| import Mathlib |
| import OAI.Probability.JammingConcavity.GaussianTwiceTerminal |
|
|
| namespace OAI |
|
|
| /-! Gaussian Step Slope. -/ |
|
|
| 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 MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal ENNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma slope_bound_from_curvature {f : ℝ → ℝ} {D K : ℝ} |
| (hf : Differentiable ℝ f) (hf' : Differentiable ℝ (deriv f)) |
| (hD : 0 ≤ D) (hK : 0 ≤ K) (hvals : |f 1-f 0| ≤ D) |
| (hb : ∀ z, |deriv (deriv f) z| ≤ K) : |
| ∀ x, |deriv f x| ≤ (D+K)*(1+|x|) := by |
| obtain ⟨c,hc,he⟩ := exists_deriv_eq_slope f (show (0:ℝ)<1 by norm_num) |
| hf.continuous.continuousOn hf.differentiableOn |
| have hfc : |deriv f c| ≤ D := by rw [he]; simpa using hvals |
| intro x |
| have hh := Convex.norm_image_sub_le_of_norm_deriv_le |
| (f := deriv f) (s := univ) (C := K) (fun z _ => hf' z) |
| (fun z _ => by simpa only [Real.norm_eq_abs] using hb z) |
| convex_univ (x := c) (y := x) (mem_univ _) (mem_univ _) |
| simp only [Real.norm_eq_abs] at hh |
| have hd : |x-c| ≤ 1+|x| := by |
| have ht := abs_sub x c |
| rw [abs_of_pos hc.1] at ht |
| linarith [hc.2] |
| have ht := abs_add_le (deriv f x-deriv f c) (deriv f c) |
| rw [sub_add_cancel] at ht |
| have hm := mul_le_mul_of_nonneg_left hd hK |
| nlinarith [mul_nonneg hD (abs_nonneg x)] |
|
|
| theorem gaussian_step_slope : GaussianStepSlopeStatement := by |
| intro A B C κ Q hQ hA hC hk hsub |
| let K := C+κ/(1-κ*Q) |
| let D := 2*|B|+A*(3+2*Q) |
| have hden : 0 < 1-κ*Q := by linarith |
| have hK : 0 ≤ K := add_nonneg hC (div_nonneg hk hden.le) |
| have hD : 0 ≤ D := by dsimp [D]; positivity |
| refine ⟨D+K,add_nonneg hD hK,?_⟩ |
| intro u hu rs hrs hsum |
| obtain ⟨hf,hf',hb⟩ := gaussian_step_bounds u A B C κ Q hQ hu rs hrs hsum |
| have hR := gaussianStepTime_nonneg (fun r hr => (hrs r hr).2.2) |
| have hdenR : 0 < 1-κ*gaussianStepTime rs := by |
| have hh := mul_le_mul_of_nonneg_left hsum hk |
| linarith |
| have hcap : κ/(1-κ*gaussianStepTime rs) ≤ κ/(1-κ*Q) := |
| div_le_div_of_nonneg_left hk hden (by nlinarith [mul_le_mul_of_nonneg_left hsum hk]) |
| have hcurv (z : ℝ) : |deriv (deriv (gaussianRowComposition rs u)) z| ≤ K := by |
| rw [abs_le] |
| have hz := (hb z).2 |
| dsimp [K] |
| constructor <;> linarith [div_nonneg hk hden.le,hz.1,hz.2] |
| have habs (z : ℝ) : |gaussianRowComposition rs u z| ≤ |B|+A*(1+z^2+Q) := by |
| rw [abs_le] |
| have hz := (hb z).1 |
| have hadd := mul_le_mul_of_nonneg_left hsum hA |
| have hp : 0 ≤ A*(1+z^2+Q) := mul_nonneg hA (by positivity) |
| constructor <;> nlinarith [abs_nonneg B,le_abs_self B] |
| have hv : |gaussianRowComposition rs u 1-gaussianRowComposition rs u 0| ≤ D := by |
| have hh := abs_sub (gaussianRowComposition rs u 1) (gaussianRowComposition rs u 0) |
| have h1 := habs 1 |
| have h0 := habs 0 |
| dsimp [D] |
| norm_num at h1 h0 |
| linarith |
| exact slope_bound_from_curvature hf hf' hD hK hv hcurv |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal ENNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma heat_affine_continuousAt {v : ℝ → ℝ} (hv : Continuous v) |
| (hg : HeatQuadraticGrowth v) {a b : ℝ → ℝ} {t₀ : ℝ} |
| (ha : ContinuousAt a t₀) (hb : ContinuousAt b t₀) : |
| ContinuousAt (fun t => ∫ z : ℝ, v (a t+b t*z) ∂gaussianReal 0 1) t₀ := by |
| let K := 1+|a t₀|+|b t₀| |
| have hK : 0 ≤ K := by dsimp [K]; positivity |
| have hea : ∀ᶠ t in 𝓝 t₀, |a t| ≤ K := |
| (ha.abs.tendsto.eventually (eventually_lt_nhds (show |a t₀| < K by |
| dsimp [K]; linarith [abs_nonneg (b t₀)]))).mono fun _ h => h.le |
| have heb : ∀ᶠ t in 𝓝 t₀, |b t| ≤ K := |
| (hb.abs.tendsto.eventually (eventually_lt_nhds (show |b t₀| < K by |
| dsimp [K]; linarith [abs_nonneg (a t₀)]))).mono fun _ h => h.le |
| obtain ⟨C,hC,hgrow⟩ := hg |
| refine continuousAt_of_dominated |
| (μ := gaussianReal 0 1) (bound := fun z => C*(1+2*K)^2*(1+|z|)^2) |
| (Eventually.of_forall fun t => (hv.measurable.comp (by fun_prop)).aestronglyMeasurable) |
| ?_ ((gaussian_abs_one_pow_integrable 2).const_mul _) ?_ |
| · filter_upwards [hea,heb] with t hat hbt |
| filter_upwards [] with z |
| have hab : 1+|a t+b t*z| ≤ (1+2*K)*(1+|z|) := |
| (heat_affine_bound (a t) (b t) z).trans |
| (mul_le_mul_of_nonneg_right (by linarith) (by positivity)) |
| rw [Real.norm_eq_abs] |
| calc |
| |v (a t+b t*z)| ≤ C*(1+|a t+b t*z|)^2 := hgrow _ |
| _ ≤ C*((1+2*K)*(1+|z|))^2 := |
| mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hab 2) hC |
| _ = _ := by ring |
| · filter_upwards [] with z |
| exact hv.continuousAt.comp (ha.add (hb.mul_const z)) |
|
|
| lemma gaussianHeat_time_continuous {v : ℝ → ℝ} (hv : Continuous v) |
| (hg : HeatQuadraticGrowth v) (x : ℝ) : |
| Continuous (fun T => gaussianHeat v T x) := by |
| rw [continuous_iff_continuousAt] |
| intro T |
| exact heat_affine_continuousAt hv hg continuousAt_const Real.continuous_sqrt.continuousAt |
|
|
| lemma gaussianRowOperator_time_continuous {u : ℝ → ℝ} {A B C κ Q a : ℝ} |
| (hu : Twice.RowTwiceTerminal u A B C κ Q) (ha : 0 ≤ a) (x : ℝ) : |
| Continuous (fun T => gaussianRowOperator a T u x) := by |
| by_cases haz : a=0 |
| · subst a |
| simpa [gaussianRowOperator] using gaussianHeat_time_continuous |
| hu.differentiable.continuous (Twice.row_terminal_quadratic_growth hu) x |
| · have he : Continuous (fun y => Real.exp (a*u y)) := by |
| exact Real.continuous_exp.comp (continuous_const.mul hu.differentiable.continuous) |
| have hg := (Twice.row_exp_growth hu ha).1 |
| have ht := gaussianHeat_time_continuous he hg x |
| have hp (T : ℝ) : gaussianHeat (fun y => Real.exp (a*u y)) T x ≠ 0 := |
| (integral_exp_pos (heat_affine_quadratic_integrable he.measurable hg x (Real.sqrt T))).ne' |
| simpa [gaussianRowOperator,haz] using (ht.log hp).const_mul (1/a) |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal ENNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma abs_sub_le_of_hasDerivAt_Ioo {f f' : ℝ → ℝ} {s t D : ℝ} |
| (hst : s < t) (hc : ContinuousOn f (Icc s t)) |
| (hd : ∀ r ∈ Ioo s t, HasDerivAt f (f' r) r) |
| (hb : ∀ r ∈ Ioo s t, |f' r| ≤ D) : |f t-f s| ≤ D*(t-s) := by |
| obtain ⟨r,hr,he⟩ := exists_hasDerivAt_eq_slope f f' hst hc hd |
| have hh := hb r hr |
| rw [he,abs_div,abs_of_pos (sub_pos.mpr hst)] at hh |
| exact (div_le_iff₀ (sub_pos.mpr hst)).mp hh |
|
|
| theorem gaussian_step_time : GaussianStepTimeStatement := by |
| intro A B C κ Q hQ hA hC hk hsub |
| obtain ⟨L,hL,hlin⟩ := gaussian_step_slope A B C κ Q hQ hA hC hk hsub |
| let K := C+κ/(1-κ*Q) |
| have hden : 0 < 1-κ*Q := by linarith |
| have hK : 0 ≤ K := add_nonneg hC (div_nonneg hk hden.le) |
| refine ⟨K+2*L^2,by positivity,?_⟩ |
| intro u hu rs hrs a s t ha ha1 hs ht htime x |
| have hR := gaussianStepTime_nonneg (fun r hr => (hrs r hr).2.2) |
| have hRQ : gaussianStepTime rs ≤ Q := by |
| have hm : 0 ≤ max s t := hs.trans (le_max_left _ _) |
