openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Estimates /SourceWeightedErrorGeneral.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Estimates.ActualDirectBuffer | |
| import OAI.MeasureTheory.Falconer.Estimates.ActualOuterMasks | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Recursion.BranchCounts | |
| import OAI.MeasureTheory.Falconer.Recursion.BranchTransitions | |
| import OAI.MeasureTheory.Falconer.Recursion.SourceBranchDepth | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SuffixDescendants | |
| import OAI.MeasureTheory.FalconerCompletion.Recursion.Extend | |
| namespace OAI | |
| /-! Quantitative estimates for the distance-set construction. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport.CurrentFalconer.DimensionalPolicy | |
| open PlanarFalconer | |
| theorem uniform_source_weighted_error_generalND (offset : ℕ) {sig kap T eps L C B : ℝ} | |
| (hsig : 0<sig) (hkap : 0<kap) (hkap1 : kap≤1) (hgap : sig<kap/1000) | |
| (hL : 0≤L) (hC : 0≤C) (heps : eps≤1) | |
| (hT : (3*L+C*(sig+sig)+1)*(20/kap+1/sig)+B≤T) : | |
| ∃ N₀ : ℕ,0<N₀ ∧ ∀ N≥N₀,∀ A : ℕ → ℝ,ProfileLipschitzND A N L → | |
| ∀ j : ℕ,∀ b : PolicyBranch A N (sourcePolicyBlock N kap) offset sig j, | |
| b.weight C sig eps*((2:ℝ)^N)^(-T)≤((2:ℝ)^N)^(-B) := by | |
| obtain ⟨N₀,hN₀,h₀⟩ := uniform_source_ceiling_slack hsig hgap | |
| refine ⟨N₀,hN₀,?_⟩ | |
| intro N hN A hA j b | |
| have hNp : 0<N := by omega | |
| have hq := sourcePolicyBlock_pos hNp hkap | |
| have hn := (source_branch_count_bounds hNp hsig hkap hkap1 (h₀ N hN).1 b).2.2 | |
| have hb : b.exponentSum C sig eps≤(j:ℝ)*(3*L+C*(sig+sig)+eps) := by | |
| unfold PolicyBranch.exponentSum | |
| have hterm : ∀ k∈Finset.range j, | |
| branchExponentND A C sig sig eps (b.state k) (b.label k) (b.state (k+1))≤3*L+C*(sig+sig)+eps := by | |
| intro k hk | |
| have hk := Finset.mem_range.mp hk | |
| have hv := b.valid k hk.le | |
| have hu := b.valid (k+1) (by omega) | |
| have hc := (b.step k hk).cost_le hA hNp hq hL hv hu | |
| have hsd := BranchState.valid_depths hv | |
| have hud := BranchState.valid_depths hu | |
| have hi := abs_le.mp (hA.increment_le hNp hL (hud.1.trans hud.2) (hsd.1.trans hsd.2)) | |
| have hloss : 0≤C*(sig+sig) := by positivity | |
| unfold branchExponentND | |
| split_ifs <;> linarith | |
| simpa only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] using Finset.sum_le_sum hterm | |
| have hexp : b.exponentSum C sig eps-T≤-B := by | |
| have h₁ := mul_le_mul_of_nonneg_left hn (show 0≤3*L+C*(sig+sig)+1 by positivity) | |
| have h₂ := mul_le_mul_of_nonneg_left heps (Nat.cast_nonneg (α:=ℝ) j) | |
| nlinarith only [hb,h₁,h₂,hT] | |
| rw [PolicyBranch.weight,←Real.rpow_add (by positivity : 0<(2:ℝ)^N)] | |
| exact Real.rpow_le_rpow_of_exponent_le (one_le_pow₀ (by norm_num)) (by simpa only [sub_eq_add_neg] using hexp) | |
| end GraphLiteralSupport.CurrentFalconer.DimensionalPolicy | |
| end | |
| end | |
| end OAI | |