import Mathlib import OAI.Probability.JammingConcavity.RowSmoothApproxFinite namespace OAI /-! Row Weighted Step. -/ noncomputable section open MeasureTheory ProbabilityTheory Set open scoped NNReal ENNReal open Set Filter open scoped Topology open MeasureTheory ProbabilityTheory Filter Set open scoped ENNReal NNReal Topology BigOperators open MeasureTheory Filter Set open scoped ENNReal NNReal BigOperators open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal Topology open scoped NNReal ENNReal Topology open scoped NNReal Topology open Set open Set Filter MeasureTheory open scoped BigOperators open scoped Topology NNReal open scoped Topology BigOperators open scoped ENNReal NNReal open MeasureTheory Set open MeasureTheory ProbabilityTheory open scoped ENNReal NNReal BigOperators Classical open Classical open scoped ENNReal NNReal Topology BigOperators MatrixOrder open scoped NNReal BigOperators open MeasureTheory Metric Set open Metric open scoped RealInnerProductSpace open Filter open Finset Set open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter Metric open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction open scoped BigOperators Classical open scoped ENNReal NNReal Topology BigOperators Matrix MatrixOrder open scoped BigOperators RealInnerProductSpace open scoped NNReal Topology BigOperators open scoped NNReal BigOperators RealInnerProductSpace open scoped ENNReal NNReal BigOperators MatrixOrder open scoped MatrixOrder open scoped NNReal open scoped BigOperators NNReal open scoped NNReal ENNReal BigOperators Topology open scoped Topology ENNReal NNReal open scoped Matrix.Norms.L2Operator MatrixOrder Topology NNReal ENNReal BigOperators open scoped Topology ENNReal NNReal BigOperators MatrixOrder Matrix.Norms.L2Operator open scoped Topology NNReal ENNReal BigOperators open scoped BigOperators NNReal Topology open scoped Topology NNReal ENNReal BigOperators MatrixOrder open scoped Topology NNReal ENNReal BigOperators MatrixOrder Matrix.Norms.L2Operator open scoped Topology NNReal ENNReal open Set Filter MeasureTheory ProbabilityTheory open scoped Topology BigOperators NNReal open Set Filter open scoped Topology namespace MicroscopicJamming lemma row_step_weighted_terminal {u v : ℝ → ℝ} {A B C κ Av Bv Cv κv Q L ε : ℝ} (hQ : 0 < Q) (hu : RowAnalyticTerminal u A B C κ Q) (hv : RowAnalyticTerminal v Av Bv Cv κv Q) (hL : 0 ≤ L) (hε : 0 ≤ ε) {rs : List (ℝ × ℝ)} (hrs : ∀ r ∈ rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) (hT : gaussianStepTime rs=Q) (hgu : ∀ t ∈ Set.Icc 0 Q,∀ x,|deriv (gaussianStepPath rs u t) x| ≤ L*(1+|x|)) (hgv : ∀ t ∈ Set.Icc 0 Q,∀ x,|deriv (gaussianStepPath rs v t) x| ≤ L*(1+|x|)) (hterm : ∀ x, |u x-v x| ≤ ε*(1+x^2)^2) : |gaussianRowComposition rs u 0-gaussianRowComposition rs v 0| ≤ ε*Real.exp ((10+8*L)*Q) := by obtain ⟨D,J,hD,hJ,hpath⟩ := gaussian_step_path A B C κ Q hQ hu.2.1 hu.2.2.1 hu.2.2.2.1 hu.2.2.2.2.1 obtain ⟨E,K,hE,hK,hpathv⟩ := gaussian_step_path Av Bv Cv κv Q hQ hv.2.1 hv.2.2.1 hv.2.2.2.1 hv.2.2.2.2.1 obtain ⟨_,htu,htimeu,hspaceu⟩ := hpath u hu rs hrs hT.le obtain ⟨_,htv,htimev,hspacev⟩ := hpathv v hv rs hrs hT.le have hxu (t : ℝ) (ht : t ∈ Set.Icc 0 Q) := hspaceu t ht.1 (hT ▸ ht.2) have hxv (t : ℝ) (ht : t ∈ Set.Icc 0 Q) := hspacev t ht.1 (hT ▸ ht.2) have hcu := gaussianStepPath_jointly_continuous hD hJ hT htimeu (fun t ht htR => ⟨(hspaceu t ht htR).1,fun x => ((hspaceu t ht htR).2.2 x).1⟩) have hcv := gaussianStepPath_jointly_continuous hE hK hT htimev (fun t ht htR => ⟨(hspacev t ht htR).1,fun x => ((hspacev t ht htR).2.2 x).1⟩) let W := |B|+A*(1+Q)+(|Bv|+Av*(1+Q)) have hWu : 0 ≤ |B|+A*(1+Q) := by have := hu.2.1; positivity have hWv : 0 ≤ |Bv|+Av*(1+Q) := by have := hv.2.1; positivity have hW : 0 ≤ W := add_nonneg hWu hWv have hbu (t : ℝ) (ht : t ∈ Set.Icc 0 Q) (x : ℝ) : |gaussianStepPath rs u t x| ≤ W*(1+x^2) := (gaussianStepPath_global_value_bound