| linarith |
| obtain ⟨hdf,hdf',hbf⟩ := gaussian_step_bounds u A B C κ Q hQ hu rs hrs hRQ |
| have hsum : κ*((Q-gaussianStepTime rs)+gaussianStepTime rs)<1 := by simpa using hsub |
| obtain ⟨hkn,hknQ,_⟩ := gaussian_curvature_update hk hR (sub_nonneg.mpr hRQ) hsum |
| have htw : Twice.RowTwiceTerminal (gaussianRowComposition rs u) |
| (A*(1+gaussianStepTime rs)) B C (κ/(1-κ*gaussianStepTime rs)) |
| (Q-gaussianStepTime rs) := by |
| refine ⟨hdf,hdf',mul_nonneg hA (by linarith),hC,hkn,hknQ,fun y => ?_⟩ |
| refine ⟨?_,(hbf y).1.2,(hbf y).2⟩ |
| have hh := (hbf y).1.1 |
| nlinarith [mul_nonneg (mul_nonneg hA hR) (sq_nonneg y)] |
| let f : ℝ → ℝ := fun r => gaussianRowOperator a r (gaussianRowComposition rs u) x |
| let d : ℝ → ℝ := fun r => (1/2:ℝ)* |
| (deriv (deriv (gaussianRowOperator a r (gaussianRowComposition rs u))) x+ |
| a*(deriv (gaussianRowOperator a r (gaussianRowComposition rs u)) x)^2) |
| have hcont : Continuous f := gaussianRowOperator_time_continuous htw ha x |
| have hdiff (r : ℝ) (hr : 0 < r) : HasDerivAt f (d r) r := |
| (Twice.gaussianRowOperator_calculus htw ha hr).2.2 x |
| have hb (r : ℝ) (hr : 0 ≤ r) (hrQ : r+gaussianStepTime rs ≤ Q) : |
| |d r| ≤ (K+2*L^2)*(1+x^2) := by |
| have hrr : ∀ v ∈ (a,r)::rs, 0 ≤ v.1 ∧ v.1 ≤ 1 ∧ 0 ≤ v.2 := by |
| intro v hv |
| rcases List.mem_cons.mp hv with h | h |
| · subst v; exact ⟨ha,ha1,hr⟩ |
| · exact hrs v h |
| have htot : gaussianStepTime ((a,r)::rs) ≤ Q := by |
| simpa [gaussianStepTime] using hrQ |
| obtain ⟨_,_,hbb⟩ := gaussian_step_bounds u A B C κ Q hQ hu ((a,r)::rs) hrr htot |
| have hj := hlin u hu ((a,r)::rs) hrr htot x |
| have hdenr : 0 < 1-κ*gaussianStepTime ((a,r)::rs) := by |
| have hh := mul_le_mul_of_nonneg_left htot hk |
| linarith |
| have hcap : κ/(1-κ*gaussianStepTime ((a,r)::rs)) ≤ κ/(1-κ*Q) := |
| div_le_div_of_nonneg_left hk hden (by nlinarith [mul_le_mul_of_nonneg_left htot hk]) |
| have hcurv : |deriv (deriv (gaussianRowOperator a r (gaussianRowComposition rs u))) x| ≤ K := by |
| rw [abs_le] |
| have hh := (hbb x).2 |
| change -C ≤ deriv (deriv (gaussianRowOperator a r (gaussianRowComposition rs u))) x ∧ |
| deriv (deriv (gaussianRowOperator a r (gaussianRowComposition rs u))) x ≤ |
| κ/(1-κ*gaussianStepTime ((a,r)::rs)) at hh |
| dsimp [K] |
| constructor <;> linarith [hh.1,hh.2,div_nonneg hk hden.le] |
| change |deriv (gaussianRowOperator a r (gaussianRowComposition rs u)) x| ≤ L*(1+|x|) at hj |
| have hsq : (deriv (gaussianRowOperator a r (gaussianRowComposition rs u)) x)^2 ≤ |
| 2*L^2*(1+x^2) := by |
| have hp := pow_le_pow_left₀ (abs_nonneg _) hj 2 |
| rw [sq_abs] at hp |
| have hx : (1+|x|)^2 ≤ 2*(1+x^2) := by nlinarith [sq_nonneg (|x|-1),sq_abs x] |
| have hx' := mul_le_mul_of_nonneg_left hx (sq_nonneg L) |
| nlinarith [sq_nonneg L] |
| have hadd := abs_add_le |
| (deriv (deriv (gaussianRowOperator a r (gaussianRowComposition rs u))) x) |
| (a*(deriv (gaussianRowOperator a r (gaussianRowComposition rs u)) x)^2) |
| rw [abs_of_nonneg (mul_nonneg ha (sq_nonneg _))] at hadd |
| have haSq := mul_le_mul_of_nonneg_right ha1 |
| (sq_nonneg (deriv (gaussianRowOperator a r (gaussianRowComposition rs u)) x)) |
| dsimp [d] |
| rw [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<1/2)] |
| nlinarith [mul_nonneg hK (sq_nonneg x)] |
| have hbound (s t : ℝ) (hs : 0 ≤ s) (hst : s<t) |
| (hT : t+gaussianStepTime rs ≤ Q) : |
| |f t-f s| ≤ (K+2*L^2)*(1+x^2)*(t-s) := by |
| apply abs_sub_le_of_hasDerivAt_Ioo hst hcont.continuousOn |
| (fun r hr => hdiff r (by linarith [hr.1])) |
| intro r hr |
| exact hb r (by linarith [hr.1]) (by linarith [hr.2]) |
| rcases lt_trichotomy s t with hst | he | hts |
| · rw [abs_of_pos (sub_pos.mpr hst)] |
| exact hbound s t hs hst (by linarith [le_max_right s t]) |
| · subst t |
| simp |
| · rw [abs_sub_comm t s,abs_of_pos (sub_pos.mpr hts)] |
| rw [abs_sub_comm (gaussianRowOperator a t (gaussianRowComposition rs u) x)] |
| exact hbound t s ht hts (by linarith [le_max_left s t]) |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal ENNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma gaussianStepPath_time_bound {A B C κ Q D : ℝ} {u : ℝ → ℝ} |
| (_hQ : 0 < Q) (_hu : RowAnalyticTerminal u A B C κ Q) (hD : 0 ≤ D) |
| (hblock : ∀ rs : List (ℝ × ℝ), (∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → |
| ∀ a s t : ℝ, 0 ≤ a → a ≤ 1 → 0 ≤ s → 0 ≤ t → |
| max s t+gaussianStepTime rs ≤ Q → ∀ x, |
| |gaussianRowOperator a t (gaussianRowComposition rs u) x- |
| gaussianRowOperator a s (gaussianRowComposition rs u) x| ≤ D*(1+x^2)*|t-s|) : |
| ∀ rs : List (ℝ × ℝ), (∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → |
| gaussianStepTime rs ≤ Q → ∀ s t, 0 ≤ s → s ≤ t → ∀ x, |
| |gaussianStepPath rs u t x-gaussianStepPath rs u s x| ≤ D*(1+x^2)*(t-s) := by |
| intro rs |
| induction rs with |
| | nil => |
| intro _ _ s t hs hst x |
| simp only [gaussianStepPath,gaussianStepRemainder,gaussianRowComposition,sub_self,abs_zero] |
| exact mul_nonneg (mul_nonneg hD (by positivity)) (sub_nonneg.mpr hst) |
| | cons r rs ih => |
| intro hrs hsum s t hs hst x |
| have hr := hrs r (by simp) |
| have htail : ∀ v ∈ rs, 0 ≤ v.1 ∧ v.1 ≤ 1 ∧ 0 ≤ v.2 := fun v hv => hrs v (by simp [hv]) |
| have hR := gaussianStepTime_nonneg (fun v hv => (htail v hv).2.2) |
| have htot : gaussianStepTime (r::rs)=r.2+gaussianStepTime rs := by simp [gaussianStepTime] |
| rw [htot] at hsum |
| have ht : 0 ≤ t := hs.trans hst |
| by_cases htR : t ≤ r.2 |
| · have hsR : s ≤ r.2 := hst.trans htR |
| rw [gaussianStepPath_cons_le r rs u htR,gaussianStepPath_cons_le r rs u hsR] |
| have hm : max (r.2-s) (r.2-t)+gaussianStepTime rs ≤ Q := by |
| rw [max_eq_left (by linarith)] |
| linarith |
| have hh := hblock rs htail r.1 (r.2-s) (r.2-t) hr.1 hr.2.1 |
| (sub_nonneg.mpr hsR) (sub_nonneg.mpr htR) hm x |
| have he : |(r.2-t)-(r.2-s)|=t-s := by rw [abs_of_nonpos (by linarith)]; ring |
| simpa only [he] using hh |
| · have hRt : r.2 < t := lt_of_not_ge htR |
| rw [gaussianStepPath_cons_gt r rs u hRt] |
| by_cases hsR : s ≤ r.2 |
| · rw [gaussianStepPath_cons_le r rs u hsR] |
| have htailQ : gaussianStepTime rs ≤ Q := by linarith |
| have hfirst := ih htail htailQ 0 (t-r.2) (by norm_num) (by linarith) x |
| rw [gaussianStepPath_zero (fun v hv => (htail v hv).2.2) u] at hfirst |
| have hsecond := hblock rs htail r.1 (r.2-s) 0 hr.1 hr.2.1 |
| (sub_nonneg.mpr hsR) (by norm_num) (by rw [max_eq_left (by linarith)]; linarith) x |
| rw [gaussianRowOperator_zero] at hsecond |
| rw [abs_of_nonpos (by linarith : 0-(r.2-s) ≤ 0)] at hsecond |
| have htri := abs_sub_le (gaussianStepPath rs u (t-r.2) x) |
| (gaussianRowComposition rs u x) (gaussianRowOperator r.1 (r.2-s) (gaussianRowComposition rs u) x) |
| nlinarith |
| · rw [gaussianStepPath_cons_gt r rs u (lt_of_not_ge hsR)] |
| have hh := ih htail (by linarith) (s-r.2) (t-r.2) (by linarith) (by linarith) x |
| convert hh using 1; ring |
|
|
| theorem gaussian_step_path : GaussianStepPathStatement := by |
| intro A B C κ Q hQ hA hC hk hsub |
| obtain ⟨D,hD,hblock⟩ := gaussian_step_time A B C κ Q hQ hA hC hk hsub |
| obtain ⟨L,hL,hlin⟩ := gaussian_step_slope A B C κ Q hQ hA hC hk hsub |
| refine ⟨D,L,hD,hL,?_⟩ |
| intro u hu rs hrs hsum |
| have hnonneg : ∀ r ∈ rs, 0 ≤ r.2 := fun r hr => (hrs r hr).2.2 |
| refine ⟨fun x => congrFun (gaussianStepPath_zero hnonneg u) x, |
| fun x => congrFun (gaussianStepPath_terminal hrs u) x,?_,?_⟩ |
| · intro s t hs ht x |
| rcases le_total s t with hst | hts |
| · rw [abs_of_nonneg (sub_nonneg.mpr hst)] |
| exact gaussianStepPath_time_bound hQ hu hD (hblock u hu) rs hrs hsum s t hs hst x |
| · rw [abs_sub_comm t s,abs_of_nonneg (sub_nonneg.mpr hts),abs_sub_comm] |
| exact gaussianStepPath_time_bound hQ hu hD (hblock u hu) rs hrs hsum t s ht hts x |
| · intro t ht htR |
| obtain ⟨hv,hvT⟩ := gaussianStepRemainder_valid hrs ht |
| rw [max_eq_left (sub_nonneg.mpr htR)] at hvT |
| have hvQ : gaussianStepTime (gaussianStepRemainder rs t) ≤ Q := by rw [hvT]; linarith |
| obtain ⟨hdf,hdf',hbf⟩ := gaussian_step_bounds u A B C κ Q hQ hu _ hv hvQ |
| refine ⟨hdf,hdf',fun x => ⟨hlin u hu _ hv hvQ x,(hbf x).2.1,?_⟩⟩ |
| change deriv (deriv (gaussianRowComposition (gaussianStepRemainder rs t) u)) x ≤ _ |
| rw [← hvT] |
| exact (hbf x).2.2 |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma gaussianStepPath_cylinder_lipschitz {u : ℝ → ℝ} {rs : List (ℝ × ℝ)} |
| {Q R D L : ℝ} (hR : 0 ≤ R) (hD : 0 ≤ D) (hL : 0 ≤ L) |
| (hQeq : gaussianStepTime rs=Q) |
| (htime : ∀ s t, 0 ≤ s → 0 ≤ t → ∀ x, |
| |gaussianStepPath rs u t x-gaussianStepPath rs u s x| ≤ D*(1+x^2)*|t-s|) |
| (hspace : ∀ t, 0 ≤ t → t ≤ gaussianStepTime rs → |
| Differentiable ℝ (gaussianStepPath rs u t) ∧ |
| ∀ x, |deriv (gaussianStepPath rs u t) x| ≤ L*(1+|x|)) : |
| LipschitzWith (⟨D*(1+R^2)+L*(1+R),by positivity⟩ : ℝ≥0) |
| (fun p : RowCylinder Q R => gaussianStepPath rs u p.1.1 p.1.2) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro p q |
| have hp := p.2 |
| have hq := q.2 |
| have hpx : |p.1.2| ≤ R := abs_le.mpr hp.2 |
| have hqx : |q.1.2| ≤ R := abs_le.mpr hq.2 |
| have hhT := htime q.1.1 p.1.1 hq.1.1 hp.1.1 p.1.2 |
| have hhS := hspace q.1.1 hq.1.1 (by rw [hQeq]; exact hq.1.2) |
| have hhx : |gaussianStepPath rs u q.1.1 p.1.2-gaussianStepPath rs u q.1.1 q.1.2| ≤ |
| L*(1+R)*|p.1.2-q.1.2| := by |
| have hc := Convex.norm_image_sub_le_of_norm_deriv_le |
| (f := gaussianStepPath rs u q.1.1) (s := Icc (-R) R) (C := L*(1+R)) |
| (fun z _ => hhS.1 z) |
| (fun z hz => by |
| rw [Real.norm_eq_abs] |
| exact (hhS.2 z).trans (mul_le_mul_of_nonneg_left (by linarith [abs_le.mpr hz]) hL)) |
| (convex_Icc (-R) R) hq.2 hp.2 |
| simpa only [Real.norm_eq_abs] using hc |
| have hprodT : |p.1.1-q.1.1| ≤ dist p q := by |
| change dist (p.1.1) (q.1.1) ≤ dist p q |
| change dist p.1.1 q.1.1 ≤ dist p.1 q.1 |
| rw [Prod.dist_eq] |
| exact le_max_left _ _ |
| have hprodX : |p.1.2-q.1.2| ≤ dist p q := by |
| change dist (p.1.2) (q.1.2) ≤ dist p q |
| change dist p.1.2 q.1.2 ≤ dist p.1 q.1 |
| rw [Prod.dist_eq] |
| exact le_max_right _ _ |
| have hp2 : p.1.2^2 ≤ R^2 := by |
| have hh := (sq_le_sq₀ (abs_nonneg p.1.2) hR).mpr hpx |
| simpa only [sq_abs] using hh |
| have hhT' : |gaussianStepPath rs u p.1.1 p.1.2-gaussianStepPath rs u q.1.1 p.1.2| ≤ |
| D*(1+R^2)*dist p q := hhT.trans |
| (mul_le_mul (mul_le_mul_of_nonneg_left (by linarith) hD) hprodT (abs_nonneg _) (by positivity)) |
| have hhS' := hhx.trans (mul_le_mul_of_nonneg_left hprodX (by positivity : 0 ≤ L*(1+R))) |
| have htri := abs_sub_le (gaussianStepPath rs u p.1.1 p.1.2) |
| (gaussianStepPath rs u q.1.1 p.1.2) (gaussianStepPath rs u q.1.1 q.1.2) |
| change |gaussianStepPath rs u p.1.1 p.1.2-gaussianStepPath rs u q.1.1 q.1.2| ≤ |
| (D*(1+R^2)+L*(1+R))*dist p q |
| nlinarith |
|
|
| lemma gaussianStepPath_cylinder_bound {u : ℝ → ℝ} {A B C κ Q R : ℝ} |
| (hQ : 0 < Q) (hR : 0 ≤ R) (hu : RowAnalyticTerminal u A B C κ Q) |
| {rs : List (ℝ × ℝ)} (hrs : ∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) |
| (hQeq : gaussianStepTime rs=Q) (p : RowCylinder Q R) : |
| |gaussianStepPath rs u p.1.1 p.1.2| ≤ |B|+A*(1+R^2+Q) := by |
| obtain ⟨hv,hvT⟩ := gaussianStepRemainder_valid hrs p.2.1.1 |
| rw [hQeq,max_eq_left (sub_nonneg.mpr p.2.1.2)] at hvT |
| have hvQ : gaussianStepTime (gaussianStepRemainder rs p.1.1) ≤ Q := by rw [hvT]; linarith [p.2.1.1] |
| have hb := (gaussian_step_bounds u A B C κ Q hQ hu _ hv hvQ).2.2 p.1.2 |
| change (-A*(1+p.1.2^2+gaussianStepTime (gaussianStepRemainder rs p.1.1)) ≤ |
| gaussianStepPath rs u p.1.1 p.1.2 ∧ gaussianStepPath rs u p.1.1 p.1.2 ≤ B) ∧ _ at hb |
| have hp := abs_le.mpr p.2.2 |
| have hp2 : p.1.2^2 ≤ R^2 := by |
| have hh := (sq_le_sq₀ (abs_nonneg p.1.2) hR).mpr hp |
| simpa only [sq_abs] using hh |
| have hmul := mul_le_mul_of_nonneg_left (show 1+p.1.2^2+gaussianStepTime (gaussianStepRemainder rs p.1.1) ≤ |
| 1+R^2+Q by linarith) hu.2.1 |
| rw [abs_le] |
| have hnn : 0 ≤ A*(1+R^2+Q) := mul_nonneg hu.2.1 (by positivity) |
| constructor <;> linarith [hb.1.1,hb.1.2,abs_nonneg B,le_abs_self B] |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma gaussianStepPath_jointly_continuous {u : ℝ → ℝ} {rs : List (ℝ × ℝ)} |
| {Q D L : ℝ} (hD : 0 ≤ D) (hL : 0 ≤ L) (hQeq : gaussianStepTime rs=Q) |
| (htime : ∀ s t, 0 ≤ s → 0 ≤ t → ∀ x, |
| |gaussianStepPath rs u t x-gaussianStepPath rs u s x| ≤ D*(1+x^2)*|t-s|) |
| (hspace : ∀ t, 0 ≤ t → t ≤ gaussianStepTime rs → |
| Differentiable ℝ (gaussianStepPath rs u t) ∧ |
| ∀ x, |deriv (gaussianStepPath rs u t) x| ≤ L*(1+|x|)) : |
| ContinuousOn (fun p : ℝ × ℝ => gaussianStepPath rs u p.1 p.2) (Icc 0 Q ×ˢ univ) := by |
| intro p hp |
| let R := |p.2|+1 |
| have hR : 0 < R := by dsimp [R]; positivity |
| have hxl : -R < p.2 := by dsimp [R]; linarith [neg_abs_le p.2] |
| have hxr : p.2 < R := by dsimp [R]; linarith [le_abs_self p.2] |
| have hc : ContinuousOn (fun p : ℝ × ℝ => gaussianStepPath rs u p.1 p.2) |
| (Icc 0 Q ×ˢ Icc (-R) R) := |
| continuousOn_iff_continuous_domRestrict.mpr |
| (gaussianStepPath_cylinder_lipschitz hR.le hD hL hQeq htime hspace).continuous |
| apply (hc p ⟨hp.1,⟨hxl.le,hxr.le⟩⟩).mono_of_mem_nhdsWithin |
| have hh : ∀ᶠ q : ℝ × ℝ in 𝓝 p, q.2 ∈ Icc (-R) R := |
| continuous_snd.continuousAt.eventually (Icc_mem_nhds hxl hxr) |
| filter_upwards [self_mem_nhdsWithin,hh.filter_mono nhdsWithin_le_nhds] with q hq hqx |
| exact ⟨hq.1,hqx⟩ |
|
|
| lemma gaussianStepPath_global_value_bound {u : ℝ → ℝ} {A B C κ Q : ℝ} |
| (hQ : 0 < Q) (hu : RowAnalyticTerminal u A B C κ Q) |
| {rs : List (ℝ × ℝ)} (hrs : ∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) |
| (hQeq : gaussianStepTime rs=Q) {t : ℝ} (ht : t ∈ Icc 0 Q) (x : ℝ) : |
| |gaussianStepPath rs u t x| ≤ (|B|+A*(1+Q))*(1+x^2) := by |
| have hx : x ∈ Icc (-|x|) |x| := abs_le.mp le_rfl |
| have hh := gaussianStepPath_cylinder_bound hQ (abs_nonneg x) hu hrs hQeq |
| (⟨(t,x),ht,hx⟩ : RowCylinder Q |x|) |
| change |gaussianStepPath rs u t x| ≤ |B|+A*(1+|x|^2+Q) at hh |
| rw [sq_abs] at hh |
| have h1 := mul_nonneg (abs_nonneg B) (sq_nonneg x) |
| have h2 := mul_nonneg (mul_nonneg hu.2.1 hQ.le) (sq_nonneg x) |
| nlinarith |
| end MicroscopicJamming |
|
|
| |
| |
|
|
| namespace MicroscopicJamming |
|
|
| def GaussianStepPDEStatement : Prop := |
| ∀ (u : ℝ → ℝ) (A B C κ Q : ℝ), 0 < Q → RowAnalyticTerminal u A B C κ Q → |
| ∀ (pre rs : List (ℝ × ℝ)) (a T : ℝ), |
| (∀ r ∈ pre ++ (a,T)::rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → |
| gaussianStepTime (pre ++ (a,T)::rs) ≤ Q → |
| ∀ t : ℝ, 0 < t → t < T → ∀ x, |
| HasDerivAt (fun s => gaussianStepPath (pre ++ (a,T)::rs) u (gaussianStepTime pre+s) x) |
| (-(1/2:ℝ)*(deriv (deriv (gaussianStepPath (pre ++ (a,T)::rs) u |
| (gaussianStepTime pre+t))) x+ |
| a*(deriv (gaussianStepPath (pre ++ (a,T)::rs) u |
| (gaussianStepTime pre+t)) x)^2)) t |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma gaussian_composition_twice {u : ℝ → ℝ} {A B C κ Q : ℝ} |
| (hQ : 0 < Q) (hu : RowAnalyticTerminal u A B C κ Q) |
| {rs : List (ℝ × ℝ)} (hrs : ∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) |
| (hsum : gaussianStepTime rs ≤ Q) : |
| Twice.RowTwiceTerminal (gaussianRowComposition rs u) |
| (A*(1+gaussianStepTime rs)) B C (κ/(1-κ*gaussianStepTime rs)) |
| (Q-gaussianStepTime rs) := by |
| obtain ⟨hd,hd',hb⟩ := gaussian_step_bounds u A B C κ Q hQ hu rs hrs hsum |
| have hR := gaussianStepTime_nonneg (fun r hr => (hrs r hr).2.2) |
| have hk := hu.2.2.2.1 |
| have hsub : κ*((Q-gaussianStepTime rs)+gaussianStepTime rs)<1 := by |
| simpa using hu.2.2.2.2.1 |
| obtain ⟨hkn,hknQ,_⟩ := gaussian_curvature_update hk hR (sub_nonneg.mpr hsum) hsub |
| refine ⟨hd,hd',mul_nonneg hu.2.1 (by linarith),hu.2.2.1,hkn,hknQ,fun y => ?_⟩ |
| refine ⟨?_,(hb y).1.2,(hb y).2⟩ |
| have hh := (hb y).1.1 |
| nlinarith [mul_nonneg (mul_nonneg hu.2.1 hR) (sq_nonneg y)] |
| end MicroscopicJamming |
|
|
| |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
|
|
| namespace MicroscopicJamming |
|
|
| lemma gaussianStepTime_append (pre rs : List (ℝ × ℝ)) : |
| gaussianStepTime (pre++rs)=gaussianStepTime pre+gaussianStepTime rs := by |
| simp [gaussianStepTime] |
|
|
| lemma gaussianStepPath_prefix (pre rs : List (ℝ × ℝ)) |
| (hp : ∀ r ∈ pre, 0 ≤ r.2) (u : ℝ → ℝ) {t : ℝ} (ht : 0 < t) : |
| gaussianStepPath (pre++rs) u (gaussianStepTime pre+t)=gaussianStepPath rs u t := by |
| induction pre with |
| | nil => simp [gaussianStepTime] |
| | cons r pre ih => |
| have hr := hp r (by simp) |
| have htail : ∀ v ∈ pre, 0 ≤ v.2 := fun v hv => hp v (by simp [hv]) |
| have hP := gaussianStepTime_nonneg htail |
| have hsum : gaussianStepTime (r::pre)=r.2+gaussianStepTime pre := by simp [gaussianStepTime] |
| simp only [List.cons_append,hsum] |
| rw [gaussianStepPath_cons_gt r (pre++rs) u (by linarith)] |
| have he : r.2+gaussianStepTime pre+t-r.2=gaussianStepTime pre+t := by ring |
| rw [he] |
| exact ih htail |
|
|
| theorem gaussian_step_pde : GaussianStepPDEStatement := by |
| intro u A B C κ Q hQ hu pre rs a T hrs hsum t ht htT x |
| have hr : 0 ≤ a ∧ a ≤ 1 ∧ 0 ≤ T := hrs (a,T) (by simp) |
| have hpre : ∀ r ∈ pre, 0 ≤ r.2 := fun r h => (hrs r (List.mem_append_left _ h)).2.2 |
| have htail : ∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2 := fun r h => |
| hrs r (List.mem_append_right _ (by simp [h])) |
| have hP := gaussianStepTime_nonneg hpre |
| have htime : gaussianStepTime (pre++(a,T)::rs)=gaussianStepTime pre+T+gaussianStepTime rs := by |
| rw [gaussianStepTime_append] |
| simp [gaussianStepTime,add_assoc] |
| have hR : gaussianStepTime rs ≤ Q := by rw [htime] at hsum; linarith |
| have htw := gaussian_composition_twice hQ hu htail hR |
| have hd := (Twice.gaussianRowOperator_calculus htw hr.1 (sub_pos.mpr htT)).2.2 x |
| have hc := hd.comp t ((hasDerivAt_const t T).sub (hasDerivAt_id t)) |
| have hpath (s : ℝ) (hs : 0 < s) (hsT : s < T) : |
| gaussianStepPath (pre++(a,T)::rs) u (gaussianStepTime pre+s)= |
| gaussianRowOperator a (T-s) (gaussianRowComposition rs u) := by |
| rw [gaussianStepPath_prefix pre ((a,T)::rs) hpre u hs] |
| exact gaussianStepPath_cons_le (a,T) rs u hsT.le |
| have hh := hc.congr_of_eventuallyEq |
| (show (fun s => gaussianStepPath (pre++(a,T)::rs) u (gaussianStepTime pre+s) x) =ᶠ[𝓝 t] |
| (fun s => gaussianRowOperator a (T-s) (gaussianRowComposition rs u) x) from by |
| filter_upwards [eventually_gt_nhds ht,eventually_lt_nhds htT] with s hs hsT |
| exact congrFun (hpath s hs hsT) x) |
| rw [hpath t ht htT] |
| simpa only [zero_sub,mul_neg_one,neg_mul] using hh |
| end MicroscopicJamming |
|
|
| |
| |
|
|
| open Set Filter |
| open scoped Topology |
|
|
| namespace MicroscopicJamming |
|
|
| def gaussianStepCoefficient : List (ℝ × ℝ) → ℝ → ℝ |
| | [], _ => 0 |
| | (a,T)::rs,t => if t ≤ T then a else gaussianStepCoefficient rs (t-T) |
|
|
| |
|
|
| def RowClassicalVarianceSolution (Q : ℝ) (u m : ℝ → ℝ) (F : ℝ → ℝ → ℝ) : Prop := |
| ContinuousOn (fun p : ℝ × ℝ => F p.1 p.2) (Icc 0 Q ×ˢ univ) ∧ |
| ContinuousOn (fun p : ℝ × ℝ => deriv (F p.1) p.2) (Icc 0 Q ×ˢ univ) ∧ |
| ContinuousOn (fun p : ℝ × ℝ => deriv (deriv (F p.1)) p.2) (Icc 0 Q ×ˢ univ) ∧ |
| (∀ t ∈ Icc 0 Q, Differentiable ℝ (F t) ∧ Differentiable ℝ (deriv (F t))) ∧ |
| (∀ t ∈ Ioo 0 Q, ∀ x, HasDerivAt (fun r => F r x) |
| (-(1/2:ℝ)*(deriv (deriv (F t)) x+m t*(deriv (F t) x)^2)) t) ∧ |
| (∀ x, F Q x=u x) ∧ |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ t ∈ Icc 0 Q, ∀ x, |
| |F t x| ≤ K*(1+x^2) ∧ |deriv (F t) x| ≤ K*(1+|x|) |
| |
| |
| |
| def RowVarianceExistenceStatement : Prop := |
| ∀ A B C κ Q : ℝ, 0 < Q → 0 ≤ A → 0 ≤ C → 0 ≤ κ → κ*Q < 1 → |
| ∃ L : ℝ, 0 ≤ L ∧ ∀ u : ℝ → ℝ, RowAnalyticTerminal u A B C κ Q → |
| ∀ m : ℝ → ℝ, ContinuousOn m (Icc 0 Q) → (∀ t ∈ Icc 0 Q, 0 ≤ m t ∧ m t ≤ 1) → |
| ∃ F : ℝ → ℝ → ℝ, RowClassicalVarianceSolution Q u m F ∧ |
| (∀ t ∈ Icc 0 Q, ∀ x, -A*(1+x^2+Q-t) ≤ F t x ∧ F t x ≤ B ∧ |
| |deriv (F t) x| ≤ L*(1+|x|) ∧ |
| -C ≤ deriv (deriv (F t)) x ∧ deriv (deriv (F t)) x ≤ κ/(1-κ*(Q-t))) ∧ |
| (∀ G : ℝ → ℝ → ℝ, RowClassicalVarianceSolution Q u m G → |
| ∀ t ∈ Icc 0 Q, ∀ x, F t x=G t x) ∧ |
| (∀ rs : ℕ → List (ℝ × ℝ), |
| (∀ n, ∀ r ∈ rs n, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → |
| (∀ n, gaussianStepTime (rs n)=Q) → |
| TendstoUniformlyOn (fun n => gaussianStepCoefficient (rs n)) m atTop (Icc 0 Q) → |
| ∀ R : ℝ, 0 < R → |
| TendstoUniformlyOn (fun n (p : ℝ × ℝ) => gaussianStepPath (rs n) u p.1 p.2) |
| (fun p => F p.1 p.2) atTop (Icc 0 Q ×ˢ Icc (-R) R)) |
| end MicroscopicJamming |
|
|
| end |
|
|
| end OAI |
|
|
