openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /TreeCellMembershipConstantOnFinerGrid.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.ResidualPairSourceSymbolEqScalarPair | |
| 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 MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff ENNReal Topology Classical | |
| variable {n : ℕ} | |
| lemma treeCell_membership_constant_on_finer_grid | |
| {P : CellPath (n+1)} {h : ℕ} (hP : P.length≤h) (X : Fin (n+1) → ℤ) | |
| {x y : EuclideanSpace ℝ (Fin (n+1))} | |
| (hx : x∈CurrentFalconer.ambientGridCell h X) | |
| (hy : y∈CurrentFalconer.ambientGridCell h X) : x∈treeCell P ↔ y∈treeCell P := by | |
| have hs := CurrentFalconer.ambientGridCell_subset_ancestor hP X | |
| have one {u v : EuclideanSpace ℝ (Fin (n+1))} | |
| (hu : u∈CurrentFalconer.ambientGridCell h X) | |
| (hv : v∈CurrentFalconer.ambientGridCell h X) (huP : u∈treeCell P) : v∈treeCell P := by | |
| have he : (fun i => X i/(2:ℤ)^(h-P.length))=(fun i => (pathIndex P i:ℤ)) := by | |
| by_contra hne | |
| exact Set.disjoint_left.mp (CurrentFalconer.ambientGridCell_disjoint hne) (hs hu) huP | |
| have hh := hs hv | |
| rw [he] at hh | |
| exact hh | |
| exact ⟨one hx hy,one hy hx⟩ | |
| lemma actualEndpointSmooth_grid_reached | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (N : ℕ) | |
| (sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) {h : ℕ} (ha : descriptorAnchor b≤h) | |
| (X : Fin (n+1) → ℤ) {x y : EuclideanSpace ℝ (Fin (n+1))} | |
| (hx : x∈CurrentFalconer.ambientGridCell h X) (hy : y∈CurrentFalconer.ambientGridCell h X) : | |
| actualEndpointSmooth ρ m N sig C J i b x=actualEndpointSmooth ρ m N sig C J i b y := by | |
| have he (P : CellPath (n+1)) (hP : P∈dyadicPaths (descriptorAnchor b)) : | |
| x∈treeCell P ↔ y∈treeCell P := | |
| treeCell_membership_constant_on_finer_grid ((mem_dyadicPaths _ _).mp hP ▸ ha) X hx hy | |
| by_cases ex : ∃ P∈dyadicPaths (descriptorAnchor b), x∈treeCell P | |
| · obtain ⟨P,hP,hxP⟩ := ex | |
| rw [actualEndpointSmooth_of_mem ρ m N sig C J i b hP hxP, actualEndpointSmooth_of_mem ρ m N sig C J i b hP ((he P hP).mp hxP)] | |
| · have hx0 : ∀ P∈dyadicPaths (descriptorAnchor b), x∉treeCell P := by | |
| intro P hP hxP | |
| exact ex ⟨P,hP,hxP⟩ | |
| rw [actualEndpointSmooth_of_absent ρ m N sig C J i b hx0, actualEndpointSmooth_of_absent ρ m N sig C J i b (fun P hP hyP => hx0 P hP ((he P hP).mpr hyP))] | |
| lemma actualAngularProduct_grid_reached | |
| (D : Finset PolicyDescriptor) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (N : ℕ) | |
| (sig C : ℝ) (J i : ℕ) {h : ℕ} (ha : ∀ b∈D, descriptorAnchor b≤h) | |
| (X : Fin (n+1) → ℤ) {x y : EuclideanSpace ℝ (Fin (n+1))} | |
| (hx : x∈CurrentFalconer.ambientGridCell h X) (hy : y∈CurrentFalconer.ambientGridCell h X) : | |
| actualAngularProduct D id L ρ m N sig C J i x=actualAngularProduct D id L ρ m N sig C J i y := by | |
| apply Subtype.ext | |
| funext w | |
| apply Finset.prod_congr rfl | |
| intro b hb | |
| have he : actualEndpointAngular (L b) ρ m N sig C J i b x=actualEndpointAngular (L b) ρ m N sig C J i b y := by | |
| unfold actualEndpointAngular | |
| rw [actualEndpointSmooth_grid_reached ρ m N sig C J i b (ha b hb) X hx hy] | |
| exact congrArg (fun f : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))) => f.1 w) he | |
| noncomputable def newlyReachedDescriptorList (A : ℕ → ℝ) (N q a t p h : ℕ) : Finset PolicyDescriptor := | |
| (unreachedDescriptorList A N q a t p).filter (fun b => descriptorAnchor b≤h) | |
| lemma actualResidualAngular_step_factorization | |
| (A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (x : EuclideanSpace ℝ (Fin (n+1))) : | |
| actualResidualAngular A N q a t p L ρ m sig C J i x = | |
| smoothMulND (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i x) | |
| (actualResidualAngular A N q a t h L ρ m sig C J i x) := by | |
| have hcomp : (unreachedDescriptorList A N q a t p).filter (fun b => ¬descriptorAnchor b≤h)= | |
| unreachedDescriptorList A N q a t h := by | |
| ext b | |
| simp only [unreachedDescriptorList,Finset.mem_filter,not_le] | |
| constructor | |
| · rintro ⟨⟨hb, _⟩, hbh⟩ | |
| exact ⟨hb,hbh⟩ | |
| · rintro ⟨hb,hbh⟩ | |
| exact ⟨⟨hb,lt_of_le_of_lt hph hbh⟩,hbh⟩ | |
| apply Subtype.ext | |
| funext w | |