hQ hu hrs hT ht x).trans (mul_le_mul_of_nonneg_right (by dsimp [W]; linarith) (by positivity)) have hbv (t : ℝ) (ht : t ∈ Set.Icc 0 Q) (x : ℝ) : |gaussianStepPath rs v t x| ≤ W*(1+x^2) := (gaussianStepPath_global_value_bound hQ hv hrs hT ht x).trans (mul_le_mul_of_nonneg_right (by dsimp [W]; linarith) (by positivity)) have hh := row_weighted_terminal_bound hQ hW hL hε hcu hcv (fun t ht => ⟨(hxu t ht).1,(hxu t ht).2.1⟩) (fun t ht => ⟨(hxv t ht).1,(hxv t ht).2.1⟩) (fun t ht x => gaussianStepPath_right_pde hQ hu hrs hT.le (hT ▸ ht) x) (fun t ht x => gaussianStepPath_right_pde hQ hv hrs hT.le (hT ▸ ht) x) (fun t _ => gaussianStepRightCoefficient_bounds hrs t) hbu hbv hgu hgv (fun x => by simpa only [hT] using (by rw [htu x,htv x]; exact hterm x : |gaussianStepPath rs u (gaussianStepTime rs) x-gaussianStepPath rs v (gaussianStepTime rs) x| ≤ ε*(1+x^2)^2)) simpa only [gaussianStepPath_zero (fun r hr => (hrs r hr).2.2)] using hh lemma row_composition_weighted_Lipschitz {u v : ℝ → ℝ} {A B C κ Av Bv Cv κv Q L H Hv ε : ℝ} (hQ : 0 < Q) (hu : RowAnalyticTerminal u A B C κ Q) (hv : RowAnalyticTerminal v Av Bv Cv κv Q) (hub : RowBoundedTerminal u L H) (hvb : RowBoundedTerminal v L Hv) (hε : 0 ≤ ε) {rs : List (ℝ × ℝ)} (hrs : ∀ r ∈ rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) (hT : gaussianStepTime rs=Q) (hterm : ∀ x, |u x-v x| ≤ ε*(1+x^2)^2) : |gaussianRowComposition rs u 0-gaussianRowComposition rs v 0| ≤ ε*Real.exp ((10+8*L)*Q) := by apply row_step_weighted_terminal hQ hu hv hub.L_nonneg hε hrs hT _ _ hterm · intro t ht x obtain ⟨K,hK⟩ := rowBoundedComposition_terminal hub (gaussianStepRemainder rs t) (gaussianStepRemainder_valid hrs ht.1).1 exact (hK.slope x).trans (by nlinarith [mul_nonneg hub.L_nonneg (abs_nonneg x)]) · intro t ht x obtain ⟨K,hK⟩ := rowBoundedComposition_terminal hvb (gaussianStepRemainder rs t) (gaussianStepRemainder_valid hrs ht.1).1 exact (hK.slope x).trans (by nlinarith [mul_nonneg hvb.L_nonneg (abs_nonneg x)]) end MicroscopicJamming open Set Filter MeasureTheory open scoped Topology namespace MicroscopicJamming abbrev rowConv (φ u : ℝ → ℝ) : ℝ → ℝ := convolution φ u (.mul ℝ ℝ) volume lemma rowConv_integrable {φ u : ℝ → ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hu : Continuous u) (x : ℝ) : Integrable (fun y => φ y*u (x-y)) := hs.convolutionExists_left_of_continuous_right (.mul ℝ ℝ) hφ.locallyIntegrable hu x lemma row_iterated_smooth {φ : ℝ → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (n : ℕ) : ContDiff ℝ (↑(⊤:ℕ∞)) (iteratedDeriv n φ) := by induction n with | zero => simpa using hφ | succ n ih => rw [iteratedDeriv_succ]; exact (contDiff_infty_iff_deriv.mp ih).2 lemma row_iterated_compact {φ : ℝ → ℝ} (hs : HasCompactSupport φ) (n : ℕ) : HasCompactSupport (iteratedDeriv n φ) := by induction n with | zero => simpa using hs | succ n ih => rw [iteratedDeriv_succ]; exact ih.deriv lemma rowConv_deriv {φ u : ℝ → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (hs : HasCompactSupport φ) (hu : Continuous u) : deriv (rowConv φ u)=rowConv (deriv φ) u := by funext x exact (hs.hasDerivAt_convolution_left (.mul ℝ ℝ) (hφ.of_le (by simp)) hu.locallyIntegrable x).deriv lemma rowConv_iterated {φ u : ℝ → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (hs : HasCompactSupport φ) (hu : Continuous u) (n : ℕ) : iteratedDeriv n (rowConv φ u)=rowConv (iteratedDeriv n φ) u := by induction n with | zero => rfl | succ n ih => rw [iteratedDeriv_succ,ih,rowConv_deriv (row_iterated_smooth hφ n) (row_iterated_compact hs n) hu,←iteratedDeriv_succ] lemma row_integral_deriv_zero {φ : ℝ → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (hs : HasCompactSupport φ) : (∫ x, deriv φ x)=0 := by have he : rowConv φ (fun _ => (1:ℝ))=fun _ => ∫ x,φ x := by funext x; simp [rowConv,convolution_def] have hd := congrFun (rowConv_deriv (u:=fun _ => (1:ℝ)) hφ hs continuous_const) 0 rw [he] at