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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 91 | import Mathlib
namespace OAI
namespace BorsukNine
open scoped BigOperators
section BlockVector
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
noncomputable def blockVector (I : Finset ι) (lam : ℝ) (r : ι → ℝ) (mu : ℝ)
(z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) : EuclideanSpace ℝ ι :=
WithLp.toLp 2 (fun i => if hi : i ∈ I then lam * r i else -mu * z ⟨i,Finset.mem_compl.mpr hi⟩)
lemma blockVector_in (I : Finset ι) (lam : ℝ) (r : ι → ℝ) (mu : ℝ)
(z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) {i : ι} (hi : i ∈ I) :
blockVector I lam r mu z i = lam * r i := by simp [blockVector,hi]
lemma blockVector_out (I : Finset ι) (lam : ℝ) (r : ι → ℝ) (mu : ℝ)
(z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) (i : ↥(Iᶜ : Finset ι)) :
blockVector I lam r mu z i.val = -mu * z i := by simp [blockVector,Finset.mem_compl.mp i.property]
lemma blockVector_continuous {X : Type*} [TopologicalSpace X] (I : Finset ι)
{lam mu : X → ℝ} {r : X → ι → ℝ} {z : X → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)}
(hl : Continuous lam) (hr : Continuous r) (hm : Continuous mu) (hz : Continuous z) :
Continuous (fun x => blockVector I (lam x) (r x) (mu x) (z x)) := by
apply (PiLp.continuous_toLp 2 _).comp
apply continuous_pi
intro i
by_cases hi : i ∈ I
· simp only [dite_eq_left hi]
exact hl.mul ((continuous_apply i).comp hr)
· simp only [dite_eq_right hi]
let j : ↥(Iᶜ : Finset ι) := ⟨i,Finset.mem_compl.mpr hi⟩
have hj : Continuous (fun x => z x j) := (PiLp.continuous_apply 2 _ j).comp hz
exact hm.neg.mul hj
lemma blockVector_continuousOn {X : Type*} [TopologicalSpace X] (I : Finset ι) {S : Set X}
{lam mu : X → ℝ} {r : X → ι → ℝ} {z : X → EuclideanSpace ℝ ↥(Iᶜ : Finset ι)}
(hl : ContinuousOn lam S) (hr : ContinuousOn r S) (hm : ContinuousOn mu S) (hz : ContinuousOn z S) :
ContinuousOn (fun x => blockVector I (lam x) (r x) (mu x) (z x)) S := by
apply continuousOn_iff_continuous_domRestrict.mpr
exact blockVector_continuous I (continuousOn_iff_continuous_domRestrict.mp hl)
(continuousOn_iff_continuous_domRestrict.mp hr) (continuousOn_iff_continuous_domRestrict.mp hm)
(continuousOn_iff_continuous_domRestrict.mp hz)
lemma blockVector_nonzero {I : Finset ι} (hI : I.Nonempty) {lam mu : ℝ} (hl : 0 < lam)
{r : ι → ℝ} (hr : ∀ i ∈ I, 0 < r i) (z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)) :
blockVector I lam r mu z ≠ 0 := by
obtain ⟨i,hi⟩ := hI
intro h
have he := congrArg (fun v : EuclideanSpace ℝ ι => v i) h
rw [blockVector_in _ _ _ _ _ hi] at he
exact (ne_of_gt (mul_pos hl (hr i hi))) he
lemma blockVector_ray {I : Finset ι} {lam mu t : ℝ} (_hl : 0 < lam) (_hm : 0 < mu)
(hlm : lam + mu = 1) (_ht : 0 < t) {a r : ι → ℝ}
(hao : ∀ i ∉ I, a i = 0) (hro : ∀ i ∉ I, r i = 0)
(has : ∑ i, a i = 1) (hrs : ∑ i, r i = 1)
{b z : EuclideanSpace ℝ ↥(Iᶜ : Finset ι)} (hbs : ∑ i, b i = 1) (hzs : ∑ i, z i = 1)
(he : blockVector I lam a mu b = t • blockVector I (1/2:ℝ) r (1/2:ℝ) z) :
t = 1 ∧ lam = 1/2 ∧ mu = 1/2 ∧ a = r ∧ b = z := by
have hei (i : ι) (hi : i ∈ I) : lam * a i = t * ((1/2:ℝ) * r i) := by
have h := congrArg (fun v : EuclideanSpace ℝ ι => v i) he
simpa only [PiLp.smul_apply,smul_eq_mul,blockVector_in _ _ _ _ _ hi] using h
have heo (i : ↥(Iᶜ : Finset ι)) : -mu * b i = t * (-(1/2:ℝ) * z i) := by
have h := congrArg (fun v : EuclideanSpace ℝ ι => v i.val) he
simpa only [PiLp.smul_apply,smul_eq_mul,blockVector_out] using h
have hia : ∑ i ∈ I, a i = 1 := by
rw [← has]; exact Finset.sum_subset (Finset.subset_univ I) (fun i _ hi => hao i hi)
have hir : ∑ i ∈ I, r i = 1 := by
rw [← hrs]; exact Finset.sum_subset (Finset.subset_univ I) (fun i _ hi => hro i hi)
have hl' : lam = t*(1/2:ℝ) := by
have hs := Finset.sum_congr (s₁ := I) (rfl) hei
simpa only [← Finset.mul_sum,hia,hir,mul_one] using hs
have hm' : -mu = t*(-(1/2:ℝ)) := by
have hs := congrArg (fun v : ↥(Iᶜ : Finset ι) → ℝ => ∑ i, v i) (funext heo)
simpa only [← Finset.mul_sum,hbs,hzs,mul_one] using hs
have ht1 : t = 1 := by linarith
have hl2 : lam = 1/2 := by linarith
have hm2 : mu = 1/2 := by linarith
refine ⟨ht1,hl2,hm2,?_,?_⟩
· funext i
by_cases hi : i ∈ I
· have h := hei i hi; rw [ht1,hl2] at h; linarith
· rw [hao i hi,hro i hi]
· apply PiLp.ext
intro i
have h := heo i; rw [ht1,hm2] at h; linarith
end BlockVector
end BorsukNine
end OAI
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