| import Mathlib |
| import OAI.Geometry.DoublingHilbert.Main |
|
|
| namespace OAI |
|
|
| open Set Metric |
| noncomputable section |
| namespace CompactBanach |
| open DoublingHilbert |
|
|
| theorem sum_scale_le {s : Finset ℕ} {R : ℝ} (hR : 0 ≤ R) |
| (hs : ∀ j ∈ s, scale j ≤ R) : ∑ j ∈ s, scale j ≤ 2 * R := by |
| classical |
| by_cases hse : s.Nonempty |
| · let m := s.min' hse |
| have hm : m ∈ s := s.min'_mem hse |
| have hmj (j : ℕ) (hj : j ∈ s) : m ≤ j := s.min'_le j hj |
| have hgeom := (summable_geometric_of_abs_lt_one (r := (1000 : ℝ)⁻¹) (by norm_num)).sum_le_tsum |
| (s.image (fun j => j - m)) (fun _ _ => by positivity) |
| rw [tsum_geometric_of_abs_lt_one (by norm_num : |(1000 : ℝ)⁻¹| < 1)] at hgeom |
| have himage : (∑ j ∈ s.image (fun j => j - m), (1000 : ℝ)⁻¹ ^ j) = |
| ∑ j ∈ s, (1000 : ℝ)⁻¹ ^ (j - m) := by |
| apply Finset.sum_image |
| intro i hi j hj he |
| dsimp at he |
| have := hmj i hi |
| have := hmj j hj |
| omega |
| rw [himage] at hgeom |
| have heq : (∑ j ∈ s, scale j) = scale m * |
| ∑ j ∈ s, (1000 : ℝ)⁻¹ ^ (j - m) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j hj |
| have hj' : m + (j - m) = j := Nat.add_sub_of_le (hmj j hj) |
| conv_lhs => rw [← hj', scale_add] |
| rw [heq] |
| calc |
| _ ≤ scale m * (1 - (1000 : ℝ)⁻¹)⁻¹ := |
| mul_le_mul_of_nonneg_left hgeom (scale_pos m).le |
| _ ≤ 2 * scale m := by nlinarith [scale_pos m] |
| _ ≤ 2 * R := by linarith [hs m hm] |
| · simp only [Finset.not_nonempty_iff_eq_empty.mp hse, Finset.sum_empty] |
| positivity |
|
|
| |
| theorem norm_map_point_sub_le {B : Type*} [NormedAddCommGroup B] [NormedSpace ℝ B] |
| (T : RealL2 →ₗ[ℝ] B) (hT : ∀ i a, ‖T (lp.single 2 i a)‖ ≤ |a|) |
| (p q : Base) (w w' : Tuple) {R : ℝ} (hR : 0 ≤ R) |
| (hf : |p.1 - q.1| ≤ R) (hs : |p.2 - q.2| ≤ R) |
| (hw : ∀ j, w j ≠ w' j → scale j ≤ R) : |
| ‖T (point p w - point q w')‖ ≤ 6 * R := by |
| classical |
| let s := w.support ∪ w'.support |
| let d := s.filter fun j => w j ≠ w' j |
| have heq : point p w - point q w' = baseVector (p - q) + |
| ∑ j ∈ d, (entryVector w j - entryVector w' j) := by |
| have hsum : (∑ j ∈ d, (entryVector w j - entryVector w' j)) = |
| ∑ j ∈ s, (entryVector w j - entryVector w' j) := by |
| apply Finset.sum_subset (Finset.filter_subset _ _) |
| intro j hj hnot |
| have he : w j = w' j := by simpa [d, hj] using hnot |
| simp [entryVector, he] |
| rw [hsum, Finset.sum_sub_distrib, |
| ← tupleVector_sum_on w s Finset.subset_union_left, |
| ← tupleVector_sum_on w' s Finset.subset_union_right, ← baseVector_sub] |
| simp only [point] |
| abel |
| have hb : ‖T (baseVector (p - q))‖ ≤ 2 * R := by |
| simp only [baseVector, map_add, Prod.fst_sub, Prod.snd_sub] |
| exact (norm_add_le _ _).trans (by linarith [hT 0 (p.1 - q.1), hT 1 (p.2 - q.2)]) |
| have he (u : Tuple) (j : ℕ) : ‖T (entryVector u j)‖ ≤ scale j := by |
| by_cases hu : u j = 0 |
| · simp [hu, (scale_pos j).le] |
| · simpa only [entryVector_of_ne_zero hu, levelVector, abs_of_pos (scale_pos j)] using |
| hT (coordinate j (u j - 1)) (scale j) |
| have hd : ‖T (∑ j ∈ d, (entryVector w j - entryVector w' j))‖ ≤ 4 * R := by |
| rw [map_sum] |
| calc |
| _ ≤ ∑ j ∈ d, ‖T (entryVector w j - entryVector w' j)‖ := norm_sum_le _ _ |
| _ ≤ ∑ j ∈ d, 2 * scale j := Finset.sum_le_sum fun j _ => by |
| rw [map_sub] |
| exact (norm_sub_le _ _).trans (by linarith [he w j, he w' j]) |
| _ = 2 * ∑ j ∈ d, scale j := by rw [Finset.mul_sum] |
| _ ≤ 4 * R := by |
| have ht := sum_scale_le (s := d) hR (fun j hj => hw j (Finset.mem_filter.mp hj).2) |
| linarith |
| rw [heq, map_add] |
| exact (norm_add_le _ _).trans (by linarith) |
|
|
| theorem norm_map_point_sub_le_dist {B : Type*} [NormedAddCommGroup B] [NormedSpace ℝ B] |
| (T : RealL2 →ₗ[ℝ] B) (hT : ∀ i a, ‖T (lp.single 2 i a)‖ ≤ |a|) |
| (p q : Base) (w w' : Tuple) : |
| ‖T (point p w - point q w')‖ ≤ 6 * dist (point p w) (point q w') := |
| norm_map_point_sub_le T hT p q w w' dist_nonneg |
| (abs_fst_sub_le_dist p q w w') (abs_snd_sub_le_dist p q w w') |
| (fun j hj => (scale_le_norm_entryVector_sub hj).trans (norm_entryVector_sub_le_dist p q w w' j)) |
|
|
| end CompactBanach |
| end |
|
|
| end OAI |
|
|