hd simpa [rowConv,convolution_def] using hd.symm lemma row_integral_iterated_zero {φ : ℝ → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (hs : HasCompactSupport φ) {n : ℕ} (hn : 1 ≤ n) : (∫ x,iteratedDeriv n φ x)=0 := by obtain ⟨k,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n≠0) rw [iteratedDeriv_succ] exact row_integral_deriv_zero (row_iterated_smooth hφ k) (row_iterated_compact hs k) lemma rowConv_iterated_bound {φ u : ℝ → ℝ} {L : ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) (hs : HasCompactSupport φ) (hu : Continuous u) (_hL : 0 ≤ L) (hlip : ∀ x y, |u x-u y| ≤ L*|x-y|) {n : ℕ} (hn : 1 ≤ n) : ∃ K : ℝ, ∀ x, |iteratedDeriv n (rowConv φ u) x| ≤ K := by let ψ := iteratedDeriv n φ have hcψ : Continuous ψ := (row_iterated_smooth hφ n).continuous have hsψ : HasCompactSupport ψ := row_iterated_compact hs n have hiψ : Integrable ψ := hcψ.integrable_of_hasCompactSupport hsψ have him : Integrable (fun y => |ψ y| *|y|) := (hcψ.abs.mul continuous_abs).integrable_of_hasCompactSupport (hsψ.abs.mul_right) refine ⟨L*(∫ y, |ψ y| *|y|),fun x => ?_⟩ rw [rowConv_iterated hφ hs hu n] have hi := rowConv_integrable hcψ hsψ hu x have he : rowConv ψ u x=∫ y,ψ y*(u (x-y)-u x) := by simp only [rowConv,convolution_def,ContinuousLinearMap.mul_apply'] rw [show (fun y => ψ y*(u (x-y)-u x))=(fun y => ψ y*u (x-y)-ψ y*u x) by funext y; ring, integral_sub hi (hiψ.mul_const (u x)),integral_mul_const,row_integral_iterated_zero hφ hs hn] simp rw [he] calc _ ≤ ∫ y, |ψ y*(u (x-y)-u x)| := abs_integral_le_integral_abs _ ≤ ∫ y,L*(|ψ y| *|y|) := integral_mono_of_nonneg (Eventually.of_forall (by intro y; positivity)) (him.const_mul L) (Eventually.of_forall (fun y => by dsimp only rw [abs_mul] have hh := hlip (x-y) x simp only [sub_sub_cancel_left,abs_neg] at hh nlinarith [mul_le_mul_of_nonneg_left hh (abs_nonneg (ψ y))])) _ = _ := integral_const_mul _ _ end MicroscopicJamming open Set Filter MeasureTheory open scoped Topology namespace MicroscopicJamming lemma rowReal_Lipschitz {u : ℝ → ℝ} {L : ℝ} (hL : 0 ≤ L) (h : ∀ x y, |u x-u y| ≤ L*|x-y|) : LipschitzWith ⟨L,hL⟩ u := by apply LipschitzWith.of_dist_le_mul intro x y change |u x-u y| ≤ L*|x-y| exact h x y lemma rowConv_lipschitz {φ u : ℝ → ℝ} {L : ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hn : ∀ x, 0 ≤ φ x) (h1 : (∫ x,φ x)=1) (hu : Continuous u) (hlip : ∀ x y, |u x-u y| ≤ L*|x-y|) : ∀ x y, |rowConv φ u x-rowConv φ u y| ≤ L*|x-y| := by intro x y have hi : Integrable φ := hφ.integrable_of_hasCompactSupport hs change |(∫ z,φ z*u (x-z))-(∫ z,φ z*u (y-z))| ≤ _ rw [←integral_sub (rowConv_integrable hφ hs hu x) (rowConv_integrable hφ hs hu y)] calc _ ≤ ∫ z,|φ z*u (x-z)-φ z*u (y-z)| := abs_integral_le_integral_abs _ ≤ ∫ z,φ z*(L*|x-y|) := integral_mono_of_nonneg (Eventually.of_forall (fun _ => abs_nonneg _)) (hi.mul_const _) (Eventually.of_forall (fun z => by dsimp only rw [←mul_sub,abs_mul,abs_of_nonneg (hn z)] have hh := hlip (x-z) (y-z) simp only [sub_sub_sub_cancel_right] at hh exact mul_le_mul_of_nonneg_left hh (hn z))) _ = _ := by rw [integral_mul_const,h1,one_mul] lemma rowConv_concave {φ u : ℝ → ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hn : ∀ x, 0 ≤ φ x) (hu : Continuous u) (hc : ConcaveOn ℝ univ u) : ConcaveOn ℝ univ (rowConv φ u) := by refine ⟨convex_univ,?_⟩ intro x _ y _ a b ha hb hab simp only [smul_eq_mul] change a*(∫ z,φ z*u (x-z))+b*(∫ z,φ z*u (y-z)) ≤ ∫ z,φ z*u (a*x+b*y-z) rw [←integral_const_mul,←integral_const_mul,←integral_add ((rowConv_integrable hφ hs hu x).const_mul a) ((rowConv_integrable hφ hs hu y).const_mul b)] apply integral_mono ((rowConv_integrable hφ hs hu x).const_mul a |>.add ((rowConv_integrable hφ hs hu y).const_mul b)) (rowConv_integrable hφ hs hu _) intro z have hh := hc.2 (mem_univ (x-z)) (mem_univ (y-z)) ha hb hab simp only [smul_eq_mul] at hh have he : a*(x-z)+b*(y-z)=a*x+b*y-z := by calc _ = a*x+b*y-(a+b)*z := by