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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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⟩; 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; subst i; simp [pointSupport] at hi
have h1 : i ≠ 1 := by intro h; subst i; simp [pointSupport] at hi
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
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