| import Mathlib |
| import OAI.Geometry.DoublingHilbert.Main |
|
|
| namespace OAI |
|
|
| open Set Metric |
| noncomputable section |
| namespace CompactBanach |
| open DoublingHilbert |
|
|
| @[simp] theorem coordinate_eq_iff (j i m h : ℕ) : |
| coordinate j i = coordinate m h ↔ j = m ∧ i = h := by |
| constructor |
| · intro hh |
| exact Prod.mk.inj (coordinate_injective hh) |
| · rintro ⟨rfl, rfl⟩ |
|
|
| @[simp] theorem point_apply_zero (p : Base) (w : Tuple) : point p w 0 = p.1 := by |
| simp only [point, lp.coeFn_add, Pi.add_apply, baseVector, tupleVector, lp.coeFn_sum, |
| Finset.sum_apply, levelVector, lp.single_apply] |
| simp |
|
|
| @[simp] theorem point_apply_one (p : Base) (w : Tuple) : point p w 1 = p.2 := by |
| simp only [point, lp.coeFn_add, Pi.add_apply, baseVector, tupleVector, lp.coeFn_sum, |
| Finset.sum_apply, levelVector, lp.single_apply] |
| simp |
|
|
| theorem point_apply_coordinate (p : Base) (w : Tuple) (j i : ℕ) : |
| point p w (coordinate j i) = |
| if w j ≠ 0 ∧ w j - 1 = i then scale j else 0 := by |
| classical |
| simp only [point, lp.coeFn_add, Pi.add_apply, baseVector, tupleVector, lp.coeFn_sum, |
| Finset.sum_apply, levelVector] |
| simp only [lp.single_apply_ne _ _ _ (coordinate_ne_zero j i), |
| lp.single_apply_ne _ _ _ (coordinate_ne_one j i), zero_add] |
| simp [lp.single_apply, Pi.single_apply, coordinate_eq_iff, ite_and, eq_comm] |
|
|
| |
| theorem exists_coordinate_difference {p q : Base} {w w' : Tuple} {j : ℕ} |
| (hj : w j ≠ w' j) : |
| ∃ i, |point p w i - point q w' i| = scale j := by |
| by_cases hw : w j = 0 |
| · have hw' : w' j ≠ 0 := by omega |
| refine ⟨coordinate j (w' j - 1), ?_⟩ |
| simp [point_apply_coordinate, hw, hw', abs_of_pos (scale_pos j)] |
| · refine ⟨coordinate j (w j - 1), ?_⟩ |
| have hn : ¬ (w' j ≠ 0 ∧ w' j - 1 = w j - 1) := by omega |
| simp [point_apply_coordinate, hw, hn, abs_of_pos (scale_pos j)] |
| |
| noncomputable def pointSupport (w : Tuple) : Finset ℕ := |
| insert 0 (insert 1 (w.support.image fun j => coordinate j (w j - 1))) |
|
|
| theorem point_apply_of_not_mem_support (p : Base) (w : Tuple) {i : ℕ} |
| (hi : i ∉ pointSupport w) : point p w i = 0 := by |
| classical |
| have h0 : i ≠ 0 := by intro h |
| have h1 : i ≠ 1 := by intro h |
| have hj (j : ℕ) (hj : j ∈ w.support) : i ≠ coordinate j (w j - 1) := by |
| intro he |
| apply hi |
| apply Finset.mem_insert_of_mem |
| apply Finset.mem_insert_of_mem |
| exact Finset.mem_image.mpr ⟨j, hj, he.symm⟩ |
| simp only [point, lp.coeFn_add, Pi.add_apply, baseVector, tupleVector, lp.coeFn_sum, |
| Finset.sum_apply, levelVector] |
| simp only [lp.single_apply_ne _ _ _ h0, lp.single_apply_ne _ _ _ h1, zero_add] |
| exact Finset.sum_eq_zero fun level hlevel => lp.single_apply_ne _ _ _ (hj level hlevel) |
|
|
| theorem finite_coordinate_support (X : Finset RealL2) |
| (hX : ∀ z ∈ X, z ∈ constructedSet) : |
| ∃ I : Finset ℕ, ∀ z ∈ X, ∀ i ∉ I, z i = 0 := by |
| classical |
| have h : ∀ z : X, ∃ I : Finset ℕ, ∀ i ∉ I, (z : RealL2) i = 0 := by |
| intro z |
| obtain ⟨p, w, _, he⟩ := hX z z.property |
| refine ⟨pointSupport w, ?_⟩ |
| intro i hi |
| rw [he] |
| exact point_apply_of_not_mem_support p w hi |
| choose I hI using h |
| refine ⟨Finset.univ.biUnion I, ?_⟩ |
| intro z hz i hi |
| apply hI ⟨z, hz⟩ i |
| intro hmem |
| exact hi (Finset.mem_biUnion.mpr ⟨⟨z, hz⟩, Finset.mem_univ _, hmem⟩) |
|
|
| end CompactBanach |
| end |
|
|
| end OAI |
|
|