ring _ = _ := by rw [hab]; ring rw [he] at hh dsimp only [Pi.add_apply] nlinarith [mul_le_mul_of_nonneg_left hh (hn z)] lemma rowConv_upper {φ u : ℝ → ℝ} {B : ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hn : ∀ x, 0 ≤ φ x) (h1 : (∫ x,φ x)=1) (hu : Continuous u) (hb : ∀ x,u x ≤ B) (x : ℝ) : rowConv φ u x ≤ B := by have hi : Integrable φ := hφ.integrable_of_hasCompactSupport hs calc _ ≤ ∫ z,φ z*B := integral_mono (rowConv_integrable hφ hs hu x) (hi.mul_const B) (fun z => mul_le_mul_of_nonneg_left (hb _) (hn z)) _ = B := by rw [integral_mul_const,h1,one_mul] lemma rowConv_approx {φ u : ℝ → ℝ} {L δ : ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hn : ∀ x, 0 ≤ φ x) (h1 : (∫ x,φ x)=1) (hu : Continuous u) (hL : 0 ≤ L) (hr : ∀ x, φ x≠0 → |x| ≤ δ) (hlip : ∀ x y, |u x-u y| ≤ L*|x-y|) (x : ℝ) : |rowConv φ u x-u x| ≤ L*δ := by have hi : Integrable φ := hφ.integrable_of_hasCompactSupport hs have he : rowConv φ u x-u x=∫ y,φ y*(u (x-y)-u x) := by change (∫ y,φ y*u (x-y))-u x= _ simp only [mul_sub] rw [integral_sub (rowConv_integrable hφ hs hu x) (hi.mul_const _),integral_mul_const,h1,one_mul] rw [he] calc _ ≤ ∫ y,|φ y*(u (x-y)-u x)| := abs_integral_le_integral_abs _ ≤ ∫ y,φ y*(L*δ) := integral_mono_of_nonneg (Eventually.of_forall (fun _ => abs_nonneg _)) (hi.mul_const _) (Eventually.of_forall (fun y => by dsimp only by_cases hz : φ y=0 · simp [hz] rw [abs_mul,abs_of_nonneg (hn y)] apply mul_le_mul_of_nonneg_left _ (hn y) have hh := hlip (x-y) x simp only [sub_sub_cancel_left,abs_neg] at hh exact hh.trans (mul_le_mul_of_nonneg_left (hr y hz) hL))) _ = _ := by rw [integral_mul_const,h1,one_mul] end MicroscopicJamming open Set Filter MeasureTheory open scoped Topology namespace MicroscopicJamming lemma row_smooth_bounded_analytic {u : ℝ → ℝ} {L B Q : ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) (hc : ConcaveOn ℝ univ u) (hL : 0 ≤ L) (hlip : ∀ x y, |u x-u y| ≤ L*|x-y|) (hb : ∀ x,u x ≤ B) (hd : ∀ n : ℕ,2 ≤ n → ∃ K : ℝ,∀ x,|iteratedDeriv n u x| ≤ K) : ∃ A C H : ℝ, RowAnalyticTerminal u A B C 0 Q ∧ RowBoundedTerminal u L H := by obtain ⟨K,hK⟩ := hd 2 le_rfl have hK0 : 0 ≤ K := (abs_nonneg _).trans (hK 0) have hslope (x : ℝ) : |deriv u x| ≤ L := by have hh := norm_deriv_le_of_lipschitz (rowReal_Lipschitz hL hlip) (x₀:=x) change |deriv u x| ≤ L at hh exact hh have hcurve (x : ℝ) : |deriv (deriv u) x| ≤ K := by simpa [iteratedDeriv_succ,iteratedDeriv_zero] using hK x have hnonpos (x : ℝ) : deriv (deriv u) x ≤ 0 := by have ha : Antitone (deriv u) := fun a b hab => hc.antitoneOn_deriv (fun z _ => hu.differentiable (by simp) z) (mem_univ a) (mem_univ b) hab exact ha.deriv_nonpos refine ⟨|u 0|+L,K,K,?_,?_,?_,hL,hK0,hslope,hcurve⟩ · refine ⟨hu,by positivity,hK0,le_rfl,by simp,hd,fun x => ⟨?_,hb x,(abs_le.mp (hcurve x)).1,hnonpos x⟩⟩ have hh := (abs_le.mp (hlip x 0)).1 simp only [sub_zero] at hh have hx : |x| ≤ 1+x^2 := by nlinarith [sq_nonneg (|x|-1),sq_abs x] nlinarith [le_abs_self (u 0),neg_abs_le (u 0),mul_le_mul_of_nonneg_left hx hL,mul_nonneg (abs_nonneg (u 0)) (sq_nonneg x)] · exact hu.differentiable (by simp) · exact (contDiff_infty_iff_deriv.mp hu).2.differentiable (by simp) lemma rowCap_lipschitz {u : ℝ → ℝ} {L : ℝ} (_hL : 0 ≤ L) (hlip : ∀ x y,|u x-u y| ≤ L*|x-y|) (B : ℝ) : ∀ x y,|min (u x) B-min (u y) B| ≤ L*|x-y| := by intro x y have h := abs_min_sub_min_le_max (u x) B (u y) B simp only [sub_self,abs_zero,max_eq_left (abs_nonneg (u x-u y))] at h exact h.trans (hlip x y) lemma rowCap_error {u : ℝ → ℝ} {L ε : ℝ} (hL : 0 ≤ L) (hε : 0<ε) (hlip : ∀ x y,|u x-u y| ≤ L*|x-y|) (x : ℝ) : |min (u x) (u 0+L^2/ε)-u x| ≤ ε*x^2 := by by_cases hx : u x ≤ u 0+L^2/ε · rw [min_eq_left hx,sub_self,abs_zero]; positivity · rw [min_eq_right (le_of_not_ge hx),abs_of_nonpos (by linarith)] have hh := (abs_le.mp (hlip x 0)).2 simp only [sub_zero] at hh have hdiv : (L^2/ε)*ε=L^2 := div_mul_cancel₀ _ hε.ne' have hab := sq_abs