| import OAI.Analysis.DiamondDistortion.MidpointHeads |
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| namespace OAI |
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| universe u v |
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| noncomputable section |
| open scoped BigOperators Classical Topology |
| open Set |
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| namespace DiamondDistortion |
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| namespace Diamond |
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| section |
| variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| theorem quotient_full_gain {t g D s : ℝ} (ht : 0 < t) (hg : 0 < g) |
| (htD : t ≤ 1/(2*D)) (hX : Midpoint.QuotientGain X t g) |
| (k : ℕ) (f : Vertex k → X) (hf : HasDistortion k f D s) : |
| 1 ≤ D/(1+g)^k := by |
| obtain ⟨hD,_,h⟩ := normalized_distortion f hf |
| have hg1 : 0 < 1+g := by positivity |
| let c : ℝ := 1/(1+g) |
| have hc : 0 ≤ c := by dsimp [c]; positivity |
| have hc1 : c ≤ 1 := (div_le_one₀ hg1).mpr (by linarith) |
| have hstep : ∀ (x : X) (z : ℕ → X) (R : ℝ),0 ≤ R → R ≤ D → 1 ≤ ‖x‖ → |
| (∀ i,‖x+z i‖ ≤ R) → (∀ i,‖x-z i‖ ≤ R) → |
| (∀ i j,i ≠ j → 2*1 ≤ ‖z i-z j‖) → ‖x‖ ≤ c*R := by |
| intro x z R hR hRD hx hp hm hs |
| have hxD := (norm_center_le (hp 0) (hm 0)).trans hRD |
| have h := Midpoint.quotient_parent_full_gain ht hg hD htD hX z hx hxD hp hm |
| (by simpa only [mul_one] using hs) |
| simpa only [c,div_eq_mul_inv,one_mul,mul_comm] using h |
| have hi := iterate_contraction hc hc1 hstep k (fun v => s⁻¹ • f v) |
| (show 0 ≤ D by linarith) le_rfl (fun u v => by simpa only [one_mul] using (h u v).1) |
| (fun e => slope_upper _ (fun u v => by simpa only [mul_one] using (h u v).2) e) |
| simpa only [c,div_pow,one_pow,div_eq_mul_inv,one_mul,mul_comm,inv_pow] using hi |
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| theorem quotient_transfer {t g D s : ℝ} (ht : 0 < t) (hg : 0 < g) |
| (htD : t ≤ 1/(2*D)) (hX : Midpoint.QuotientGain X t g) |
| (k : ℕ) (f : Vertex k → X) (hf : HasDistortion k f D s) : |
| (1+g/2)^k ≤ D := by |
| have h := quotient_full_gain ht hg htD hX k f hf |
| have hp : 0 < (1+g)^k := by positivity |
| have hpow : (1+g)^k ≤ D := by simpa only [one_mul] using (le_div_iff₀ hp).mp h |
| exact (pow_le_pow_left₀ (by positivity : (0:ℝ) ≤ 1+g/2) (by linarith : 1+g/2 ≤ 1+g) k).trans hpow |
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| / |
| bounded by the specified multiple of its own slope. The recursion restricts |
| to old vertices and therefore includes every stage, not just the last one. -/ |
| def StableParent (c : ℝ) : (k : ℕ) → (Vertex k → X) → Prop |
| | 0,_ => False |
| | n+1,f => (∃ e : Edge n,∀ (i : ℕ) (b : Bool), |
| ‖slope f (e,i,b)‖ ≤ c*‖slope (fun v => f (Sum.inl v)) e‖) ∨ |
| StableParent c n (fun v => f (Sum.inl v)) |
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| lemma absence_stable_growth {c s : ℝ} (hc : 0 ≤ c) (k : ℕ) (f : Vertex k → X) |
| (hl : ∀ u v,s*distance k u v ≤ ‖f u-f v‖) (hn : ¬ StableParent c k f) : |
| ∃ e : Edge k,s*c^k ≤ ‖slope f e‖ := by |
| induction k with |
| | zero => exact ⟨(),by simpa using slope_lower f hl ()⟩ |
| | succ n ih => |
| simp only [StableParent,not_or] at hn |
| obtain ⟨e,he⟩ := ih (fun v => f (Sum.inl v)) (old_lower f hl) hn.2 |
| have hne : ¬ ∀ (i : ℕ) (b : Bool), |
| ‖slope f (e,i,b)‖ ≤ c*‖slope (fun v => f (Sum.inl v)) e‖ := |
| fun h => hn.1 ⟨e,h⟩ |
| push Not at hne |
| obtain ⟨i,b,hchild⟩ := hne |
| refine ⟨(e,i,b),?_⟩ |
| have hm := mul_le_mul_of_nonneg_left he hc |
| calc |
| s*c^(n+1) = c*(s*c^n) := by ring |
| _ ≤ c*‖slope (fun v => f (Sum.inl v)) e‖ := hm |
| _ ≤ ‖slope f (e,i,b)‖ := hchild.le |
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| theorem stable_parent {D δ : ℝ} (hδ : 0 < δ) (k : ℕ) (f : Vertex k → X) |
| (hf : HasDistortion k f D 1) (hdepth : D < (1+δ)^k) : |
| StableParent (1+δ) k f := by |
| by_contra hn |
| obtain ⟨e,he⟩ := absence_stable_growth (by positivity : 0 ≤ 1+δ) k f |
| (fun u v => (hf.2.2 u v).1) hn |
| have hb : ‖slope f e‖ ≤ D := slope_upper f |
| (fun u v => by simpa only [mul_one] using (hf.2.2 u v).2) e |
| simp only [one_mul] at he |
| linarith |
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| / |
| lens, retaining the radius and separation as independent parameters. -/ |
| def EnlargedLensExcluded (X : Type*) [NormedAddCommGroup X] (R β : ℝ) : Prop := |
| ∀ u : X,‖u‖=1 → ¬ ∃ z : ℕ → X, |
| (∀ i,max ‖u+z i‖ ‖u-z i‖ ≤ 1+β) ∧ |
| (∀ i j,i ≠ j → R ≤ ‖z i-z j‖) |
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| lemma enlarged_parent {D β L : ℝ} (_hD : 1 ≤ D) (hβ : 0 < β) |
| (hX : EnlargedLensExcluded X (2/D) β) {x : X} (z : ℕ → X) |
| (hx : 1 ≤ ‖x‖) (hxD : ‖x‖ ≤ D) |
| (hp : ∀ i,‖x+z i‖ ≤ L) (hm : ∀ i,‖x-z i‖ ≤ L) |
| (hs : ∀ i j,i ≠ j → 2 ≤ ‖z i-z j‖) : ‖x‖ < L/(1+β) := by |
| by_contra! hn |
| let r := ‖x‖ |
| have hr : 0 < r := by dsimp [r]; linarith |
| have hβ1 : 0 < 1+β := by positivity |
| have hL : L ≤ r*(1+β) := (div_le_iff₀ hβ1).mp hn |
| have hunit : ‖r⁻¹ • x‖=1 := by |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hr.le),inv_mul_cancel₀ hr.ne'] |
| apply hX (r⁻¹ • x) hunit |
| refine ⟨fun i => r⁻¹ • z i,?_,?_⟩ |
| · intro i |
| apply max_le |
| · rw [← smul_add,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hr.le)] |
| calc |
| r⁻¹*‖x+z i‖ ≤ r⁻¹*(r*(1+β)) := |
| mul_le_mul_of_nonneg_left ((hp i).trans hL) (inv_nonneg.mpr hr.le) |
| _ = 1+β := by rw [← mul_assoc,inv_mul_cancel₀ hr.ne',one_mul] |
| · rw [← smul_sub,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hr.le)] |
| calc |
| r⁻¹*‖x-z i‖ ≤ r⁻¹*(r*(1+β)) := |
| mul_le_mul_of_nonneg_left ((hm i).trans hL) (inv_nonneg.mpr hr.le) |
| _ = 1+β := by rw [← mul_assoc,inv_mul_cancel₀ hr.ne',one_mul] |
| · intro i j hij |
| rw [← smul_sub,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hr.le)] |
| calc |
| 2/D ≤ 2/r := div_le_div_of_nonneg_left (by norm_num) hr hxD |
| _ = r⁻¹*2 := by ring |
| _ ≤ r⁻¹*‖z i-z j‖ := mul_le_mul_of_nonneg_left (hs i j hij) (inv_nonneg.mpr hr.le) |
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| theorem enlarged_backward {D β s : ℝ} (hβ : 0 < β) |
| (hX : EnlargedLensExcluded X (2/D) β) (k : ℕ) (f : Vertex k → X) |
| (hf : HasDistortion k f D s) : 1 ≤ D/(1+β)^k := by |
| obtain ⟨hD,_,h⟩ := normalized_distortion f hf |
| have hβ1 : 0 < 1+β := by positivity |
| let c : ℝ := 1/(1+β) |
| have hc : 0 ≤ c := by dsimp [c]; positivity |
