| import OAI.Analysis.DiamondDistortion.Backward |
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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 RecursiveApplications |
| open Midpoint |
| variable {X ι : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| def c6 : ℝ := 2*Real.exp (-(Real.exp 1-1))/(4:ℝ)^6 |
| def c3 : ℝ := (6*Real.exp 1)⁻¹^3 |
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| def q6 (ε : ℝ) : ℝ := Real.sqrt (1-c6*ε^6/512) |
| def theta (ε : ℝ) : ℝ := Real.sqrt (1-c3*ε^3/(2*Real.sqrt 2)) |
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| lemma c6_pos : 0 < c6 := by unfold c6; positivity |
| lemma c6_le_one : c6 ≤ 1 := by |
| have he : 1 ≤ Real.exp 1 := Real.one_le_exp (by norm_num) |
| have hl : Real.exp (-(Real.exp 1-1)) ≤ 1 := Real.exp_le_one_iff.mpr (by linarith) |
| simp only [neg_sub] at hl |
| unfold c6 |
| norm_num |
| linarith |
| lemma c3_pos : 0 < c3 := by unfold c3; positivity |
| lemma c3_le_one : c3 ≤ 1 := by |
| have he : 1 ≤ Real.exp 1 := Real.one_le_exp (by norm_num) |
| have hi : (6*Real.exp 1)⁻¹ ≤ 1 := (inv_le_one₀ (by positivity)).mpr (by linarith) |
| exact pow_le_one₀ (by positivity) hi |
|
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| / |
| algebraic rewrite of the displayed sixth and cubic estimates. -/ |
| def PowerInput (A : Heads ι X) (n m : ℕ) (c : ℝ) : Prop := |
| ∀ (H : ι) (p y : X) (R : ℝ),A.projection H p=p → A.projection H y=0 → |
| 0 < R → ‖p+y‖ ≤ R → ‖p-y‖ ≤ R → |
| R^m*‖p‖^2+c*‖y‖^n ≤ R^(m+2) |
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| lemma power_lens (A : Heads ι X) {n m : ℕ} {c ε : ℝ} (hc : 0 ≤ c) (hε : 0 < ε) |
| (hinput : PowerInput A n m c) : |
| Diamond.UnitLensBound X ε (Real.sqrt (1-c*(ε/2)^n)) := by |
| intro x z hp hm hs |
| have hh := positive_limit_le |
| (fun η => (1+2*η)^m*(max (‖x‖-η) 0)^2+c*(max (ε/2-η) 0)^n) |
| (fun η => (1+2*η)^(m+2)) (by fun_prop) (by fun_prop) (fun η hη => by |
| obtain ⟨H,p,y,hP,hY,hpL,hyL,hpR,hmR⟩ := A.pure_pair z hη hp hm hs |
| have h := hinput H p y (1+2*η) hP hY (by positivity) hpR hmR |
| have hp' := (sq_le_sq₀ (le_max_right _ _) (norm_nonneg _)).mpr |
| (max_le hpL (norm_nonneg _)) |
| have hy' := pow_le_pow_left₀ (le_max_right (ε/2-η) 0) |
| (max_le hyL (norm_nonneg y)) n |
| exact (add_le_add |
| (mul_le_mul_of_nonneg_left hp' (by positivity : 0 ≤ (1+2*η)^m)) |
| (mul_le_mul_of_nonneg_left hy' hc)).trans h) |
| simp only [mul_zero,add_zero,sub_zero,one_pow,one_mul, |
| max_eq_left (norm_nonneg x),max_eq_left (half_pos hε).le] at hh |
| have hrad : 0 ≤ 1-c*(ε/2)^n := by nlinarith [sq_nonneg ‖x‖] |
| exact (Real.le_sqrt (norm_nonneg _) hrad).mpr (by linarith) |
|
|
| / |
| real division, for which the entire right side is zero when S=0. -/ |
| def SixthInput (A : Heads ι X) : Prop := |
| ∀ (H : ι) (p y : X),A.projection H p=p → A.projection H y=0 → |
| let S := ‖p+y‖^2+‖p-y‖^2 |
| c6*‖y‖^6/S^2 ≤ S-2*‖p‖^2 |
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| lemma sixth_to_power (A : Heads ι X) (hinput : SixthInput A) : PowerInput A 6 4 (c6/8) := by |
| intro H p y R hP hY hR hp hm |
| let S := ‖p+y‖^2+‖p-y‖^2 |
| have hS : 0 ≤ S := by dsimp [S]; positivity |
| have hcap : S ≤ 2*R^2 := by |
| have hp2 := (sq_le_sq₀ (norm_nonneg _) hR.le).mpr hp |
| have hm2 := (sq_le_sq₀ (norm_nonneg _) hR.le).mpr hm |
| dsimp [S] |
| linarith |
| by_cases hzero : S=0 |
| · have hplus : ‖p+y‖=0 := by |
| have he : ‖p+y‖^2+‖p-y‖^2=0 := hzero |
| nlinarith [sq_nonneg ‖p-y‖,norm_nonneg (p+y)] |
| have hminus : ‖p-y‖=0 := by |
| have he : ‖p+y‖^2+‖p-y‖^2=0 := hzero |
| nlinarith [sq_nonneg ‖p+y‖,norm_nonneg (p-y)] |
| have hp0 := Diamond.norm_center_le (le_of_eq hplus) (le_of_eq hminus) |
| have hy0 := Diamond.norm_deviation_le (le_of_eq hplus) (le_of_eq hminus) |
| have hpn : ‖p‖=0 := le_antisymm hp0 (norm_nonneg _) |
| have hyn : ‖y‖=0 := le_antisymm hy0 (norm_nonneg _) |
| rw [hpn,hyn] |
| simpa using (pow_nonneg hR.le 6) |
| have hSpos : 0 < S := lt_of_le_of_ne hS (Ne.symm hzero) |
| have hh := hinput H p y hP hY |
| change c6*‖y‖^6/S^2 ≤ S-2*‖p‖^2 at hh |
| have henergy : c6*‖y‖^6 ≤ (S-2*‖p‖^2)*S^2 := |
| (div_le_iff₀ (sq_pos_of_pos hSpos)).mp hh |
| have hdef : 0 ≤ S-2*‖p‖^2 := |
| (by have := c6_pos; positivity : 0 ≤ c6*‖y‖^6/S^2).trans hh |
| have hS2 : S^2 ≤ (2*R^2)^2 := (sq_le_sq₀ hS (by positivity)).mpr hcap |
| have hmul := mul_le_mul_of_nonneg_left hS2 hdef |
| have hmul2 := mul_le_mul_of_nonneg_right |
| (show S-2*‖p‖^2 ≤ 2*R^2-2*‖p‖^2 by linarith) (sq_nonneg (2*R^2)) |
| have hfin := henergy.trans (hmul.trans hmul2) |
| norm_num |
| nlinarith only [hfin] |
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|
| lemma q6_range {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : 0 ≤ q6 ε ∧ q6 ε < 1 := by |
| have hp : 0 < c6*ε^6/512 := by have := c6_pos; positivity |
| have hpow : ε^6 ≤ 1 := pow_le_one₀ hε.le hε1 |
| have hl : c6*ε^6 ≤ 1 := (mul_le_of_le_one_left (by positivity) c6_le_one).trans hpow |
| have hrad : 0 ≤ 1-c6*ε^6/512 := by linarith |
| have hsq := Real.sq_sqrt hrad |
| constructor |
| · exact Real.sqrt_nonneg _ |
| · unfold q6 |
| nlinarith [Real.sqrt_nonneg (1-c6*ε^6/512)] |
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| theorem sixth_lens (A : Heads ι X) (hinput : SixthInput A) {ε : ℝ} (hε : 0 < ε) |
| (_hε1 : ε ≤ 1) : Diamond.UnitLensBound X ε (q6 ε) := by |
| have h := power_lens A (show 0 ≤ c6/8 by have := c6_pos; positivity) hε (sixth_to_power A hinput) |
| have he : 1-(c6/8)*(ε/2)^6=1-c6*ε^6/512 := by ring |
| simpa only [he,q6] using h |
|
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| theorem sixth_depth (A : Heads ι X) (hinput : SixthInput A) {D s : ℝ} |
| (k : ℕ) (f : Diamond.Vertex k → X) (hf : Diamond.HasDistortion k f D s) : |
| D⁻¹ ≤ (q6 D⁻¹)^k := by |
| have hD : 0 < D := lt_of_lt_of_le (by norm_num) hf.1 |
| have hα : 0 < D⁻¹ := inv_pos.mpr hD |
| have hα1 : D⁻¹ ≤ 1 := (inv_le_one₀ hD).mpr hf.1 |
| have hlens : Diamond.UnitLensBound X (2*D⁻¹) (q6 D⁻¹) := by |
| intro x z hp hm hs |
| exact sixth_lens A hinput hα hα1 x z hp hm |