| have he := (Finset.prod_filter_mul_prod_filter_not | |
| (unreachedDescriptorList A N q a t p) (fun b => descriptorAnchor b≤h) | |
| (fun b => (actualEndpointAngular (L b) ρ m N sig C J i b x).1 w)).symm | |
| rw [hcomp] at he | |
| exact he | |
| lemma actualResidualSourceSymbol_grid_factorization | |
| (A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (X : Fin (n+1) → ℤ) {x : EuclideanSpace ℝ (Fin (n+1))} | |
| (hx : x∈CurrentFalconer.ambientGridCell h X) (w : EuclideanSpace ℝ (Fin (n+1))) : | |
| (actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1 w = | |
| (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i | |
| (CurrentFalconer.ambientGridCenter h X)).1 w * | |
| (actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1 w := by | |
| have he := actualResidualAngular_step_factorization A N q a t p h hph L ρ m sig C J i x | |
| have hc := actualAngularProduct_grid_reached (newlyReachedDescriptorList A N q a t p h) | |
| L ρ m N sig C J i (fun b hb => (Finset.mem_filter.mp hb).2) X hx | |
| (CurrentFalconer.ambientGridCenter_mem h X) | |
| change f x*(Q.1 w*(actualResidualAngular A N q a t p L ρ m sig C J i x).1 w)=_ | |
| rw [he] | |
| change f x*(Q.1 w*((actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i x).1 w * _))=_ | |
| rw [hc] | |
| change f x*(Q.1 w*(_*_))=_*(f x*(Q.1 w*_)) | |
| dsimp only [actualResidualAngular] | |
| ring | |
| lemma actualResidualTransform_grid_factorization | |
| (A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h) | |
| (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ) | |
| (f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (X : Fin (n+1) → ℤ) (M r : ℝ) (w : EuclideanSpace ℝ (Fin (n+1))) : | |
| sourceAngularTransformND ρ (CurrentFalconer.ambientGridCell h X) M | |
| (fun x => (actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1) r w = | |
| (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i | |
| (CurrentFalconer.ambientGridCenter h X)).1 w * | |
| sourceAngularTransformND ρ (CurrentFalconer.ambientGridCell h X) M | |
| (fun x => (actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1) r w := by | |
| unfold sourceAngularTransformND | |
| rw [show (∫ x in CurrentFalconer.ambientGridCell h X, spherePlaneWaveND r x w*(actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1 w ∂ρ) = ∫ x in CurrentFalconer.ambientGridCell h X, (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i (CurrentFalconer.ambientGridCenter h X)).1 w * (spherePlaneWaveND r x w*(actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1 w) ∂ρ by apply setIntegral_congr_fun (CurrentFalconer.measurableSet_ambientGridCell h X) intro x hx dsimp only rw [actualResidualSourceSymbol_grid_factorization A N q a t p h hph L ρ m sig C J i f Q X hx w] | |
| ring] | |
| rw [integral_const_mul] | |
| exact (mul_smul_comm _ _ _).symm | |
| lemma uniform_newlyReachedSymbol_bound | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (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 p h : ℕ), 0<q → | |
| ∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (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 (n+1))), ‖w‖=1 → | |
| ‖(actualAngularProduct (newlyReachedDescriptorList A N q a t p h) 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 p h hq L ρ m C J i hL htotal x w hw | |
| have hsub : newlyReachedDescriptorList A N q a t p h ⊆ policyDescriptorList A N q a t := (Finset.filter_subset _ _).trans (Finset.filter_subset _ _) | |
| have hcard : ((newlyReachedDescriptorList A N q a t p h).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∈newlyReachedDescriptorList A N q a t p h, (L b).length ≤ total := | |
| (Finset.sum_le_sum_of_subset_of_nonneg hsub (by intros; exact Nat.zero_le _)).trans htotal | |
| have hh := hDb (newlyReachedDescriptorList A N q a t p h) 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 | |
| theorem actualResidual_graph_reached_removalND | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (B total : ℕ) (hW : ∀ L∈W, L.length≤B) {sig : ℝ} (hsig : 0<sig) | |
| (Cq : ℝ) (hCq : 0≤Cq) : | |
| ∃ D : ℝ, 1≤D ∧ ∀ᶠ N : ℕ in atTop, | |
| ∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ], | |
| ∀ (A : ℕ → ℝ) (q a t p h : ℕ), 0<q → p≤h → | |
| ∀ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) | |
