| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SuffixDescendants | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveEnvironment | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.RecursiveUnit | |
| 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 MeasureTheory Filter PlanarFalconer | |
| open scoped ContDiff Classical | |
| variable {n : ℕ} | |
| noncomputable def recursiveLoss (n : ℕ) (C : ℝ) : ℝ := |C|+100*(n+2:ℝ) | |
| lemma recursiveLoss_pos (n : ℕ) (C : ℝ) : 0<recursiveLoss n C := by | |
| unfold recursiveLoss | |
| positivity | |
| lemma recursiveLoss_graph (n : ℕ) (C : ℝ) : C+36*(n+1:ℝ)+5≤2*recursiveLoss n C := by | |
| unfold recursiveLoss | |
| nlinarith [le_abs_self C,abs_nonneg C,(Nat.cast_nonneg n : (0:ℝ)≤(n:ℝ))] | |
| lemma recursiveLoss_dim (n : ℕ) (C : ℝ) : 4*(n+2:ℝ)≤2*recursiveLoss n C := by | |
| unfold recursiveLoss | |
| nlinarith [abs_nonneg C,(Nat.cast_nonneg n : (0:ℝ)≤(n:ℝ))] | |
| lemma recursiveLoss_test (n : ℕ) (C : ℝ) : C≤2*recursiveLoss n C := by | |
| unfold recursiveLoss | |
| nlinarith [le_abs_self C,abs_nonneg C,(Nat.cast_nonneg n : (0:ℝ)≤(n:ℝ))] | |
| lemma weighted_child_cap {N : ℕ} {a b x M : ℝ} (hx : 0≤x) (hab : a≤b) | |
| (h : ((2:ℝ)^N)^b*x≤M) : ((2:ℝ)^N)^a*x≤M := | |
| (mul_le_mul_of_nonneg_right (Real.rpow_le_rpow_of_exponent_le | |
| (one_le_pow₀ (by norm_num)) hab) hx).trans h | |
| lemma farSeparated_centre {a : ℕ} {K : ℝ} {X Y : CellPath (n+1)} | |
| (hX : X.length=a) (hY : Y.length=a) | |
| (h : Attempt4FullFar.farSeparatedGridND a K (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ))) : | |
| CentreSeparated X Y a K := by | |
| unfold CentreSeparated treeCenter | |
| rw [hX,hY] | |
| change K/8/(2:ℝ)^a≤_ ∧ _≤_ | |
| simpa only [Attempt4FullFar.farSeparatedGridND,div_div,ambientGridCenter,Attempt4FullFar.ambientGridCenter] using h | |
| lemma RecursiveEnvironmentND.distance_mem (e : RecursiveEnvironmentND n) (K : ℝ) | |
| (W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))) | |
| (total : ℕ) (Ψ : ℝ → ℝ) (a t i : ℕ) (X Y : CellPath (n+1)) | |
| (hX : X∈dyadicPaths a) (hY : Y∈dyadicPaths a) | |
| (hox : 0<e.ρ (treeCell X)) (hoy : 0<e.ρ (treeCell Y)) | |
| (hsep : Attempt4FullFar.farSeparatedGridND a K (fun j => (pathIndex X j:ℤ)) (fun j => (pathIndex Y j:ℤ))) : | |
| e.energies K W total Ψ (.D a t i) ((e.distanceData a t i X Y).coincidence.toReal) := | |
| ⟨X,Y,hX,hY,hox,hoy,hsep,rfl⟩ | |
| end GraphLiteralSupport.CurrentFalconer.TreeND | |
| end | |
| end | |
| end OAI | |