| import OAI.Combinatorics.HalvingLines.ChainDrawing |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace PlanarHalving.Batches |
| open PlaneGraph RankCoordinates |
| variable {R : ℕ} |
|
|
| lemma ConvexPath.extension_vertex (C : ConvexPath R) {l u : ℝ} |
| (hsl : ∀ b : Fin R, C.point b.castSucc ≠ C.point b.succ → |
| l ≤ slope (C.point b.castSucc) (C.point b.succ) ∧ slope (C.point b.castSucc) (C.point b.succ) ≤ u) |
| (c : Fin (R+1)) : C.extension l u (C.point c).1 = (C.point c).2 := by |
| induction c using Fin.induction with |
| | zero => |
| apply le_antisymm (C.extension_below_vertex hsl 0) |
| have hh := le_envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) (.inr false) (C.point 0).1 |
| simp only [ConvexPath.extensionSlopes, ConvexPath.extensionIntercepts] at hh |
| change _ ≤ envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) (C.point 0).1 |
| linarith only [hh] |
| | succ b ih => |
| by_cases hb : C.point b.castSucc = C.point b.succ |
| · rw [← hb] |
| exact ih |
| · exact C.extension_edge hsl b hb (right_mem_segment ℝ _ _) |
|
|
| lemma ConvexPath.extension_left (C : ConvexPath R) {l u : ℝ} (hlu : l ≤ u) |
| (hsl : ∀ b : Fin R, C.point b.castSucc ≠ C.point b.succ → |
| l ≤ slope (C.point b.castSucc) (C.point b.succ) ∧ slope (C.point b.castSucc) (C.point b.succ) ≤ u) |
| {x : ℝ} (hx : x ≤ (C.point 0).1) : |
| C.extension l u x = (C.point 0).2 + l*(x-(C.point 0).1) := by |
| apply le_antisymm |
| · apply envelope_le |
| intro i |
| have hi := le_envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) i (C.point 0).1 |
| change _ ≤ C.extension l u (C.point 0).1 at hi |
| rw [C.extension_vertex hsl] at hi |
| have hm := mul_nonpos_of_nonneg_of_nonpos |
| (sub_nonneg.mpr (C.extension_slopes hlu hsl i).1) (sub_nonpos.mpr hx) |
| nlinarith only [hi,hm] |
| · have hh := le_envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) (.inr false) x |
| simp only [ConvexPath.extensionSlopes, ConvexPath.extensionIntercepts] at hh |
| change _ ≤ envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) x |
| linarith only [hh] |
|
|
| lemma ConvexPath.extension_right (C : ConvexPath R) {l u : ℝ} (hlu : l ≤ u) |
| (hsl : ∀ b : Fin R, C.point b.castSucc ≠ C.point b.succ → |
| l ≤ slope (C.point b.castSucc) (C.point b.succ) ∧ slope (C.point b.castSucc) (C.point b.succ) ≤ u) |
| {x : ℝ} (hx : (C.point (Fin.last R)).1 ≤ x) : |
| C.extension l u x = (C.point (Fin.last R)).2 + u*(x-(C.point (Fin.last R)).1) := by |
| apply le_antisymm |
| · apply envelope_le |
| intro i |
| have hi := le_envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) i (C.point (Fin.last R)).1 |
| change _ ≤ C.extension l u (C.point (Fin.last R)).1 at hi |
| rw [C.extension_vertex hsl] at hi |
| have hm := mul_nonpos_of_nonpos_of_nonneg |
| (sub_nonpos.mpr (C.extension_slopes hlu hsl i).2) (sub_nonneg.mpr hx) |
| nlinarith only [hi,hm] |
| · have hh := le_envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) (.inr true) x |
| simp only [ConvexPath.extensionSlopes, ConvexPath.extensionIntercepts] at hh |
| change _ ≤ envelope (C.extensionSlopes l u) (C.extensionIntercepts l u) x |
| linarith only [hh] |
|
|
| lemma finite_intermediate {R : ℕ} (f : Fin (R+1) → ℝ) {x : ℝ} |
| (h₀ : f 0 ≤ x) (h₁ : x < f (Fin.last R)) : |
| ∃ b : Fin R, f b.castSucc ≤ x ∧ x < f b.succ := by |
| have haux : ∀ c : Fin (R+1), x < f c → ∃ b : Fin R, f b.castSucc ≤ x ∧ x < f b.succ := by |
| intro c |
| induction c using Fin.induction with |
| | zero => exact fun h => (not_lt_of_ge h₀ h).elim |
| | succ b ih => |
| intro h |
| by_cases hb : f b.castSucc ≤ x |
| · exact ⟨b,hb,h⟩ |
| · exact ih (lt_of_not_ge hb) |
| exact haux (Fin.last R) h₁ |
|
|
| lemma ConvexPath.extension_middle (C : ConvexPath R) {l u : ℝ} |
| (hsl : ∀ b : Fin R, C.point b.castSucc ≠ C.point b.succ → |
| l ≤ slope (C.point b.castSucc) (C.point b.succ) ∧ slope (C.point b.castSucc) (C.point b.succ) ≤ u) |
| {x : ℝ} (h₀ : (C.point 0).1 ≤ x) (h₁ : x < (C.point (Fin.last R)).1) : |
| ∃ b : Fin R, C.point b.castSucc ≠ C.point b.succ ∧ |
| (x,C.extension l u x) ∈ segment ℝ (C.point b.castSucc) (C.point b.succ) := by |
| obtain ⟨b,hb₀,hb₁⟩ := finite_intermediate (fun c => (C.point c).1) h₀ h₁ |
| have hbx : (C.point b.castSucc).1 < (C.point b.succ).1 := hb₀.trans_lt hb₁ |
| have hb : C.point b.castSucc ≠ C.point b.succ := fun h => hbx.ne (congrArg Prod.fst h) |
| let t := (x-(C.point b.castSucc).1)/((C.point b.succ).1-(C.point b.castSucc).1) |
| have ht : t ∈ Set.Icc (0:ℝ) 1 := by |
| refine ⟨div_nonneg (sub_nonneg.mpr hb₀) (sub_nonneg.mpr hbx.le), ?_⟩ |
| exact (div_le_one (sub_pos.mpr hbx)).mpr (by linarith only [hb₁]) |
| let z : Point := (1-t) • C.point b.castSucc + t • C.point b.succ |
| have hz : z ∈ segment ℝ (C.point b.castSucc) (C.point b.succ) := by |
| rw [segment_eq_image] |
| exact ⟨t,ht,rfl⟩ |
| have hzx : z.1 = x := by |
| dsimp [z,t] |
| field_simp [ne_of_gt (sub_pos.mpr hbx)] |
| ring |
| have hzy := C.extension_edge hsl b hb hz |
| rw [hzx] at hzy |
| have he : (x,C.extension l u x) = z := Prod.ext hzx.symm hzy |
| exact ⟨b,hb,he ▸ hz⟩ |
|
|
| lemma ConvexPath.extension_cases (C : ConvexPath R) {l u : ℝ} (hlu : l ≤ u) |
| (hsl : ∀ b : Fin R, C.point b.castSucc ≠ C.point b.succ → |
| l ≤ slope (C.point b.castSucc) (C.point b.succ) ∧ slope (C.point b.castSucc) (C.point b.succ) ≤ u) |
| {z : Point} (hz : C.extension l u z.1 = z.2) : |
| (z.1 ≤ (C.point 0).1 ∧ z.2 = (C.point 0).2 + l*(z.1-(C.point 0).1)) ∨ |
| ((C.point (Fin.last R)).1 ≤ z.1 ∧ z.2 = (C.point (Fin.last R)).2 + u*(z.1-(C.point (Fin.last R)).1)) ∨ |
| (∃ b : Fin R, C.point b.castSucc ≠ C.point b.succ ∧ z ∈ segment ℝ (C.point b.castSucc) (C.point b.succ)) := by |
| by_cases hl : z.1 ≤ (C.point 0).1 |
| · exact Or.inl ⟨hl, hz.symm.trans (C.extension_left hlu hsl hl)⟩ |
| by_cases hu : (C.point (Fin.last R)).1 ≤ z.1 |
| · exact Or.inr (Or.inl ⟨hu,hz.symm.trans (C.extension_right hlu hsl hu)⟩) |
| obtain ⟨b,hb,hz'⟩ := C.extension_middle hsl (le_of_not_ge hl) (lt_of_not_ge hu) |
| rw [hz] at hz' |
| exact Or.inr (Or.inr ⟨b,hb,hz'⟩) |
|
|
| end PlanarHalving.Batches |
| namespace PlanarHalving.Spatial |
| open Set Filter Metric |
| open scoped Topology |
|
|
| theorem subsequence_of_asymptotic_continuity |
| {X Y : Type*} [MetricSpace X] [SecondCountableTopology X] |
| [MetricSpace Y] [CompleteSpace Y] |
| (f : ℕ → X → Y) (G : Set X) (hG : G.Nonempty) (K : Set Y) (hK : IsCompact K) |
| (hbounded : ∀ n p, p ∈ G → f n p ∈ K) |
| (hcont : ∀ p ∈ G, ∀ ε > 0, ∃ r > 0, |
| ∀ᶠ n in atTop, ∀ q, dist q p < r → dist (f n q) (f n p) ≤ ε) : |
| ∃ (φ : ℕ → ℕ) (l : G → Y), StrictMono φ ∧ |
| (∀ p, l p ∈ K) ∧ |
| (∀ p : G, ∀ u : ℕ → X, Tendsto u atTop (𝓝 (p : X)) → |
| Tendsto (fun n => f (φ n) (u n)) atTop (𝓝 (l p))) := by |
| classical |
| let : Nonempty G := hG.to_subtype |
| let : CompactSpace K := isCompact_iff_compactSpace.mp hK |
| obtain ⟨q,hq⟩ := TopologicalSpace.exists_dense_seq G |
| let F : ℕ → ℕ → K := fun n j => ⟨f n (q j), hbounded n (q j) (q j).property⟩ |
| obtain ⟨g,φ,hφ,hg⟩ := CompactSpace.tendsto_subseq F |
| have hgrid (j : ℕ) : Tendsto (fun n => f (φ n) (q j)) atTop (𝓝 (g j : Y)) := |
| continuous_subtype_val.tendsto _ |>.comp ((tendsto_pi_nhds.mp hg) j) |
| have hcauchy (p : G) : CauchySeq (fun n => f (φ n) p) := by |
| rw [Metric.cauchySeq_iff] |
| intro ε hε |
| obtain ⟨r,hr,hrf⟩ := hcont p p.property (ε/4) (by positivity) |
| obtain ⟨j,hj⟩ := hq.exists_dist_lt p hr |
| have hj' : dist (q j : X) p < r := by simpa only [Subtype.dist_eq, dist_comm] using hj |
| obtain ⟨N,hN⟩ := eventually_atTop.mp (hφ.tendsto_atTop.eventually hrf) |
| obtain ⟨N',hN'⟩ := Metric.cauchySeq_iff.mp (hgrid j).cauchySeq (ε/4) (by positivity) |
| refine ⟨max N N',fun m hm n hn => ?_⟩ |