| (m : ℝ → ℝ) (C : ℝ) (J i : ℕ) | |
| (f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (Q S : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))), | |
| Measurable f → Measurable g → (∀ x, ‖f x‖≤1) → (∀ x, ‖g x‖≤1) → | |
| (∀ b∈policyDescriptorList A N q a t, L₁ b∈W) → | |
| (∀ b∈policyDescriptorList A N q a t, L₂ b∈W) → | |
| (∑ b∈policyDescriptorList A N q a t, (L₁ b).length≤total) → | |
| (∑ b∈policyDescriptorList A N q a t, (L₂ b).length≤total) → | |
| (∀ w, ‖w‖=1 → ‖Q.1 w‖≤Cq) → (∀ w, ‖w‖=1 → ‖S.1 w‖≤Cq) → | |
| ∀ (X Y : Finset (Fin (n+1) → ℤ)) (hXY : (X.product Y).Nonempty) (M K v : ℝ) (Ψ : ℝ → ℝ), | |
| Integrable (fun r => Ψ (r/(2:ℝ)^v)) → (∀ r, 0≤Ψ r) → | |
| Attempt4Near.graphChildMaxND (n+1) (Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v Ψ X Y hXY | |
| (originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t p L₁ ρ m sig C J i f Q x).1)) | |
| (originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t p L₂ κ m sig C J i g S x).1)) ≤ | |
| D^(4*total)* | |
| Attempt4Near.graphChildMaxND (n+1) (Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v Ψ X Y hXY | |
| (originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1)) | |
| (originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1)) := by | |
| obtain ⟨D,hD,hDb⟩ := uniform_newlyReachedSymbol_bound W B total hW sig hsig | |
| obtain ⟨E,hE,hEb⟩ := actual_residual_transform_uniform_package W B total hW hsig Cq hCq | |
| refine ⟨D,hD,?_⟩ | |
| filter_upwards [hDb,hEb] with N hDb hEb | |
| intro ρ κ hρ hκ A q a t p h hq hph L₁ L₂ m C J i f g Q S hf hg hfb hgb hL₁ hL₂ htotal₁ htotal₂ hQ hS X Y hXY M K v Ψ hΨ hΨpos | |
| obtain ⟨hbm,hbn,hbt⟩ := hEb ρ A q a t h hq L₁ m C J i f Q hf hfb hL₁ htotal₁ hQ | |
| obtain ⟨hcm,hcn,hct⟩ := hEb κ A q a t h hq L₂ m C J i g S hg hgb hL₂ htotal₂ hS | |
| let ψ := fun X w => (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L₁ ρ m N sig C J i (CurrentFalconer.ambientGridCenter h X)).1 w | |
| let φ := fun Y w => (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L₂ κ m N sig C J i (CurrentFalconer.ambientGridCenter h Y)).1 w | |
| have hψ (X : Fin (n+1) → ℤ) : ∀ᵐ w ∂Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), ‖ψ X w‖≤D^total := by | |
| filter_upwards [Attempt4FullFar.canonicalSphereMeasureND_ae_norm] with w hw | |
| exact hDb A q a t p h hq L₁ ρ m C J i hL₁ htotal₁ _ w hw | |
| have hφ (Y : Fin (n+1) → ℤ) : ∀ᵐ w ∂Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), ‖φ Y w‖≤D^total := by | |
| filter_upwards [Attempt4FullFar.canonicalSphereMeasureND_ae_norm] with w hw | |
| exact hDb A q a t p h hq L₂ κ m C J i hL₂ htotal₂ _ w hw | |
| have hh := Attempt4Near.graphChildMaxND_bounded_factors (n+1) | |
| (Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v X Y hXY | |
| (originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1)) | |
| (originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1)) | |
| (fun X _ => (hbt (Attempt4FullFar.ambientGridCell h X) M).1) | |
| (fun Y _ => (hct (Attempt4FullFar.ambientGridCell h Y) K).1) | |
| (fun X => |M⁻¹| * (ρ (Attempt4FullFar.ambientGridCell h X)).toReal*E) | |
| (fun Y => |K⁻¹| * (κ (Attempt4FullFar.ambientGridCell h Y)).toReal*E) | |
| (fun X _ => (hbt (Attempt4FullFar.ambientGridCell h X) M).2) | |
| (fun Y _ => (hct (Attempt4FullFar.ambientGridCell h Y) K).2) | |
| ψ φ (fun X _ => hψ X) (fun Y _ => hφ Y) hΨ hΨpos | |
| have hpow : (D^total)^2*(D^total)^2=D^(4*total) := by | |
| rw [← pow_add,← pow_mul] | |
| congr 1 | |
| omega | |
| rw [hpow] at hh | |
| have hUeq : originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t p L₁ ρ m sig C J i f Q x).1) = | |
| fun X z => ψ X z.2 * originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1) X z := by | |
| funext X z | |
| exact actualResidualTransform_grid_factorization A N q a t p h hph L₁ ρ m sig C J i f Q X M z.1 z.2 | |
| have hWeq : originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t p L₂ κ m sig C J i g S x).1) = | |
| fun Y z => φ Y z.2 * originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1) Y z := by | |
| funext Y z | |
| exact actualResidualTransform_grid_factorization A N q a t p h hph L₂ κ m sig C J i g S Y K z.1 z.2 | |
| rw [hUeq,hWeq] | |
| exact hh | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |