| import Mathlib |
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| namespace OAI |
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| open Set |
| open scoped BigOperators |
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| namespace SimplexChord |
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| variable {ι : Type*} [Fintype ι] |
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| abbrev standardSimplex : Set (ι → ℝ) := |
| {weights | (∀ coordinate, 0 ≤ weights coordinate) ∧ ∑ coordinate, weights coordinate = 1} |
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| theorem positive_mass_eq_negative_mass (b : ι → ℝ) (hb : ∑ i, b i = 0) : |
| (∑ i, max 0 (b i)) = ∑ i, max 0 (-b i) := by |
| have hpoint : ∀ i, max 0 (b i) - max 0 (-b i) = b i := by |
| intro i |
| rcases le_total 0 (b i) with hi | hi |
| · rw [max_eq_right hi, max_eq_left (neg_nonpos.mpr hi)] |
| ring |
| · rw [max_eq_left hi, max_eq_right (neg_nonneg.mpr hi)] |
| ring |
| have h := congrArg (fun f : ι → ℝ => ∑ i, f i) (funext hpoint) |
| simp only [Finset.sum_sub_distrib] at h |
| linarith |
|
|
| theorem negative_mass_pos (b : ι → ℝ) (hb : ∑ i, b i = 0) (hne : b ≠ 0) : |
| 0 < ∑ i, max 0 (-b i) := by |
| have hn : 0 ≤ ∑ i, max 0 (-b i) := |
| Finset.sum_nonneg (fun i _ => le_max_left _ _) |
| by_contra hnot |
| have hz : (∑ i, max 0 (-b i)) = 0 := le_antisymm (le_of_not_gt hnot) hn |
| have hcoords : ∀ i, max 0 (-b i) = 0 := by |
| simpa using (Finset.sum_eq_zero_iff_of_nonneg |
| (fun i (_ : i ∈ (Finset.univ : Finset ι)) => le_max_left 0 (-b i))).mp hz |
| have hbpos : ∀ i, 0 ≤ b i := by |
| intro i |
| have h := le_max_right 0 (-b i) |
| rw [hcoords i] at h |
| linarith |
| have hbzero : ∀ i, b i = 0 := by |
| simpa using (Finset.sum_eq_zero_iff_of_nonneg |
| (fun i (_ : i ∈ (Finset.univ : Finset ι)) => hbpos i)).mp hb |
| exact hne (funext hbzero) |
|
|
| theorem endpoints_iff (b x : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) : |
| (x ∈ standardSimplex ∧ (fun i => x i + s * b i) ∈ standardSimplex) ↔ |
| (∀ i, s * max 0 (-b i) ≤ x i) ∧ ∑ i, x i = 1 := by |
| constructor |
| · rintro ⟨hx, hy⟩ |
| refine ⟨fun i => ?_, hx.2⟩ |
| rcases le_total 0 (-b i) with hi | hi |
| · rw [max_eq_right hi] |
| have h := hy.1 i |
| linarith |
| · rw [max_eq_left hi, mul_zero] |
| exact hx.1 i |
| · rintro ⟨hx, hsum⟩ |
| have hxnonneg : ∀ i, 0 ≤ x i := fun i => |
| (mul_nonneg hs (le_max_left _ _)).trans (hx i) |
| refine ⟨⟨hxnonneg, hsum⟩, ⟨fun i => ?_, ?_⟩⟩ |
| · have hle := mul_le_mul_of_nonneg_left (le_max_right 0 (-b i)) hs |
| have h := hx i |
| nlinarith |
| · simp only [Finset.sum_add_distrib, ← Finset.mul_sum, hsum, hb, mul_zero, |
| add_zero] |
|
|
| theorem chord_bound (b x : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) (hx : x ∈ standardSimplex) |
| (hy : (fun i => x i + s * b i) ∈ standardSimplex) : |
| s * (∑ i, max 0 (-b i)) ≤ 1 := by |
| have h := (endpoints_iff b x hb hs).mp ⟨hx, hy⟩ |
| calc |
| s * (∑ i, max 0 (-b i)) = ∑ i, s * max 0 (-b i) := Finset.mul_sum .. |
| _ ≤ ∑ i, x i := Finset.sum_le_sum (fun i _ => h.1 i) |
| _ = 1 := h.2 |
|
|
| theorem endpoint_unique (b x : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) (hsc : s * (∑ i, max 0 (-b i)) = 1) |
| (hx : x ∈ standardSimplex) |
| (hy : (fun i => x i + s * b i) ∈ standardSimplex) : |
| x = fun i => s * max 0 (-b i) := by |
| have h := (endpoints_iff b x hb hs).mp ⟨hx, hy⟩ |
| have hsum : (∑ i, (x i - s * max 0 (-b i))) = 0 := by |
| rw [Finset.sum_sub_distrib, ← Finset.mul_sum, h.2, hsc] |
| ring |
| have hz := (Finset.sum_eq_zero_iff_of_nonneg |
| (fun i (_ : i ∈ (Finset.univ : Finset ι)) => sub_nonneg.mpr (h.1 i))).mp hsum |
| funext i |
| exact sub_eq_zero.mp (hz i (Finset.mem_univ i)) |
|
|
| theorem intersection_homothety (b : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) (hsc : s * (∑ i, max 0 (-b i)) < 1) : |
| {x | x ∈ standardSimplex ∧ |
| (fun i => x i + s * b i) ∈ standardSimplex} = |
| (fun y i => s * max 0 (-b i) + |
| (1 - s * (∑ j, max 0 (-b j))) * y i) '' standardSimplex := by |
| let c := ∑ i, max 0 (-b i) |
| have hden : 0 < 1 - s * c := sub_pos.mpr hsc |
| ext x |
| constructor |
| · intro hx |
| have h := (endpoints_iff b x hb hs).mp hx |
| let y : ι → ℝ := fun i => (x i - s * max 0 (-b i)) / (1 - s * c) |
| refine ⟨y, ⟨?_, ?_⟩, ?_⟩ |
| · intro i |
| exact div_nonneg (sub_nonneg.mpr (h.1 i)) hden.le |
| · change (∑ i, (x i - s * max 0 (-b i)) / (1 - s * c)) = 1 |
| rw [← Finset.sum_div, Finset.sum_sub_distrib, ← Finset.mul_sum, h.2] |
| exact div_self (ne_of_gt hden) |
| · funext i |
| change s * max 0 (-b i) + (1 - s * c) * |
| ((x i - s * max 0 (-b i)) / (1 - s * c)) = x i |
| rw [mul_div_cancel₀ _ (ne_of_gt hden)] |
| ring |
| · rintro ⟨y, hy, rfl⟩ |
| apply (endpoints_iff b _ hb hs).mpr |
| refine ⟨fun i => ?_, ?_⟩ |
| · have hn := mul_nonneg hden.le (hy.1 i) |
| dsimp [c] at hn |
| linarith |
| · simp only [Finset.sum_add_distrib, ← Finset.mul_sum, hy.2, mul_one] |
| ring |
|
|
| theorem maximal_chord (b : ι → ℝ) (hb : ∑ i, b i = 0) (hne : b ≠ 0) : |
| let c := ∑ i, max 0 (-b i) |
| let x : ι → ℝ := fun i => max 0 (-b i) / c |
| x ∈ standardSimplex ∧ |
| (fun i => x i + (1 / c) * b i) ∈ standardSimplex ∧ |
| ∀ (y : ι → ℝ) (s : ℝ), 0 ≤ s → y ∈ standardSimplex → |
| (fun i => y i + s * b i) ∈ standardSimplex → s ≤ 1 / c := by |
| dsimp only |
| have hc := negative_mass_pos b hb hne |
| have hmem := (endpoints_iff b (fun i => max 0 (-b i) / ∑ j, max 0 (-b j)) |
| hb (one_div_nonneg.mpr hc.le)).mpr |
| (And.intro (fun i => by rw [one_div, div_eq_mul_inv]; ring_nf; rfl) |
| (by rw [← Finset.sum_div]; exact div_self (ne_of_gt hc))) |
| refine ⟨hmem.1, hmem.2, ?_⟩ |
| intro y s hs hy hz |
| exact (le_div_iff₀ hc).mpr (chord_bound b y hb hs hy hz) |
|
|
| theorem exists_chord_iff (b : ι → ℝ) (hb : ∑ i, b i = 0) (hne : b ≠ 0) |
| {s : ℝ} (hs : 0 ≤ s) : |
| (∃ x : ι → ℝ, x ∈ standardSimplex ∧ |
| (fun i => x i + s * b i) ∈ standardSimplex) ↔ |
| s * (∑ i, max 0 (-b i)) ≤ 1 := by |
| constructor |
| · rintro ⟨x, hx, hy⟩ |
| exact chord_bound b x hb hs hx hy |
| · intro hsc |
| have hc := negative_mass_pos b hb hne |
| refine ⟨fun i => max 0 (-b i) / ∑ j, max 0 (-b j), |
| (endpoints_iff b _ hb hs).mpr ⟨?_, ?_⟩⟩ |
| · intro i |
| apply (le_div_iff₀ hc).mpr |
| have h := mul_le_mul_of_nonneg_right hsc (le_max_left 0 (-b i)) |
| nlinarith |
| · rw [← Finset.sum_div] |
| exact div_self (ne_of_gt hc) |
|
|
| theorem intersection_empty (b : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) (hsc : 1 < s * (∑ i, max 0 (-b i))) : |
| {x | x ∈ standardSimplex ∧ |
| (fun i => x i + s * b i) ∈ standardSimplex} = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro x hx |
| exact (not_le_of_gt hsc) (chord_bound b x hb hs hx.1 hx.2) |
|
|
| theorem intersection_singleton (b : ι → ℝ) (hb : ∑ i, b i = 0) |
| {s : ℝ} (hs : 0 ≤ s) (hsc : s * (∑ i, max 0 (-b i)) = 1) : |
| {x | x ∈ standardSimplex ∧ |
| (fun i => x i + s * b i) ∈ standardSimplex} = |
| {fun i => s * max 0 (-b i)} := by |
| ext x |
| constructor |
| · intro hx |
| exact Set.mem_singleton_iff.mpr (endpoint_unique b x hb hs hsc hx.1 hx.2) |
| · intro hx |
| rw [Set.mem_singleton_iff] at hx |
| subst x |
| apply (endpoints_iff b _ hb hs).mpr |
| exact ⟨fun _ => le_rfl, by rw [← Finset.mul_sum, hsc]⟩ |
|
|
| theorem abs_mass_eq_twice (b : ι → ℝ) (hb : ∑ i, b i = 0) : |
| (∑ i, |b i|) = 2 * ∑ i, max 0 (-b i) := by |
| have hpoint : ∀ i, |b i| = max 0 (b i) + max 0 (-b i) := by |
| intro i |
| rcases le_total 0 (b i) with hi | hi |
| · rw [abs_of_nonneg hi, max_eq_right hi, max_eq_left (neg_nonpos.mpr hi)] |
| ring |
| · rw [abs_of_nonpos hi, max_eq_left hi, max_eq_right (neg_nonneg.mpr hi)] |
| ring |
| simp_rw [hpoint] |
| rw [Finset.sum_add_distrib, positive_mass_eq_negative_mass b hb] |
| ring |
|
|
| theorem sum_sq_le_twice_mass_sq (b : ι → ℝ) (hb : ∑ i, b i = 0) : |
| (∑ i, (b i)^2) ≤ 2 * (∑ i, max 0 (-b i))^2 := by |
| have hpoint : ∀ i, (b i)^2 = (max 0 (b i))^2 + (max 0 (-b i))^2 := by |
| intro i |
| rcases le_total 0 (b i) with hi | hi |
| · rw [max_eq_right hi, max_eq_left (neg_nonpos.mpr hi)] |
| ring |
| · rw [max_eq_left hi, max_eq_right (neg_nonneg.mpr hi)] |
| ring |
| simp_rw [hpoint] |
| rw [Finset.sum_add_distrib] |
| have hp := Finset.sum_sq_le_sq_sum_of_nonneg |
| (fun i (_ : i ∈ (Finset.univ : Finset ι)) => le_max_left 0 (b i)) |
| have hn := Finset.sum_sq_le_sq_sum_of_nonneg |
| (fun i (_ : i ∈ (Finset.univ : Finset ι)) => le_max_left 0 (-b i)) |
| rw [positive_mass_eq_negative_mass b hb] at hp |
| linarith |
|
|
| end SimplexChord |
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| end OAI |
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