| have hc1 : c ≤ 1 := (div_le_one₀ hβ1).mpr (by linarith) |
| have hstep : ∀ (x : X) (z : ℕ → X) (R : ℝ),0 ≤ R → R ≤ D → 1 ≤ ‖x‖ → |
| (∀ i,‖x+z i‖ ≤ R) → (∀ i,‖x-z i‖ ≤ R) → |
| (∀ i j,i ≠ j → 2*1 ≤ ‖z i-z j‖) → ‖x‖ ≤ c*R := by |
| intro x z R hR hRD hx hp hm hs |
| have hxD := (norm_center_le (hp 0) (hm 0)).trans hRD |
| have h := (enlarged_parent hD hβ hX z hx hxD hp hm |
| (by simpa only [mul_one] using hs)).le |
| simpa only [c,div_eq_mul_inv,one_mul,mul_comm] using h |
| have hi := iterate_contraction hc hc1 hstep k (fun v => s⁻¹ • f v) |
| (show 0 ≤ D by linarith) le_rfl (fun u v => by simpa only [one_mul] using (h u v).1) |
| (fun e => slope_upper _ (fun u v => by simpa only [mul_one] using (h u v).2) e) |
| simpa only [c,div_pow,one_pow,div_eq_mul_inv,one_mul,mul_comm,inv_pow] using hi |
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| theorem stable_parent_obstruction {D δ s : ℝ} (hδ : 0 < δ) |
| (hX : EnlargedLensExcluded X (2/D) δ) (k : ℕ) (f : Vertex k → X) |
| (hf : HasDistortion k f D s) : (1+δ)^k ≤ D := by |
| have h := enlarged_backward hδ hX k f hf |
| simpa only [one_mul] using (le_div_iff₀ (by positivity : 0 < (1+δ)^k)).mp h |
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| end |
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| variable {X ι : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| / |
| def stageSup {k : ℕ} (f : Vertex k → X) : ℝ := |
| sSup (Set.range (fun e : Edge k => ‖slope f e‖)) |
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| lemma slope_le_stageSup {k : ℕ} (f : Vertex k → X) {D : ℝ} |
| (hb : ∀ e,‖slope f e‖ ≤ D) (e : Edge k) : ‖slope f e‖ ≤ stageSup f := by |
| exact le_csSup ⟨D,by rintro _ ⟨e,rfl⟩; exact hb e⟩ ⟨e,rfl⟩ |
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| lemma stageSup_le {k : ℕ} (f : Vertex k → X) {D : ℝ} |
| (hb : ∀ e,‖slope f e‖ ≤ D) : stageSup f ≤ D := by |
| unfold stageSup |
| apply csSup_le (s := Set.range (fun e : Edge k => ‖slope f e‖)) |
| ⟨‖slope f (baseEdge k)‖,⟨baseEdge k,rfl⟩⟩ |
| rintro _ ⟨e,rfl⟩ |
| exact hb e |
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| lemma parametric_Q_one {b lam : ℝ} (hb : 0 < b) : 1 ≤ Real.sqrt (1+b*lam^2) := by |
| have h : (1:ℝ)^2 ≤ 1+b*lam^2 := by nlinarith [mul_nonneg hb.le (sq_nonneg lam)] |
| exact (Real.le_sqrt (by norm_num) (by positivity)).mpr h |
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| theorem parametric_stage (A : Midpoint.Heads ι X) {a b lam D : ℝ} |
| (ha : 0 < a) (hb : 0 < b) (hlam : 0 < lam) (hinput : A.ParametricInput a b) |
| (n : ℕ) (f : Vertex (n+1) → X) (hf : HasDistortion (n+1) f D 1) : |
| let Q := Real.sqrt (1+b*lam^2) |
| stageSup (fun v => f (Sum.inl v)) ≤ Q*stageSup f-a*lam ∧ |
| Q*stageSup f-a*lam ≤ stageSup f-(a*lam-(Q-1)*D) := by |
| let Q := Real.sqrt (1+b*lam^2) |
| have hbnd : ∀ e,‖slope f e‖ ≤ D := slope_upper f |
| (fun u v => by simpa only [mul_one] using (hf.2.2 u v).2) |
| have hR : stageSup f ≤ D := stageSup_le f hbnd |
| have hQ : 1 ≤ Q := parametric_Q_one hb |
| constructor |
| · apply stageSup_le |
| intro e |
| have hp : ∀ i,‖slope (fun v => f (Sum.inl v)) e+deviation f e i‖ ≤ stageSup f := by |
| intro i |
| rw [slope_plus_deviation] |
| exact slope_le_stageSup f hbnd _ |