x have hs := two_mul_le_add_sq L (ε*|x|) simp only [mul_pow,sq_abs] at hs have hp := mul_nonneg hL (abs_nonneg x) have hh2 : ε*(u x-u 0) ≤ ε*(L*|x|) := mul_le_mul_of_nonneg_left hh hε.le apply (mul_le_mul_iff_right₀ hε).mp nlinarith [mul_nonneg hε.le hp] lemma row_lipschitz_analytic_approx_one {u : ℝ → ℝ} {L ε Q : ℝ} (hc : ConcaveOn ℝ univ u) (hL : 0 ≤ L) (hlip : ∀ x y,|u x-u y| ≤ L*|x-y|) (hε : 0<ε) : ∃ v : ℝ → ℝ, ∃ A B C H : ℝ, RowAnalyticTerminal v A B C 0 Q ∧ RowBoundedTerminal v L H ∧ ConcaveOn ℝ univ v ∧ (∀ x,|v x-u x| ≤ ε*(L+1)*(1+x^2)^2) := by let φ : ContDiffBump (0:ℝ) := ⟨ε/2,ε,half_pos hε,half_lt_self hε⟩ let v (x : ℝ) := min (u x) (u 0+L^2/ε) have hvl := rowCap_lipschitz hL hlip (u 0+L^2/ε) have hvc : Continuous v := (rowReal_Lipschitz hL hvl).continuous have hvs : ConcaveOn ℝ univ v := hc.inf (concaveOn_const (c:=u 0+L^2/ε) convex_univ) have hφc : Continuous (φ.normed volume) := φ.continuous_normed have hφs : HasCompactSupport (φ.normed volume) := φ.hasCompactSupport_normed have hφn : ∀ x,0 ≤ φ.normed volume x := φ.nonneg_normed have hφ1 : (∫ x,φ.normed volume x)=1 := φ.integral_normed let w := rowConv (φ.normed volume) v have hw : ContDiff ℝ (↑(⊤:ℕ∞)) w := hφs.contDiff_convolution_left (.mul ℝ ℝ) φ.contDiff_normed hvc.locallyIntegrable have hwc := rowConv_concave hφc hφs hφn hvc hvs have hwl := rowConv_lipschitz hφc hφs hφn hφ1 hvc hvl obtain ⟨A,C,H,hwA,hwB⟩ := row_smooth_bounded_analytic (Q:=Q) hw hwc hL hwl (rowConv_upper hφc hφs hφn hφ1 hvc (fun x => min_le_right _ _)) (fun n hn => rowConv_iterated_bound φ.contDiff_normed hφs hvc hL hvl (by omega)) refine ⟨w,A,u 0+L^2/ε,C,H,hwA,hwB,hwc,fun x => ?_⟩ have hr (y : ℝ) (hy : φ.normed volume y≠0) : |y| ≤ ε := by have hh : y ∈ Function.support (φ.normed volume) := hy rw [φ.support_normed_eq] at hh have hh' : |y|<ε := by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hh exact hh'.le have h1 := rowConv_approx hφc hφs hφn hφ1 hvc hL hr hvl x have h2 := rowCap_error hL hε hlip x have ht : |w x-u x| ≤ |w x-v x|+|v x-u x| := by simpa only [sub_add_sub_cancel] using abs_add_le (w x-v x) (v x-u x) have hx : 0 ≤ x^2 := sq_nonneg x have he : ε*(L+x^2) ≤ ε*(L+1)*(1+x^2)^2 := by have hh : L+x^2 ≤ (L+1)*(1+x^2)^2 := by nlinarith [mul_nonneg hL hx,mul_nonneg hL (sq_nonneg (x^2)),sq_nonneg (x^2)] simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh hε.le dsimp [w] at ht ⊢ dsimp [v] at h2 ht linarith end MicroscopicJamming open Set Filter MeasureTheory ProbabilityTheory open scoped Topology namespace MicroscopicJamming lemma rowBounded_lip {u : ℝ → ℝ} {L H : ℝ} (hu : RowBoundedTerminal u L H) : ∀ x y,|u x-u y| ≤ L*|x-y| := by intro x y have hh := Convex.norm_image_sub_le_of_norm_deriv_le (fun z (_ : z∈(univ : Set ℝ)) => hu.diff z) (fun z _ => by simpa only [Real.norm_eq_abs] using hu.slope z) convex_univ (mem_univ y) (mem_univ x) simpa only [Real.norm_eq_abs] using hh lemma row_pointwise_lip {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {L : ℝ} (hlip : ∀ n x y,|v n x-v n y| ≤ L*|x-y|) (hlim : ∀ x,Tendsto (fun n => v n x) atTop (𝓝 (u x))) : ∀ x y,|u x-u y| ≤ L*|x-y| := by intro x y exact le_of_tendsto ((hlim x).sub (hlim y)).abs (Eventually.of_forall (fun n => hlip n x y)) lemma row_linear_limit_bound {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {L : ℝ} (hlip : ∀ n x y,|v n x-v n y| ≤ L*|x-y|) (hlim : Tendsto (fun n => v n 0) atTop (𝓝 (u 0))) : ∃ M : ℝ,0 ≤ M ∧ ∀ n x,|v n x| ≤ M+L*|x| := by obtain ⟨M,hM⟩ := hlim.abs.bddAbove_range refine ⟨max M 0,le_max_right _ _,fun n x => ?