| (fun i j hij => (by linarith : D⁻¹ ≤ 2*D⁻¹).trans (hs i j hij)) |
| have hnorm := Diamond.upper_one_normalization f hf |
| obtain ⟨hq,hq1⟩ := q6_range hα hα1 |
| exact Diamond.backward hα hα1 hq hq1 hlens k (fun v => (D*s)⁻¹ • f v) |
| (fun u v => (hnorm u v).1) (fun u v => (hnorm u v).2) |
|
|
| / |
| def CubicInput (A : Heads ι X) : Prop := |
| ∀ (H : ι) (p y : X) (R : ℝ),A.projection H p=p → A.projection H y=0 → |
| 0 < R → ‖p+y‖ ≤ R → ‖p-y‖ ≤ R → |
| ‖p‖^2+c3/(2*Real.sqrt 2*R)*‖y‖^3 ≤ R^2 |
|
|
| lemma cubic_to_power (A : Heads ι X) (hinput : CubicInput A) : |
| PowerInput A 3 1 (c3/(2*Real.sqrt 2)) := by |
| intro H p y R hP hY hR hp hm |
| have h := mul_le_mul_of_nonneg_left (hinput H p y R hP hY hR hp hm) hR.le |
| have he : R*(‖p‖^2+c3/(2*Real.sqrt 2*R)*‖y‖^3)= |
| R*‖p‖^2+c3/(2*Real.sqrt 2)*‖y‖^3 := by field_simp |
| rw [he] at h |
| simpa only [pow_one,show (1+2:ℕ)=3 from rfl,show R*R^2=R^3 by ring] using h |
|
|
| lemma theta_range {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : 0 ≤ theta ε ∧ theta ε < 1 := by |
| have hsqrt : 1 ≤ Real.sqrt 2 := (Real.le_sqrt (by norm_num) (by norm_num)).mpr (by norm_num) |
| have hp : 0 < c3*ε^3/(2*Real.sqrt 2) := by have := c3_pos; positivity |
| have hpow : ε^3 ≤ 1 := pow_le_one₀ hε.le hε1 |
| have hl : c3*ε^3 ≤ 1 := (mul_le_of_le_one_left (by positivity) c3_le_one).trans hpow |
| have hratio : c3*ε^3/(2*Real.sqrt 2) ≤ 1 := |
| (div_le_one₀ (by positivity)).mpr (by linarith) |
| have hrad : 0 ≤ 1-c3*ε^3/(2*Real.sqrt 2) := by linarith |
| have hsq := Real.sq_sqrt hrad |
| constructor |
| · exact Real.sqrt_nonneg _ |
| · unfold theta |
| nlinarith [Real.sqrt_nonneg (1-c3*ε^3/(2*Real.sqrt 2))] |
|
|
| / |
| directly; dual separability is unnecessary for this stronger transfer from |
| the same displayed pure-head estimate. The weak-null clause is proved below. -/ |
| theorem cubic_depth (A : Heads ι X) (hinput : CubicInput A) {D s : ℝ} |
| (k : ℕ) (f : Diamond.Vertex k → X) (hf : Diamond.HasDistortion k f D s) : |
| 1 ≤ D*(theta D⁻¹)^k := by |
| have hD : 0 < D := lt_of_lt_of_le (by norm_num) hf.1 |
| have hα : 0 < D⁻¹ := inv_pos.mpr hD |
| have hα1 : D⁻¹ ≤ 1 := (inv_le_one₀ hD).mpr hf.1 |
| have hpow := power_lens A (show 0 ≤ c3/(2*Real.sqrt 2) by have := c3_pos; positivity) |
| (show 0 < 2*D⁻¹ by positivity) (cubic_to_power A hinput) |
| have he : 1-c3/(2*Real.sqrt 2)*(2*D⁻¹/2)^3=1-c3*(D⁻¹)^3/(2*Real.sqrt 2) := by ring |
| have hlens : Diamond.UnitLensBound X (2*D⁻¹) (theta D⁻¹) := by simpa only [he,theta] using hpow |
| have hnorm := Diamond.upper_one_normalization f hf |
| obtain ⟨hθ,hθ1⟩ := theta_range hα hα1 |
| have h := Diamond.backward hα hα1 hθ hθ1 hlens k (fun v => (D*s)⁻¹ • f v) |
| (fun u v => (hnorm u v).1) (fun u v => (hnorm u v).2) |
| have hh := mul_le_mul_of_nonneg_left h hD.le |
| simpa only [mul_inv_cancel₀ hD.ne'] using hh |
| |
| def ZeroRootInput (A : Heads ι X) : Prop := |
| ∀ (H : ι) (p y : X),A.projection H p=p → A.projection H y=0 → y ≠ 0 → |
| ‖p‖+‖y‖^3/(8*(2*‖p‖+‖y‖)^2) ≤ (‖p+y‖+‖p-y‖)/2 |
