| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.History |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
|
|
| namespace RepeatedLocalization |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
|
|
| theorem repeated_localization : MainStatement := by |
| refine ⟨1/2,by norm_num,32,?_⟩ |
| intro C₀ hC₀ |
| let A := blockInformationConstant+C₀ |
| let K := 2*Real.log 2+1 |
| let D := A+K |
| let C := 16*D |
| have hA : 0 < A := add_pos blockInformationConstant_pos hC₀ |
| have hK : 0 < K := by dsimp [K]; positivity |
| have hD : 0 < D := add_pos hA hK |
| have hC : 0 < C := by dsimp [C]; positivity |
| refine ⟨C,hC,?_⟩ |
| intro n N t hn hN rules halphabet |
| have hn0 : 0 < n := by omega |
| have hnR : (1:ℝ) ≤ n := by exact_mod_cast (show 1 ≤ n by omega) |
| let L := A*(n+1) |
| have hc : UniformBlockCost n N L := actual_uniform_blockCost hn hN halphabet |
| obtain ⟨s,f,hf,hs,⟨R⟩⟩ := history_analysis_realized hn0 rules L hc |
| have hi := s.information_of_potential (show 8 ≤ n by omega) hs |
| have hLK : L+2*Real.log 2+1 ≤ D*(n+1) := by |
| dsimp [L,D,K] |
| have hnnonneg : (0:ℝ) ≤ n := Nat.cast_nonneg n |
| nlinarith [mul_nonneg hK.le hnnonneg] |
| have hinfo : s.information ≤ 4*(t:ℝ)*D*(n+1) := by |
| have H := mul_le_mul_of_nonneg_left hLK (by positivity : 0 ≤ 4*(t:ℝ)) |
| nlinarith |
| have hDt : 0 ≤ D*(t:ℝ) := mul_nonneg hD.le (Nat.cast_nonneg t) |
| have hfinal : s.information ≤ C*(n+1)*t := by |
| apply hinfo.trans |
| dsimp [C] |
| nlinarith [mul_nonneg hDt (show 0 ≤ (n:ℝ)+1 by positivity)] |
| refine ⟨R.Q,R.valid,?_,?_⟩ |
| · exact (R.information_le hn0 rules hf).trans (ENNReal.ofReal_le_ofReal hfinal) |
| · rw [R.level_eq hn0 rules hf] |
| have hhalf : (1/2:ℝ) ≤ Real.log 2 := by |
| have H := Real.one_sub_inv_le_log_of_pos (show (0:ℝ) < 2 by norm_num) |
| norm_num at H ⊢ |
| exact H |
| have hd := s.depth_nonneg |
| have hl := s.information_lower hn0 |
| have hsmall : (n:ℝ)/2*s.depth ≤ s.information := by |
| have H := mul_le_mul_of_nonneg_right hhalf (mul_nonneg (Nat.cast_nonneg n) hd) |
| nlinarith |
| apply le_of_mul_le_mul_left (a := (n:ℝ)/2) (a0 := by positivity) |
| calc |
| (n:ℝ)/2*s.depth ≤ s.information := hsmall |
| _ ≤ 4*(t:ℝ)*D*(n+1) := hinfo |
| _ ≤ (n:ℝ)/2*(C*t) := by |
| dsimp [C] |
| nlinarith [mul_nonneg hDt (sub_nonneg.mpr hnR)] |
| end RepeatedLocalization |
|
|
| end OAI |
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|