_⟩ have hh := hlip n x 0 simp only [sub_zero] at hh have h0 : |v n 0| ≤ M := hM ⟨n,rfl⟩ have ht : |v n x| ≤ |v n x-v n 0|+|v n 0| := by simpa only [sub_add_cancel] using abs_add_le (v n x-v n 0) (v n 0) linarith [le_max_left M 0] lemma row_linear_operator_limit {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {M L a T : ℝ} (_hM : 0 ≤ M) (hL : 0 ≤ L) (ha : 0 ≤ a) (ha1 : a ≤ 1) (hu : Continuous u) (hv : ∀ n,Continuous (v n)) (hb : ∀ n x,|v n x| ≤ M+L*|x|) (hlim : ∀ x,Tendsto (fun n => v n x) atTop (𝓝 (u x))) (x : ℝ) : Tendsto (fun n => gaussianRowOperator a T (v n) x) atTop (𝓝 (gaussianRowOperator a T u x)) := by let E (y : ℝ) := Real.exp (M+L*|y|) have hg : RowExpGrowth E := ⟨Real.exp M,L,(Real.exp_pos _).le,hL,fun y => by rw [show E y=Real.exp (M+L*|y|) from rfl,abs_of_pos (Real.exp_pos _),Real.exp_add]⟩ have hi : Integrable (fun z => E (x+Real.sqrt T*z)) (gaussianReal 0 1) := hg.integrable_affine (by fun_prop) _ _ have hvb (n : ℕ) (y : ℝ) : ‖v n y‖ ≤ E y := by rw [Real.norm_eq_abs] exact (hb n y).trans (by have hh := Real.add_one_le_exp (M+L*|y|); dsimp [E]; linarith) have heb (n : ℕ) (y : ℝ) : ‖Real.exp (a*v n y)‖ ≤ E y := by rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] apply Real.exp_le_exp.mpr calc a*v n y ≤ a*|v n y| := mul_le_mul_of_nonneg_left (le_abs_self _) ha _ ≤ |v n y| := mul_le_of_le_one_left (abs_nonneg _) ha1 _ ≤ _ := hb n y by_cases hz : a=0 · simp only [gaussianRowOperator,hz,ite_true] exact tendsto_integral_of_dominated_convergence (fun z => E (x+Real.sqrt T*z)) (fun n => ((hv n).comp (by fun_prop)).aestronglyMeasurable) hi (fun n => Eventually.of_forall (fun z => hvb n _)) (Eventually.of_forall (fun z => hlim _)) · have he : Tendsto (fun n => gaussianHeat (fun y => Real.exp (a*v n y)) T x) atTop (𝓝 (gaussianHeat (fun y => Real.exp (a*u y)) T x)) := by exact tendsto_integral_of_dominated_convergence (fun z => E (x+Real.sqrt T*z)) (fun n => (by fun_prop)) hi (fun n => Eventually.of_forall (fun z => heb n _)) (Eventually.of_forall (fun z => (Real.continuous_exp.tendsto _).comp ((hlim (x+Real.sqrt T*z)).const_mul a))) have hui : Integrable (fun z => Real.exp (a*u (x+Real.sqrt T*z))) (gaussianReal 0 1) := by apply hi.mono' (by fun_prop) filter_upwards [] with z exact le_of_tendsto (((Real.continuous_exp.tendsto _).comp ((hlim (x+Real.sqrt T*z)).const_mul a)).norm) (Eventually.of_forall (fun n => heb n _)) have hp : 0 Real.exp (a*u y)) T x := integral_exp_pos hui simpa only [gaussianRowOperator,hz,ite_false] using (he.log hp.ne').const_mul (1/a) lemma row_composition_terminal_limit {u : ℝ → ℝ} {v : ℕ → ℝ → ℝ} {L : ℝ} {H : ℕ → ℝ} (hv : ∀ n,RowBoundedTerminal (v n) L (H n)) (hlim : ∀ x,Tendsto (fun n => v n x) atTop (𝓝 (u x))) (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) : ∀ x,Tendsto (fun n => gaussianRowComposition rs (v n) x) atTop (𝓝 (gaussianRowComposition rs u x)) := by induction rs with | nil => exact hlim | cons r rs ih => have htail : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2 := fun r hr => hrs r (List.mem_cons_of_mem _ hr) have hlimr := ih htail have hbr (n : ℕ) := rowBoundedComposition_terminal (hv n) rs htail have hlip (n : ℕ) := rowBounded_lip (hbr n).choose_spec have hlc := row_pointwise_lip hlip hlimr obtain ⟨M,hM,hb⟩ := row_linear_limit_bound hlip (hlimr 0) intro x exact row_linear_operator_limit hM (hv 0).L_nonneg (hrs r (by simp)).1 (hrs r (by simp)).2.1 (rowReal_Lipschitz (hv 0).L_nonneg hlc).continuous (fun n => (hbr n).choose_spec.diff.continuous) hb hlimr x end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming lemma row_lipschitz_profile_conclusions {u : ℝ → ℝ} {L Q : ℝ} (hc : ConcaveOn ℝ univ u) (hL : 0 ≤ L) (hlip : ∀ x y,|u x-u y| ≤ L*|x-y|) (hQ : 0 ≤ Q) : RowProfileConclusions u Q := by rcases eq_or_lt_of_le hQ with h0 | hQ · subst Q; exact rowProfile_zero u let δ (n : ℕ) := 1/((n:ℝ)+1) have hδ (n : ℕ) : 0<δ n := by dsimp [δ]; positivity have hδlim : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ) choose v A B C H hv hvb hvc herr using fun n => row_lipschitz_analytic_approx_one (Q:=Q) hc hL hlip (hδ n) let e (n : ℕ) := δ n*(L+1) have hen (n : ℕ) : 0 ≤ e n := by dsimp [e]; positivity have helim : Tendsto e atTop (𝓝 0) := by simpa only [zero_mul] using hδlim.mul_const (L+1) have