| |
| lemma zero_root_parent (A : Heads ι X) (hinput : ZeroRootInput A) |
| {α R : ℝ} (hα : 0 < α) (hα1 : α ≤ 1) (_hR : 0 ≤ R) (hR1 : R ≤ 1) |
| {x : X} (z : ℕ → X) (hp : ∀ i,‖x+z i‖ ≤ R) (hm : ∀ i,‖x-z i‖ ≤ R) |
| (hs : ∀ i j,i ≠ j → 2*α ≤ ‖z i-z j‖) : ‖x‖ ≤ R-α^3/4608 := by |
| let c : ℝ := α^3/2304 |
| let η : ℝ := c/6 |
| have hc : 0 < c := by dsimp [c]; positivity |
| have hη : 0 < η := by dsimp [η]; positivity |
| have hα2 : α^2 ≤ 1 := pow_le_one₀ hα.le hα1 |
| have hα3 : α^3 ≤ α := by nlinarith [mul_le_mul_of_nonneg_left hα2 hα.le] |
| have hηα : η ≤ α/2 := by dsimp [η,c]; nlinarith |
| have hηhalf : η ≤ 1/2 := hηα.trans (by linarith) |
| obtain ⟨H,p,y,hP,hY,hpL,hyL,hpR,hmR⟩ := A.pure_pair z hη hp hm hs |
| have hy : α/2 ≤ ‖y‖ := by linarith |
| have hypos : 0 < ‖y‖ := (half_pos hα).trans_le hy |
| have hy0 : y ≠ 0 := norm_pos_iff.mp hypos |
| have hp2 : ‖p‖ ≤ 2 := (Diamond.norm_center_le hpR hmR).trans (by linarith) |
| have hy2 : ‖y‖ ≤ 2 := (Diamond.norm_deviation_le hpR hmR).trans (by linarith) |
| have hdenpos : 0 < 8*(2*‖p‖+‖y‖)^2 := by positivity |
| have hden : 8*(2*‖p‖+‖y‖)^2 ≤ 288 := by |
| have hsum : 2*‖p‖+‖y‖ ≤ 6 := by linarith |
| have hsq := (sq_le_sq₀ (by positivity : 0 ≤ 2*‖p‖+‖y‖) (by norm_num : (0:ℝ) ≤ 6)).mpr hsum |
| linarith |
| have hy3 := pow_le_pow_left₀ (half_pos hα).le hy 3 |
| have hgain : c ≤ ‖y‖^3/(8*(2*‖p‖+‖y‖)^2) := by |
| apply (le_div_iff₀ hdenpos).mpr |
| have hm := mul_le_mul_of_nonneg_left hden hc.le |
| dsimp [c] at hm ⊢ |
| nlinarith only [hm,hy3] |
| have hi := hinput H p y hP hY hy0 |
| have hu : (‖p+y‖+‖p-y‖)/2 ≤ R+2*η := by linarith |
| have hchain := hi.trans hu |
| dsimp [η,c] at hchain hpL hgain |
| linarith |
| |
| theorem zero_root_depth (A : Heads ι X) (hinput : ZeroRootInput A) {D s : ℝ} |
| (k : ℕ) (f : Diamond.Vertex k → X) (hf : Diamond.HasDistortion k f D s) : |
| (k:ℝ) ≤ 4608*D^3*(1-D⁻¹) := by |
| have hD : 0 < D := lt_of_lt_of_le (by norm_num) hf.1 |
| have hα : 0 < D⁻¹ := inv_pos.mpr hD |
| have hα1 : D⁻¹ ≤ 1 := (inv_le_one₀ hD).mpr hf.1 |
| have hnorm := Diamond.upper_one_normalization f hf |
| have hstep : ∀ (x : X) (z : ℕ → X) (R : ℝ),0 ≤ R → R ≤ 1 → D⁻¹ ≤ ‖x‖ → |
| (∀ i,‖x+z i‖ ≤ R) → (∀ i,‖x-z i‖ ≤ R) → |
| (∀ i j,i ≠ j → 2*D⁻¹ ≤ ‖z i-z j‖) → ‖x‖ ≤ R-(D⁻¹)^3/4608 := by |
| intro x z R hR hR1 hx hp hm hs |
| exact zero_root_parent A hinput hα hα1 hR hR1 z hp hm hs |
| have hi := Diamond.iterate_additive (show 0 ≤ (D⁻¹)^3/4608 by positivity) hstep k |
| (fun v => (D*s)⁻¹ • f v) (by norm_num : (0:ℝ) ≤ 1) le_rfl |
| (fun u v => (hnorm u v).1) |
| (fun e => Diamond.slope_upper _ (fun u v => by simpa using (hnorm u v).2) e) |
| have hfin : (k:ℝ)*(D⁻¹)^3/4608 ≤ 1-D⁻¹ := by linarith |
| have hm := mul_le_mul_of_nonneg_right hfin (show 0 ≤ 4608*D^3 by positivity) |
| have he : ((k:ℝ)*(D⁻¹)^3/4608)*(4608*D^3)=(k:ℝ) := by field_simp |
| rw [he] at hm |
| simpa only [mul_comm] using hm |
| |
| end RecursiveApplications |
| |
| end DiamondDistortion |
| |
| end |
| |
| end OAI |
| |