| have hm : ∀ i,‖slope (fun v => f (Sum.inl v)) e-deviation f e i‖ ≤ stageSup f := by |
| intro i |
| rw [slope_minus_deviation] |
| exact slope_le_stageSup f hbnd _ |
| have hs : ∀ i j,i ≠ j → 2 ≤ ‖deviation f e i-deviation f e j‖ := by |
| intro i j hij |
| simpa only [mul_one] using deviation_separated f (fun u v => (hf.2.2 u v).1) e hij |
| have h := A.parametric_lens ha hb hlam hinput (deviation f e) hp hm hs |
| norm_num at h |
| linarith |
| · change Q*stageSup f-a*lam ≤ stageSup f-(a*lam-(Q-1)*D) |
| nlinarith [mul_le_mul_of_nonneg_left hR (sub_nonneg.mpr hQ)] |
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| / |
| theorem parametric (A : Midpoint.Heads ι X) {a b lam D s : ℝ} |
| (ha : 0 < a) (hb : 0 < b) (hlam : 0 < lam) (hinput : A.ParametricInput a b) |
| (hΔ : 0 < a*lam-(Real.sqrt (1+b*lam^2)-1)*D) |
| (k : ℕ) (f : Vertex k → X) (hf : HasDistortion k f D s) : |
| 1+(k:ℝ)*(a*lam-(Real.sqrt (1+b*lam^2)-1)*D) ≤ D := by |
| obtain ⟨hD,_,h⟩ := normalized_distortion f hf |
| let Q := Real.sqrt (1+b*lam^2) |
| let δ := a*lam-(Q-1)*D |
| have hQ : 1 ≤ Q := parametric_Q_one hb |
| have hstep : ∀ (x : X) (z : ℕ → X) (R : ℝ),0 ≤ R → R ≤ D → 1 ≤ ‖x‖ → |
| (∀ i,‖x+z i‖ ≤ R) → (∀ i,‖x-z i‖ ≤ R) → |
| (∀ i j,i ≠ j → 2*1 ≤ ‖z i-z j‖) → ‖x‖ ≤ R-δ := by |
| intro x z R hR hRD hx hp hm hs |
| have hh := A.parametric_lens ha hb hlam hinput z hp hm hs |
| norm_num at hh |
| dsimp [δ] |
| change ‖x‖+a*lam ≤ Q*R at hh |
| nlinarith [mul_le_mul_of_nonneg_left hRD (sub_nonneg.mpr hQ)] |
| exact iterate_additive hΔ.le hstep k (fun v => s⁻¹ • f v) |
| (show 0 ≤ D by linarith) le_rfl (fun u v => by simpa only [one_mul] using (h u v).1) |
| (fun e => slope_upper _ (fun u v => by simpa only [mul_one] using (h u v).2) e) |
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| theorem parametric_twentyfour (A : Midpoint.Heads ι X) (hinput : A.ParametricInput (1/2) 3) |
| {D s : ℝ} (k : ℕ) (f : Vertex k → X) (hf : HasDistortion k f D s) : |
| 1+(k:ℝ)/(24*D) ≤ D := by |
| have hD : 0 < D := lt_of_lt_of_le (by norm_num) hf.1 |
| let lam := 1/(6*D) |
| have hlam : 0 < lam := by dsimp [lam]; positivity |
| have hQ : Real.sqrt (1+3*lam^2) ≤ 1+3*lam^2/2 := by |
| apply (Real.sqrt_le_left (by positivity)).mpr |
| nlinarith [sq_nonneg (3*lam^2/2)] |
| have hden : (6*D) ≠ 0 := by positivity |
| have hlD : lam*(6*D)=1 := by dsimp [lam]; field_simp |
| have hgain : 1/(24*D) ≤ (1/2)*lam-(Real.sqrt (1+3*lam^2)-1)*D := by |
| have hs := mul_le_mul_of_nonneg_right hQ hD.le |
| have he : (1/2)*lam-(3*lam^2/2)*D = 1/(24*D) := by |
| dsimp [lam] |
| field_simp |
| ring |
| linarith |
| have hΔ : 0 < (1/2)*lam-(Real.sqrt (1+3*lam^2)-1)*D := |
| lt_of_lt_of_le (by positivity) hgain |
| have h := parametric A (by norm_num : (0:ℝ)<1/2) (by norm_num : (0:ℝ)<3) hlam hinput hΔ k f hf |
| have hm := mul_le_mul_of_nonneg_left hgain (show 0 ≤ (k:ℝ) by positivity) |
| have he : (k:ℝ)/(24*D)=(k:ℝ)*(1/(24*D)) := by ring |
| linarith |
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| end Diamond |
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| end DiamondDistortion |
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| end |
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| end OAI |
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