hlim (x : ℝ) : Tendsto (fun n => v n x) atTop (𝓝 (u x)) := by rw [tendsto_iff_dist_tendsto_zero] apply squeeze_zero (g:=fun n => e n*(1+x^2)^2) (fun _ => dist_nonneg) (fun n => by rw [Real.dist_eq]; exact herr n x) simpa only [zero_mul] using helim.mul_const ((1+x^2)^2) let K := Real.exp ((10+8*L)*Q) have hbound (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) (ht : gaussianStepTime rs=Q) (n m : ℕ) : |gaussianRowComposition rs (v n) 0-gaussianRowComposition rs (v m) 0| ≤ (e n+e m)*K := by apply row_composition_weighted_Lipschitz hQ (hv n) (hv m) (hvb n) (hvb m) (add_nonneg (hen n) (hen m)) hrs ht intro x have hh := abs_sub_le (v n x) (u x) (v m x) rw [abs_sub_comm (u x)] at hh have h1 : |v n x-u x| ≤ e n*(1+x^2)^2 := herr n x have h2 : |v m x-u x| ≤ e m*(1+x^2)^2 := herr m x nlinarith have hfinite (rs : List (ℝ × ℝ)) (hrs : ∀ r∈rs,0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) := row_composition_terminal_limit hvb hlim rs hrs 0 apply row_terminal_profile_passage hQ hv hvb hvc (show Tendsto (fun n => e n*K) atTop (𝓝 0) by simpa only [zero_mul] using helim.mul_const K) _ (fun rs hrs _ => hfinite rs hrs) intro rs hrs ht n have hh := le_of_tendsto_of_tendsto ((tendsto_const_nhds.sub (hfinite rs hrs)).abs) ((tendsto_const_nhds.add helim).mul_const K) (Eventually.of_forall (hbound rs hrs ht n)) simpa only [add_zero,abs_sub_comm] using hh end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming def rowCapSlope (R x : ℝ) := min (max (-1-x) 0) R def rowHingeCap (β R x : ℝ) := β*(rowCapSlope R x*(1+x)+(rowCapSlope R x)^2/2) lemma rowCapSlope_bounds {R : ℝ} (hR : 0 ≤ R) (x : ℝ) : 0 ≤ rowCapSlope R x ∧ rowCapSlope R x ≤ R := ⟨le_min (le_max_right _ _) hR,min_le_right _ _⟩ lemma rowHingeCap_opt {β R s : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) (hs : 0 ≤ s) (hsR : s ≤ R) (x : ℝ) : rowHingeCap β R x ≤ β*(s*(1+x)+s^2/2) := by apply mul_le_mul_of_nonneg_left _ hβ dsimp [rowCapSlope] by_cases hx : -1-x ≤ 0 · rw [max_eq_right hx,min_eq_left hR] nlinarith [mul_nonneg hs (show 0 ≤ 1+x by linarith),sq_nonneg s] · rw [max_eq_left (le_of_not_ge hx)] by_cases hxR : -1-x ≤ R · rw [min_eq_left hxR]; nlinarith [sq_nonneg (s-(-1-x))] · rw [min_eq_right (le_of_not_ge hxR)] nlinarith [sq_nonneg (R-s),mul_nonneg (show 0 ≤ R-s by linarith) (show 0 ≤ -1-x-R by linarith)] lemma rowHingeCap_concave {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) : ConcaveOn ℝ univ (rowHingeCap β R) := by refine ⟨convex_univ,?_⟩ intro x _ y _ a b ha hb hab simp only [smul_eq_mul] let s := rowCapSlope R (a*x+b*y) have hs := rowCapSlope_bounds hR (a*x+b*y) have hx := rowHingeCap_opt hβ hR hs.1 hs.2 x have hy := rowHingeCap_opt hβ hR hs.1 hs.2 y have hc : rowHingeCap β R (a*x+b*y)=a*(β*(s*(1+x)+s^2/2))+b*(β*(s*(1+y)+s^2/2)) := by dsimp [rowHingeCap,s] nlinarith [congrArg (fun t : ℝ => t*β*rowCapSlope R (a*x+b*y)) hab, congrArg (fun t : ℝ => t*β*(rowCapSlope R (a*x+b*y))^2) hab] rw [hc] exact add_le_add (mul_le_mul_of_nonneg_left hx ha) (mul_le_mul_of_nonneg_left hy hb) lemma rowHingeCap_sub_le {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) (x y : ℝ) : rowHingeCap β R x-rowHingeCap β R y ≤ β*R*|x-y| := by have hs := rowCapSlope_bounds hR y have hh := rowHingeCap_opt hβ hR hs.1 hs.2 x have hb : β*rowCapSlope R y ≤ β*R := mul_le_mul_of_nonneg_left hs.2 hβ have ht := mul_le_mul_of_nonneg_left (le_abs_self (x-y)) (mul_nonneg hβ hs.1) have ht' := mul_le_mul_of_nonneg_right hb (abs_nonneg (x-y)) dsimp [rowHingeCap] at hh ⊢ nlinarith lemma rowHingeCap_lip {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) (x y : ℝ) : |rowHingeCap β R x-rowHingeCap β R y| ≤ β*R*|x-y| := by rw [abs_le] have h1 := rowHingeCap_sub_le hβ hR x y have h2 := rowHingeCap_sub_le hβ hR y x rw [abs_sub_comm y x] at h2 constructor <;> linarith lemma rowHingeCap_upper {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) (x : ℝ) : rowHingeCap