atTop, ∀ (A : ℕ → ℝ) (q a t r : ℕ), 0<q → | |
| ∀ (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 : ℕ), | |
| (∀ b ∈ policyDescriptorList A N q a t, L b∈W) → | |
| (∑ b∈policyDescriptorList A N q a t, (L b).length ≤ total) → | |
| ∀ (x w : EuclideanSpace ℝ (Fin d)), ‖w‖=1 → | |
| ‖(actualAngularProduct (reachedDescriptorList A N q a t r) id L ρ m N sig C J i x).1 w‖ ≤ D^total := by | |
| obtain ⟨D,hD,hDb⟩ := uniform_actualAngularProduct_derivatives | |
| (ι:=PolicyDescriptor) W 0 B total 3 hW 8 sig hsig | |
| refine ⟨D,hD,?_⟩ | |
| filter_upwards [hDb,Filter.eventually_ge_atTop 1] with N hDb hN | |
| intro A q a t r hq L ρ m C J i hL htotal x w hw | |
| have hsub : reachedDescriptorList A N q a t r ⊆ policyDescriptorList A N q a t := Finset.filter_subset _ _ | |
| have hcard : ((reachedDescriptorList A N q a t r).card:ℝ)≤8*(N:ℝ)^3 := by | |
| calc | |
| _ ≤ ((policyDescriptorList A N q a t).card:ℝ) := by exact_mod_cast Finset.card_le_card hsub | |
| _ ≤ _ := actualPolicy_card_bound A hN hq | |
| have hsum : ∑ b∈reachedDescriptorList A N q a t r, (L b).length ≤ total := | |
| (Finset.sum_le_sum_of_subset_of_nonneg hsub (by intros; exact Nat.zero_le _)).trans htotal | |
| have hh := hDb (reachedDescriptorList A N q a t r) id L ρ m C J i | |
| ((policyDescriptorList A N q a t).sup descriptorLength) hcard | |
| (fun b hb => hL b (hsub hb)) hsum (fun b hb => Finset.le_sup (f:=descriptorLength) (hsub hb)) x 0 le_rfl w hw | |
| simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hh | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |