| import OAI.Analysis.C0Absorption.Detection |
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| namespace OAI |
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| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
| open NormedSpace |
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| namespace C0Absorption |
| noncomputable section |
| open Set Filter Topology |
| open scoped NNReal BigOperators ZeroAtInfty |
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| abbrev BandGrid (j : ℕ) := Fin j → GridIndex (2^j) |
| abbrev PrefixCube (j : ℕ) := Fin j → Icc (-1 : ℝ) 1 |
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| def bandPoint (j : ℕ) (ξ : BandGrid j) : PrefixCube j := |
| fun k => gridPoint (2^j) (by positivity) (ξ k) |
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| def prefixRestrict (j : ℕ) (s : C0Ball) : PrefixCube j := fun k => |
| ⟨s.val k, abs_le.mp ((c0_norm_apply_le s.val k).trans (by |
| have hh := s.property |
| simpa only [Metric.mem_closedBall,dist_zero_right] using hh))⟩ |
|
|
| theorem prefixRestrict_lipschitz (j : ℕ) : LipschitzWith 1 (prefixRestrict j) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| rw [NNReal.coe_one,one_mul] |
| apply (dist_pi_le_iff dist_nonneg).mpr |
| intro k |
| exact BoundedContinuousFunction.dist_coe_le_dist (f := x.val.toBCF) (g := y.val.toBCF) k |
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| def bandBump (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ := |
| max 0 (1-dist s (bandPoint j ξ)/dyadic j) |
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| def bandVector (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ := |
| min 1 (max 0 (2-dist s (bandPoint j ξ)/dyadic j)) |
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| def bandDenominator (j : ℕ) (s : PrefixCube j) : ℝ := ∑ ξ : BandGrid j, bandBump j ξ s |
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| def bandRow (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ := |
| bandBump j ξ s / bandDenominator j s |
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| theorem bandBump_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : |
| 0 ≤ bandBump j ξ s := le_max_left _ _ |
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| theorem bandVector_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : |
| 0 ≤ bandVector j ξ s := le_min zero_le_one (le_max_left _ _) |
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| theorem bandVector_le_one (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : |
| bandVector j ξ s ≤ 1 := min_le_left _ _ |
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| theorem bandVector_eq_one {j : ℕ} {ξ : BandGrid j} {s : PrefixCube j} |
| (h : bandBump j ξ s ≠ 0) : bandVector j ξ s=1 := by |
| have hh : 0 < 1-dist s (bandPoint j ξ)/dyadic j := by |
| by_contra hn |
| exact h (max_eq_left (by linarith)) |
| unfold bandVector |
| rw [min_eq_left] |
| exact le_max_of_le_right (by linarith) |
|
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| theorem bandVector_support {j : ℕ} {ξ : BandGrid j} {s : PrefixCube j} |
| (h : bandVector j ξ s ≠ 0) : dist s (bandPoint j ξ) < 2*dyadic j := by |
| have hmax : max 0 (2-dist s (bandPoint j ξ)/dyadic j) ≠ 0 := by |
| intro hh |
| exact h (by simp only [bandVector,hh,min_eq_right zero_le_one]) |
| have hh : 0 < 2-dist s (bandPoint j ξ)/dyadic j := by |
| by_contra hn |
| exact hmax (max_eq_left (by linarith)) |
| exact (div_lt_iff₀ (dyadic_pos j)).mp (by linarith) |
|
|
| theorem const_dist_div_lipschitz {X : Type*} [PseudoMetricSpace X] (q : X) (a : ℝ) (j : ℕ) : |
| LipschitzWith (2^j) (fun s => a-dist s q/dyadic j) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro s t |
| rw [Real.dist_eq,sub_sub_sub_cancel_left,← sub_div,abs_div,abs_of_pos (dyadic_pos j)] |
| calc |
| _ ≤ dist t s/dyadic j := div_le_div_of_nonneg_right (abs_dist_sub_le t s q) (dyadic_pos j).le |
| _ = (2^j : ℝ≥0)*dist s t := by simp only [dyadic,div_inv_eq_mul,NNReal.coe_pow,NNReal.coe_ofNat,dist_comm]; ring |
|
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| theorem bandBump_lipschitz (j : ℕ) (ξ : BandGrid j) : |
| LipschitzWith (2^j) (bandBump j ξ) := |
| (const_dist_div_lipschitz (bandPoint j ξ) 1 j).const_max 0 |
|
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| theorem bandVector_lipschitz (j : ℕ) (ξ : BandGrid j) : |
| LipschitzWith (2^j) (bandVector j ξ) := |
| ((const_dist_div_lipschitz (bandPoint j ξ) 2 j).const_max 0).const_min 1 |
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| theorem bandDenominator_ge (j : ℕ) (s : PrefixCube j) : 1/2 ≤ bandDenominator j s := by |
| classical |
| let ξ : BandGrid j := fun k => nearestGridIndex (2^j) (by positivity) (s k) |
| have hd : dist s (bandPoint j ξ) ≤ dyadic j/2 := by |
| apply (dist_pi_le_iff (div_nonneg (dyadic_pos j).le (by norm_num) : 0 ≤ dyadic j/2)).mpr |
| intro k |
| have hh := nearestGrid_dist (2^j) (by positivity) (s k) |
| change dist (s k) (gridPoint _ _ (nearestGridIndex _ _ (s k))) ≤ _ |
| refine hh.trans_eq ?_ |
| push_cast |
| unfold dyadic |
| ring |
| have hb : 1/2 ≤ bandBump j ξ s := by |
| apply le_max_of_le_right |
| have hh : dist s (bandPoint j ξ)/dyadic j ≤ 1/2 := (div_le_iff₀ (dyadic_pos j)).mpr (by linarith) |
| linarith |
| exact hb.trans (Finset.single_le_sum (fun a _ => bandBump_nonneg j a s) (Finset.mem_univ ξ)) |
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| theorem bandDenominator_pos (j : ℕ) (s : PrefixCube j) : 0 < bandDenominator j s := |
| lt_of_lt_of_le (by norm_num) (bandDenominator_ge j s) |
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| theorem bandRow_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : |
| 0 ≤ bandRow j ξ s := div_nonneg (bandBump_nonneg j ξ s) (bandDenominator_pos j s).le |
|
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| theorem bandRow_sum (j : ℕ) (s : PrefixCube j) : ∑ ξ : BandGrid j, bandRow j ξ s=1 := by |
| simp only [bandRow,← Finset.sum_div] |
| change bandDenominator j s/bandDenominator j s=1 |
| exact div_self (bandDenominator_pos j s).ne' |
|
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| theorem bandRow_vector (j : ℕ) (s : PrefixCube j) : |
| ∑ ξ : BandGrid j, bandRow j ξ s*bandVector j ξ s=1 := by |
| rw [← bandRow_sum j s] |
| apply Finset.sum_congr rfl |
| intro ξ _ |
| by_cases h : bandBump j ξ s=0 |
| · simp only [bandRow,h,zero_div,zero_mul] |
| · rw [bandVector_eq_one h,mul_one] |
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| theorem normalized_sum_distance {A : Type*} [Fintype A] (b c : A → ℝ) |
| (_hb : ∀ i, 0 ≤ b i) (hc : ∀ i, 0 ≤ c i) |
| (hB : 1/2 ≤ ∑ i,b i) (hC : 1/2 ≤ ∑ i,c i) : |
| (∑ i, |b i/(∑ i,b i)-c i/(∑ i,c i)|) ≤ 4*∑ i,|b i-c i| := by |
| let B := ∑ i,b i |
| let C := ∑ i,c i |
| have hB' : 0 < B := lt_of_lt_of_le (by norm_num) hB |
| have hC' : 0 < C := lt_of_lt_of_le (by norm_num) hC |
| have hdist : |C-B| ≤ ∑ i,|b i-c i| := by |
| rw [abs_sub_comm] |
| change |(∑ i,b i)-(∑ i,c i)| ≤ _ |
| rw [← Finset.sum_sub_distrib] |
| exact Finset.abs_sum_le_sum_abs _ _ |
| have hrow : ∑ i,c i/C=1 := by rw [← Finset.sum_div]; exact div_self hC'.ne' |
| have he (i : A) : b i/B-c i/C=(b i-c i)/B+(c i/C)*((C-B)/B) := by |
| field_simp |
| ; ring |
| calc |
| _ ≤ ∑ i, (|b i-c i|/B+(c i/C)*(|C-B|/B)) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| rw [he] |
| refine (abs_add_le _ _).trans_eq ?_ |
| simp only [abs_mul,abs_div,abs_of_pos hB',abs_of_nonneg (hc i),abs_of_pos hC'] |
| _ = (∑ i,|b i-c i|)/B+|C-B|/B := by |
| rw [Finset.sum_add_distrib,← Finset.sum_div,← Finset.sum_mul,hrow,one_mul] |
| _ ≤ (∑ i,|b i-c i|)/B+(∑ i,|b i-c i|)/B := |
| add_le_add_right ((div_le_div_iff_of_pos_right hB').mpr hdist) _ |
| _ ≤ 4*∑ i,|b i-c i| := by |
| have hh : (∑ i,|b i-c i|)/B ≤ 2*∑ i,|b i-c i| := by |
| apply (div_le_iff₀ hB').mpr |
| have ha : 0 ≤ ∑ i,|b i-c i| := Finset.sum_nonneg (fun i _ => abs_nonneg _) |
| nlinarith |
| linarith |
|
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| def bandL (j : ℕ) : ℝ≥0 := 4*Fintype.card (BandGrid j)*2^j |
|
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| theorem bandRow_distance (j : ℕ) (s t : PrefixCube j) : |
| (∑ ξ : BandGrid j,|bandRow j ξ s-bandRow j ξ t|) ≤ bandL j*dist s t := by |
| refine (normalized_sum_distance (fun ξ => bandBump j ξ s) (fun ξ => bandBump j ξ t) |
| (fun ξ => bandBump_nonneg j ξ s) (fun ξ => bandBump_nonneg j ξ t) |
| (bandDenominator_ge j s) (bandDenominator_ge j t)).trans ?_ |
| calc |
| _ ≤ 4*∑ ξ : BandGrid j, (2 : ℝ)^j*dist s t := by |
| gcongr with ξ |
| simpa only [Real.dist_eq,NNReal.coe_pow,NNReal.coe_ofNat] using |
| (bandBump_lipschitz j ξ).dist_le_mul s t |
| _ = bandL j*dist s t := by simp [bandL]; ring |
|
|
| end |
| end C0Absorption |
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| end OAI |
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