β R x ≤ 0 := by simpa using rowHingeCap_opt hβ hR (le_refl 0) hR x lemma rowHingeCap_lower {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 ≤ R) (x : ℝ) : -β*rowQuadraticHinge x ≤ rowHingeCap β R x := by dsimp [rowHingeCap,rowQuadraticHinge] by_cases hx : -1-x ≤ 0 · simp [rowCapSlope,max_eq_right hx,min_eq_left hR] · rw [max_eq_left (le_of_not_ge hx)] nlinarith [mul_nonneg hβ (sq_nonneg (rowCapSlope R x-(-1-x)))] lemma rowHinge_quadratic {β : ℝ} (hβ : 0 ≤ β) (x : ℝ) : -β*(1+x^2) ≤ -β*rowQuadraticHinge x := by dsimp [rowQuadraticHinge] by_cases hx : -1-x ≤ 0 · rw [max_eq_right hx]; nlinarith [mul_nonneg hβ (sq_nonneg x)] · rw [max_eq_left (le_of_not_ge hx)] nlinarith [mul_nonneg hβ (sq_nonneg (x-1))] lemma rowHingeCap_eq {β R : ℝ} (hR : 0 ≤ R) {x : ℝ} (hx : -1-x ≤ R) : rowHingeCap β R x = -β*rowQuadraticHinge x := by dsimp [rowHingeCap,rowCapSlope,rowQuadraticHinge] rw [min_eq_left (max_le hx hR)] by_cases h : -1-x ≤ 0 · simp [max_eq_right h] · rw [max_eq_left (le_of_not_ge h)]; ring lemma rowHingeCap_error {β R : ℝ} (hβ : 0 ≤ β) (hR : 0 < R) (x : ℝ) : |rowHingeCap β R x-(-β*rowQuadraticHinge x)| ≤ (2*β/R^2)*(1+x^2)^2 := by rw [abs_of_nonneg (sub_nonneg.mpr (rowHingeCap_lower hβ hR.le x))] by_cases hx : -1-x ≤ R · rw [rowHingeCap_eq hR.le hx,sub_self]; positivity · have hR2 : 0.2 ha.le nlinarith [sq_nonneg (x-1)] calc _ ≤ (β*(1+x^2))*R^2 := mul_le_mul_of_nonneg_right hb hR2.le _ ≤ (β*(1+x^2))*(2*(1+x^2)) := mul_le_mul_of_nonneg_left hs (by positivity) _ = _ := by ring end MicroscopicJamming open Set Filter MeasureTheory open scoped Topology namespace MicroscopicJamming lemma rowConv_quadratic_lower {φ u : ℝ → ℝ} {A : ℝ} (hφ : Continuous φ) (hs : HasCompactSupport φ) (hn : ∀ x,0 ≤ φ x) (h1 : (∫ x,φ x)=1) (hu : Continuous u) (hA : 0 ≤ A) (hb : ∀ x,-A*(1+x^2) ≤ u x) (hr : ∀ x,φ x≠0 → |x| ≤ 1) (x : ℝ) : -(3*A)*(1+x^2) ≤ rowConv φ u x := by have hi : Integrable φ := hφ.integrable_of_hasCompactSupport hs calc _ = ∫ y,φ y*(-(3*A)*(1+x^2)) := by rw [integral_mul_const,h1,one_mul] _ ≤ _ := integral_mono (hi.mul_const _) (rowConv_integrable hφ hs hu x) (fun y => by by_cases hz : φ y=0 · simp [hz] apply mul_le_mul_of_nonneg_left _ (hn y) have hy : y^2 ≤ 1 := by have hh:=hr y hz; nlinarith [sq_abs y,abs_nonneg y] have hh : 1+(x-y)^2 ≤ 3*(1+x^2) := by nlinarith [sq_nonneg (x+y)] have hm := mul_le_mul_of_nonneg_left hh hA linarith [hb (x-y)]) lemma row_hinge_analytic_approx_one {β R δ Q : ℝ} (hβ : 0<β) (hR : 0 (rowHinge_quadratic hβ.le y).trans (rowHingeCap_lower hβ.le hR.le y)) (fun y hy => (hr y hy).trans hδ1) x obtain ⟨A,C,H,hvA,hvB⟩ := row_smooth_bounded_analytic (Q:=Q) hv hvc hL hvl hvu (fun n hn => rowConv_iterated_bound φ.contDiff_normed hφs huc hL hul (by omega)) refine ⟨v,C,H,⟨hvA.1,by positivity,hvA.2.2.1,hvA.2.2.2.1,hvA.2.2.2.2.1,hvA.2.2.2.2.2.1,fun x => ⟨hlow x,(hvA.2.2.2.2.2.2 x).2⟩⟩,hvB,hvc,fun x => ?_⟩ have hm := rowConv_approx hφc hφs hφn hφ1 huc hL hr hul x have he := rowHingeCap_error hβ.le hR x have ht : |v x-(-β*rowQuadraticHinge x)| ≤ |v x-u x|+|u x-(-β*rowQuadraticHinge x)| := by simpa only [sub_add_sub_cancel] using abs_add_le (v x-u x) (u x-(-β*rowQuadraticHinge x)) have hh : β*R*δ ≤ β*R*δ*(1+x^2)^2 := by have hx : 1 ≤ (1+x^2)^2 := by nlinarith [sq_nonneg x,sq_nonneg (x^2)] exact le_mul_of_one_le_right (by positivity) hx dsimp [v,u] at hm ht ⊢ nlinarith end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming lemma row_concave_quadratic_slope {f : ℝ → ℝ} {K : ℝ} (hf : Differentiable ℝ f) (hc : ConcaveOn ℝ univ f) (hK : 0 ≤ K) (hb : ∀ x,|f x| ≤ K*(1+x^2)) (x : ℝ) : |deriv f x| ≤ (8*K)*(1+|x|) := by let d := 1+|x| have hd : 0 by simpa using (hb y).2.2) change |deriv (gaussianRowComposition (gaussianStepRemainder rs t) u) x| ≤ _ exact row_concave_quadratic_slope hd hc (by have := hu.2.1; positivity) (gaussianStepPath_global_value_bound hQ hu hrs hT ht) x end MicroscopicJamming end end OAI