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8e4d533 | 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 | import Mathlib
namespace OAI
namespace BorsukNine
open Set MeasureTheory
open scoped Topology
section NullImage
variable {D T : Type*} [NormedAddCommGroup D] [NormedSpace ℝ D]
[NormedAddCommGroup T] [NormedSpace ℝ T] [FiniteDimensional ℝ T]
[MeasurableSpace T] [BorelSpace T]
lemma differentiable_image_null_of_projection (μ : Measure T) [μ.IsAddHaarMeasure]
(p : T →L[ℝ] D) (hp : Function.Surjective p) (hpn : ¬ Function.Injective p)
{f : D → T} {S : Set D} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x) : μ (f '' S) = 0 := by
let g : T → T := fun x => f (p x)
let g' : T → T →L[ℝ] T := fun x => (fderiv ℝ f (p x)).comp p
have hd : ∀ x ∈ p ⁻¹' S, HasFDerivWithinAt g (g' x) (p ⁻¹' S) x := by
intro x hx
exact ((hf (p x) hx).hasFDerivAt.comp x p.hasFDerivAt).hasFDerivWithinAt
have hdet : ∀ x ∈ p ⁻¹' S, (g' x).det = 0 := by
intro x _
change ((g' x).toLinearMap).det = 0
apply LinearMap.det_eq_zero_iff_ker_ne_bot.mpr
apply mt LinearMap.ker_eq_bot.mp
intro hinj
exact hpn (fun u v huv => hinj (by change fderiv ℝ f (p x) (p u) = fderiv ℝ f (p x) (p v); rw [huv]))
have he : g '' (p ⁻¹' S) = f '' S := by
ext y
constructor
· rintro ⟨x,hx,rfl⟩
exact ⟨p x,hx,rfl⟩
· rintro ⟨x,hx,rfl⟩
obtain ⟨z,rfl⟩ := hp x
exact ⟨z,hx,rfl⟩
rw [← he]
exact addHaar_image_eq_zero_of_det_fderivWithin_eq_zero μ hd hdet
end NullImage
noncomputable def initialProjection (n m : ℕ) (h : n ≤ m) :
EuclideanSpace ℝ (Fin m) →L[ℝ] EuclideanSpace ℝ (Fin n) :=
LinearMap.toContinuousLinearMap
{ toFun := fun x => WithLp.toLp 2 (fun i => x (Fin.castLE h i))
map_add' := by intros; rfl
map_smul' := by intros; rfl }
lemma initialProjection_apply (n m : ℕ) (h : n ≤ m) (x : EuclideanSpace ℝ (Fin m)) (i : Fin n) :
initialProjection n m h x i = x (Fin.castLE h i) := rfl
lemma initialProjection_surjective (n m : ℕ) (h : n ≤ m) :
Function.Surjective (initialProjection n m h) := by
intro x
refine ⟨WithLp.toLp 2 (fun j : Fin m => if h : j.val < n then x ⟨j.val,h⟩ else 0),?_⟩
apply PiLp.ext
intro i
change (if hj : i.val < n then x ⟨i.val,hj⟩ else 0) = x i
simp only [dite_eq_left i.isLt]
lemma initialProjection_not_injective (n m : ℕ) (h : n < m) :
¬ Function.Injective (initialProjection n m h.le) := by
intro hh
have hl := LinearMap.finrank_le_finrank_of_injective
(f := (initialProjection n m h.le).toLinearMap) hh
simp only [finrank_euclideanSpace_fin] at hl
omega
lemma differentiable_image_null_of_dim_lt (n m : ℕ) (h : n < m)
{f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)}
{S : Set (EuclideanSpace ℝ (Fin n))} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x) :
volume (f '' S) = 0 :=
differentiable_image_null_of_projection volume (initialProjection n m h.le)
(initialProjection_surjective n m h.le) (initialProjection_not_injective n m h) hf
lemma open_exists_not_mem_differentiable_image (n m : ℕ) (h : n < m)
{f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)}
{S : Set (EuclideanSpace ℝ (Fin n))} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x)
{U : Set (EuclideanSpace ℝ (Fin m))} (hU : IsOpen U) (hne : U.Nonempty) :
∃ y ∈ U, y ∉ f '' S := by
by_contra hn
push Not at hn
have hl : volume U ≤ volume (f '' S) := measure_mono hn
rw [differentiable_image_null_of_dim_lt n m h hf] at hl
exact (hU.measure_ne_zero volume hne) (le_antisymm hl (zero_le))
end BorsukNine
end OAI
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