| have hm' := hN m (le_trans (le_max_left _ _) hm) (q j) hj' |
| have hn' := hN n (le_trans (le_max_left _ _) hn) (q j) hj' |
| have hmn := hN' m (le_trans (le_max_right _ _) hm) n (le_trans (le_max_right _ _) hn) |
| have htri := dist_triangle4 (f (φ m) p) (f (φ m) (q j)) (f (φ n) (q j)) (f (φ n) p) |
| rw [dist_comm (f (φ m) p) (f (φ m) (q j))] at htri |
| linarith only [htri,hm',hn',hmn,hε] |
| choose l hl using fun p => cauchySeq_tendsto_of_complete (hcauchy p) |
| refine ⟨φ,l,hφ,?_,?_⟩ |
| · intro p |
| exact hK.isClosed.mem_of_tendsto (hl p) (Eventually.of_forall (fun n => hbounded (φ n) p p.property)) |
| · intro p u hu |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| obtain ⟨r,hr,hrf⟩ := hcont p p.property (ε/3) (by positivity) |
| have hu' : ∀ᶠ n in atTop, dist (u n) (p : X) < r := (Metric.tendsto_nhds.mp hu) r hr |
| have hl' : ∀ᶠ n in atTop, dist (f (φ n) p) (l p) < ε/3 := |
| (Metric.tendsto_nhds.mp (hl p)) (ε/3) (by positivity) |
| apply eventually_atTop.mp |
| filter_upwards [hφ.tendsto_atTop.eventually hrf,hu',hl'] with n hn hnu hnl |
| have hh := hn (u n) hnu |
| have ht := dist_triangle (f (φ n) (u n)) (f (φ n) p) (l p) |
| linarith only [hh,ht,hnl,hε] |
|
|
| lemma asymptotic_continuity_of_radial |
| {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| (f : ℕ → X → Y) (G : Set X) (r e : ℕ → ℝ) |
| (hr : ∀ j, 0 < r j) (he : Tendsto e atTop (𝓝 0)) |
| (hb : ∀ p ∈ G, ∀ j, ∀ᶠ n in atTop, |
| ∀ q, dist q p < r j → dist (f n q) (f n p) ≤ e j) : |
| ∀ p ∈ G, ∀ ε > 0, ∃ δ > 0, |
| ∀ᶠ n in atTop, ∀ q, dist q p < δ → dist (f n q) (f n p) ≤ ε := by |
| intro p hp ε hε |
| obtain ⟨j,hj⟩ := ((tendsto_order.mp he).2 ε hε).exists |
| refine ⟨r j,hr j,?_⟩ |
| filter_upwards [hb p hp j] with n hn q hq |
| exact (hn q hq).trans hj.le |
|
|
| lemma lower_distance_of_limit |
| {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| (f : ℕ → X → Y) (φ : ℕ → ℕ) (hφ : StrictMono φ) |
| {p q : X} {x y : Y} (hp : Tendsto (fun n => f (φ n) p) atTop (𝓝 x)) |
| (hq : Tendsto (fun n => f (φ n) q) atTop (𝓝 y)) |
| {a : ℝ} (ha : ∀ᶠ n in atTop, a ≤ dist (f n p) (f n q)) : a ≤ dist x y := by |
| exact ge_of_tendsto (hp.dist hq) (hφ.tendsto_atTop.eventually ha) |
|
|
| end PlanarHalving.Spatial |
| namespace PlanarHalving.Spatial |
| open Set Filter Metric |
| open scoped Topology |
|
|
| lemma geometric_scale {q h d : ℝ} (hq : 0 < q) (hq1 : q < 1) |
| (_hh : 0 < h) (hd : 0 < d) (hdh : d ≤ h) : |
| ∃ j : ℕ, 1 ≤ j ∧ h*q^j < d ∧ q*d ≤ h*q^j := by |
| have ht : Tendsto (fun j : ℕ => h*q^j) atTop (𝓝 0) := by |
| simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one hq.le hq1).const_mul h |
| have hex : ∃ j : ℕ, h*q^j < d := ((tendsto_order.mp ht).2 d hd).exists |
| let j := Nat.find hex |
| have hj : h*q^j < d := Nat.find_spec hex |
| have hj0 : j ≠ 0 := by |
| intro h |
| simp only [h,pow_zero,mul_one] at hj |
| exact (not_lt_of_ge hdh) hj |
| have hj1 : 1 ≤ j := Nat.one_le_iff_ne_zero.mpr hj0 |
| have hprev : d ≤ h*q^(j-1) := by |
| exact le_of_not_gt (Nat.find_min hex (show j-1 < j by omega)) |
| refine ⟨j,hj1,hj,?_⟩ |
| have hp : q^j = q^(j-1)*q := by |
| rw [← pow_succ] |
| congr 1 |
| omega |
| rw [hp] |
| nlinarith only [mul_le_mul_of_nonneg_left hprev hq.le] |
|
|
| lemma geometric_lower_power {q h d s : ℝ} (hq : 0 < q) (hq1 : q < 1) |
| (hh : 0 < h) (hd : 0 < d) (hdh : d ≤ h) (hs : 0 ≤ s) : |
| ∃ j : ℕ, 1 ≤ j ∧ h*q^j < d ∧ |
| q^s * (q/h)^s * d^s ≤ (q^s)^(j+1) := by |
| obtain ⟨j,hj,hjd,hdj⟩ := geometric_scale hq hq1 hh hd hdh |
| refine ⟨j,hj,hjd,?_⟩ |
| have hdiv : q/h*d ≤ q^j := by |
| have hv : q*d/h ≤ q^j := (div_le_iff₀ hh).mpr (by nlinarith only [hdj]) |
| convert hv using 1 |
| ring |
| have hp := Real.rpow_le_rpow (by positivity : 0 ≤ q/h*d) hdiv hs |
| rw [Real.mul_rpow (by positivity) hd.le] at hp |
| rw [pow_succ, Real.rpow_pow_comm hq.le] |
| have ht := mul_le_mul_of_nonneg_left hp (Real.rpow_nonneg hq.le s) |
| nlinarith only [ht] |
|
|
| lemma inverse_holder_of_radial_limits |
| {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| (f : ℕ → X → Y) (φ : ℕ → ℕ) (hφ : StrictMono φ) |
| (G : Set X) (l : G → Y) |
| (hl : ∀ p : G, Tendsto (fun n => f (φ n) p) atTop (𝓝 (l p))) |
| {q h s : ℝ} (hq : 0 < q) (hq1 : q < 1) (hh : 0 < h) (hs : 0 < s) |
| (hdiam : ∀ p p' : G, dist (p : X) p' ≤ h) |
| (hlower : ∀ p : G, ∀ j : ℕ, 1 ≤ j → ∀ᶠ n in atTop, |
| ∀ p' : G, h*q^j < dist (p : X) p' → (q^s)^(j+1) ≤ dist (f n p) (f n p')) : |
| ∀ p p' : G, (q^s * (q/h)^s) * dist (p : X) p' ^ s ≤ dist (l p) (l p') := by |
| intro p p' |
| by_cases hpp : p = p' |
| · subst p' |
| simp [Real.zero_rpow hs.ne'] |
| · have hd : 0 < dist (p : X) p' := dist_pos.mpr (Subtype.coe_ne_coe.mpr hpp) |
| obtain ⟨j,hj,hjd,hdj⟩ := geometric_lower_power hq hq1 hh hd (hdiam p p') hs.le |
| apply hdj.trans |
| apply lower_distance_of_limit f φ hφ (hl p) (hl p') |
| filter_upwards [hlower p j hj] with n hn |
| exact hn p' hjd |
|
|
| end PlanarHalving.Spatial |
| namespace PlanarHalving.Limits |
| open MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| theorem finiteMeasure_subsequence |
| {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] |
| [MeasurableSpace X] [BorelSpace X] |
| (μ : ℕ → ℕ → FiniteMeasure X) (C : ℕ → ℝ≥0) |
| (hbound : ∀ n j, (μ n j).mass ≤ C j) : |
| ∃ (φ : ℕ → ℕ) (ν : ℕ → FiniteMeasure X), StrictMono φ ∧ |
| ∀ j, Tendsto (fun n => μ (φ n) j) atTop (𝓝 (ν j)) := by |
| let F : ℕ → ∀ j : ℕ, (Icc (0 : ℝ≥0) (C j)) × ProbabilityMeasure X := |
| fun n j => (⟨(μ n j).mass,zero_le,hbound n j⟩,(μ n j).normalize) |
| obtain ⟨g,φ,hφ,hg⟩ := CompactSpace.tendsto_subseq F |
| refine ⟨φ,fun j => (g j).1.val • (g j).2.toFiniteMeasure,hφ,?_⟩ |
| intro j |
| have hj := (tendsto_pi_nhds.mp hg) j |
| have hm := continuous_subtype_val.tendsto _ |>.comp ((continuous_fst.tendsto (g j)).comp hj) |
| have hn := (ProbabilityMeasure.toFiniteMeasure_isEmbedding X).continuous.tendsto _ |>.comp |
| ((continuous_snd.tendsto (g j)).comp hj) |
| have ht := hm.smul hn |
| convert ht using 1 |
| funext n |
| exact (μ (φ n) j).self_eq_mass_smul_normalize |
|
|
| end PlanarHalving.Limits |
|
|
| namespace PlanarHalving.RankCoordinates |
| open Set |
|
|
| lemma variation_le_two {f : ℕ → ℝ} {n : ℕ} |
| (hf : ∀ i ≤ n, 0 ≤ f i ∧ f i ≤ 1) |
| (hqc : ∀ i j k, i ≤ j → j ≤ k → k ≤ n → min (f i) (f k) ≤ f j) : |
| ∑ i ∈ Finset.range n, |f (i+1)-f i| ≤ 2 := by |
| classical |
| obtain ⟨k,hk,hmax⟩ := (Finset.range (n+1)).exists_max_image f (by simp) |
| have hkn : k ≤ n := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using hk |
| have hleft : ∑ i ∈ Finset.range k, |f (i+1)-f i| = f k-f 0 := by |
| rw [← Finset.sum_range_sub f k] |
| apply Finset.sum_congr rfl |
| intro i hi |
| have hik : i < k := Finset.mem_range.mp hi |
| apply abs_of_nonneg |
| apply sub_nonneg.mpr |
| have hq := hqc i (i+1) k (by omega) (by omega) hkn |
| rw [min_eq_left (hmax i (by simp; omega))] at hq |
| exact hq |
| have hright : ∑ i ∈ Finset.Ico k n, |f (i+1)-f i| = f k-f n := by |
| calc |
| _ = - ∑ i ∈ Finset.Ico k n, (f (i+1)-f i) := by |
| rw [← Finset.sum_neg_distrib] |
| apply Finset.sum_congr rfl |
| intro i hi |
| obtain ⟨hki,hin⟩ := Finset.mem_Ico.mp hi |
| apply abs_of_nonpos |
| apply sub_nonpos.mpr |
| have hq := hqc k i (i+1) hki (by omega) (by omega) |
| rw [min_eq_right (hmax (i+1) (by simp; omega))] at hq |
| exact hq |
| _ = _ := by rw [Finset.sum_Ico_sub f hkn]; ring |
| rw [← Finset.sum_range_add_sum_Ico (fun i => |f (i+1)-f i|) hkn,hleft,hright] |
| have h0 := (hf 0 (Nat.zero_le n)).1 |
| have hn := (hf n le_rfl).1 |
| have hk1 := (hf k hkn).2 |
| linarith only [h0,hn,hk1] |
|
|
| lemma min_le_of_quasiconcave {f : ℝ → ℝ} (hf : QuasiconcaveOn ℝ univ f) |
| {x y z : ℝ} (hxy : x ≤ y) (hyz : y ≤ z) : min (f x) (f z) ≤ f y := by |
| have hc := (hf (min (f x) (f z))).ordConnected |
| have hx : x ∈ univ ∩ {u | min (f x) (f z) ≤ f u} := by |
| refine ⟨mem_univ _,?_⟩ |
| change min (f x) (f z) ≤ f x |
| exact min_le_left _ _ |
| have hz : z ∈ univ ∩ {u | min (f x) (f z) ≤ f u} := by |
| refine ⟨mem_univ _,?_⟩ |
| change min (f x) (f z) ≤ f z |
| exact min_le_right _ _ |
| exact (hc.out hx hz ⟨hxy,hyz⟩).2 |
|
|
| lemma concave_cdf_variation {f F : ℝ → ℝ} (hf : ConcaveOn ℝ univ f) |
| (hF : Monotone F) (hFrange : ∀ x, F x ∈ Icc 0 1) |
| (x : ℕ → ℝ) (hx : Monotone x) (n : ℕ) : |
| ∑ i ∈ Finset.range n, |F (f (x (i+1)))-F (f (x i))| ≤ 2 := by |
| apply variation_le_two (fun i _ => hFrange (f (x i))) |
| intro i j k hij hjk _ |
| rw [← hF.map_min] |
| exact hF (min_le_of_quasiconcave hf.quasiconcaveOn (hx hij) (hx hjk)) |
|
|
| lemma sigmoid_variation {f : ℝ → ℝ} (hf : ConcaveOn ℝ univ f) |
| {ε : ℝ} (hε : 0 < ε) (x : ℕ → ℝ) (hx : Monotone x) (n : ℕ) : |
| ∑ i ∈ Finset.range n, |
| |Real.sigmoid (f (x (i+1))/ε)-Real.sigmoid (f (x i)/ε)| ≤ 2 := by |
| apply concave_cdf_variation hf |
| (fun _ _ h => Real.sigmoid_strictMono.monotone (div_le_div_of_nonneg_right h hε.le)) |
| (fun _ => ⟨(Real.sigmoid_pos _).le,(Real.sigmoid_lt_one _).le⟩) x hx n |
|
|
| lemma smoothed_variation {m : ℕ} (hm : 0 < m) (g : Fin m → ℝ → ℝ) |
| (hg : ∀ j, ConvexOn ℝ univ (g j)) {f : ℝ → ℝ} (hf : ConcaveOn ℝ univ f) |
| {ε : ℝ} (hε : 0 < ε) (x : ℕ → ℝ) (hx : Monotone x) (n : ℕ) : |
| ∑ i ∈ Finset.range n, |
| |smoothed g ε (x (i+1),f (x (i+1)))-smoothed g ε (x i,f (x i))| ≤ 2 := by |
| have hmR : (0 : ℝ) < m := Nat.cast_pos.mpr hm |
| have hterm (i : ℕ) : |
| |smoothed g ε (x (i+1),f (x (i+1)))-smoothed g ε (x i,f (x i))| ≤ |
| (∑ j, |Real.sigmoid ((f (x (i+1))-g j (x (i+1)))/ε) - |
| Real.sigmoid ((f (x i)-g j (x i))/ε)|) / m := by |
| unfold smoothed |
| rw [← sub_div,abs_div,abs_of_pos hmR,← Finset.sum_sub_distrib] |
| exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) hmR.le |
| calc |
| _ ≤ ∑ i ∈ Finset.range n, (∑ j, |Real.sigmoid ((f (x (i+1))-g j (x (i+1)))/ε) - |
| Real.sigmoid ((f (x i)-g j (x i))/ε)|) / m := Finset.sum_le_sum (fun i _ => hterm i) |
| _ = (∑ j : Fin m, ∑ i ∈ Finset.range n, |
| |Real.sigmoid ((f (x (i+1))-g j (x (i+1)))/ε) - |
| Real.sigmoid ((f (x i)-g j (x i))/ε)|) / m := by |
| rw [← Finset.sum_div,Finset.sum_comm] |
| _ ≤ (∑ _j : Fin m, (2 : ℝ))/m := by |
| apply div_le_div_of_nonneg_right _ hmR.le |
| exact Finset.sum_le_sum (fun j _ => sigmoid_variation (hf.sub (hg j)) hε x hx n) |
| _ = 2 := by simp; field_simp |
| |
| end PlanarHalving.RankCoordinates |
| |
| end OAI |
| |
