openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /ResidualAngularTerminal.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularHTrivialBound | |
| 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 _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ContDiff ENNReal Topology | |
| variable {n : ℕ} | |
| theorem actual_residual_angular_terminal | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) | |
| (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) : | |
| ∃ H : ℝ, 0≤H ∧ ∀ᶠ N : ℕ in atTop, | |
| ∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ], | |
| ∀ (A : ℕ → ℝ) (q a₀ t₀ a : ℕ), 0<q → 0<N → a≤N → | |
| ∀ (words₁ words₂ : 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 P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))) | |
| (X Y : CellPath (n+1)) (v lam sρ sκ : ℝ) (Bρ Bκ : ℕ), | |
| RegularizedProfileData ρ N Bρ sρ lam m → RegularizedProfileData κ N Bκ sκ lam m → | |
| X.length=a → Y.length=a → v-a≤100*(q:ℝ) → | |
| Measurable f → Measurable g → (∀ x, ‖f x‖≤1) → (∀ x, ‖g x‖≤1) → | |
| (∀ b∈policyDescriptorList A N q a₀ t₀, words₁ b∈W) → | |
| (∀ b∈policyDescriptorList A N q a₀ t₀, words₂ b∈W) → | |
| (∑ b∈policyDescriptorList A N q a₀ t₀, (words₁ b).length≤total) → | |
| (∑ b∈policyDescriptorList A N q a₀ t₀, (words₂ b).length≤total) → | |
| (∀ w, ‖w‖=1 → ‖Q.1 w‖≤Cq) → (∀ w, ‖w‖=1 → ‖P.1 w‖≤Cq) → | |
| let b := fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₁ ρ m sig C J i f Q x).1 w | |
| let c := fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₂ κ m sig C J i g P x).1 w | |
| Attempt4Near.sourceAngularHND (n+1) (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) a v Ψ | |
| (fun p => sourceAngularTransformND ρ (treeCell X) (((2:ℝ)^N)^(-m a)) b p.1 p.2) | |
| (fun p => sourceAngularTransformND κ (treeCell Y) (((2:ℝ)^N)^(-m a)) c p.1 p.2) ≤ | |
| H*((2:ℝ)^N)^(100*(n+1:ℝ)*q/N+4*(n+2:ℝ)*lam) := by | |
| obtain ⟨E,hE,he⟩ := actual_residual_transform_uniform_package W B total hW hsig Cq hCq | |
| let μ := canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))) | |
| let I : ℝ := ∫ r, |Ψ r| | |
| let H := I*E^4*(μ.real univ)^2 | |
| have hI : 0≤I := integral_nonneg (fun _ => abs_nonneg _) | |
| refine ⟨H,by dsimp [H]; positivity,?_⟩ | |
| filter_upwards [he] with N he | |
| intro ρ κ hρ hκ A q a₀ t₀ a hq hN ha words₁ words₂ m C J i f g Q P X Y v lam sρ sκ Bρ Bκ | |
| hdρ hdκ hX hY hv hfm hgm hfb hgb hw₁ hw₂ ht₁ ht₂ hQ hP | |
| let R : ℝ := (2:ℝ)^N | |
| let S : ℝ := R^((n+2:ℝ)*lam) | |
| have hS : 0≤S := by dsimp [S]; positivity | |
| have hu := (he ρ A q a₀ t₀ a hq words₁ m C J i f Q hfm hfb hw₁ ht₁ hQ).2.2 (treeCell X) (R^(-m a)) | |
| have hw := (he κ A q a₀ t₀ a hq words₂ m C J i g P hgm hgb hw₂ ht₂ hP).2.2 (treeCell Y) (R^(-m a)) | |
| have hmass₁ : |(R^(-m a))⁻¹| *(ρ (treeCell X)).toReal*E ≤ S*E := by | |
| rw [abs_of_pos (inv_pos.mpr (Real.rpow_pos_of_pos (by dsimp [R]; positivity) _))] | |
| apply mul_le_mul_of_nonneg_right _ hE | |
| rw [mul_comm,← div_eq_mul_inv] | |
| simpa only [R,S,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using hdρ.normalized_cell_mass_le ha hX | |
| have hmass₂ : |(R^(-m a))⁻¹| *(κ (treeCell Y)).toReal*E ≤ S*E := by | |
| rw [abs_of_pos (inv_pos.mpr (Real.rpow_pos_of_pos (by dsimp [R]; positivity) _))] | |
| apply mul_le_mul_of_nonneg_right _ hE | |
| rw [mul_comm,← div_eq_mul_inv] | |
| simpa only [R,S,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using hdκ.normalized_cell_mass_le ha hY | |
| have hb₁ r : ∀ᵐ w ∂μ, ‖sourceAngularTransformND ρ (treeCell X) (R^(-m a)) | |
| (fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₁ ρ m sig C J i f Q x).1 w) r w‖≤S*E := | |
| (hu.2 r).mono (fun _ hw => hw.trans hmass₁) | |
| have hb₂ r : ∀ᵐ w ∂μ, ‖sourceAngularTransformND κ (treeCell Y) (R^(-m a)) | |
| (fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₂ κ m sig C J i g P x).1 w) r w‖≤S*E := | |
| (hw.2 r).mono (fun _ hw => hw.trans hmass₂) | |
| have hh := Attempt4Near.sourceAngularHND_trivial_bound (n+1) μ a v hΨ hu.1 hw.1 hb₁ hb₂ | |
| have hS4 : S^4=R^(4*(n+2:ℝ)*lam) := by | |
| rw [← Real.rpow_mul_natCast (by dsimp [R]; positivity : 0≤R)] | |
| congr 1 | |
| norm_num | |
| ring | |
| have hcoeff : (2:ℝ)^((↑(n+1):ℝ)*(v-a))*I*((S*E)^2*μ.real univ)*((S*E)^2*μ.real univ) = | |
| H*((2:ℝ)^((n+1:ℝ)*(v-a))*R^(4*(n+2:ℝ)*lam)) := by | |
| rw [← hS4] | |
| dsimp [H] | |
| push_cast | |
| ring | |
| change _ ≤ _ at hh | |
| rw [hcoeff] at hh | |
| apply hh.trans | |
| apply mul_le_mul_of_nonneg_left _ (by dsimp [H]; positivity) | |
| have hexp : (2:ℝ)^((n+1:ℝ)*(v-a)) ≤ R^(100*(n+1:ℝ)*q/N) := by | |
| dsimp [R] | |
| rw [← Real.rpow_natCast,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)] | |
| have hn : (N:ℝ)≠0 := by exact_mod_cast (Nat.ne_of_gt hN) | |
| have heq : (N:ℝ)*(100*(n+1:ℝ)*q/N)=100*(n+1:ℝ)*q := by field_simp | |
| rw [heq] | |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) | |
| nlinarith [mul_le_mul_of_nonneg_left hv (show 0≤(n+1:ℝ) by positivity)] | |
| rw [Real.rpow_add (by dsimp [R]; positivity : 0<R)] | |
| exact mul_le_mul_of_nonneg_right hexp (by positivity) | |
| end CurrentFalconer.TreeND | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |