import Mathlib import OAI.Probability.JammingConcavity.RowParameterData namespace OAI /-! Row Derivative Approx. -/ 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 Set Filter open scoped Topology NNReal namespace MicroscopicJamming namespace Parameter lemma second_remainder_representation {F : ℝ → ℝ} {I : Set ℝ} (hI : IsOpen I) (h0 : (0:ℝ)∈I) {M₂ M₃ : ℝ} (hM₂ : 0≤M₂) (hM₃ : 0≤M₃) (hb₂ : ∀ a∈I,∀ b∈I,∀ c∈I,a≠b → a≠c → b≠c → |dd2 F a b c|≤M₂) (hb₃ : ∀ a∈I,∀ b∈I,∀ c∈I,∀ d∈I, a≠b → a≠c → a≠d → b≠c → b≠d → c≠d → |dd3 F a b c d|≤M₃) : ∃ g : ℝ → ℝ, LipschitzWith ⟨M₃,hM₃⟩ g ∧ (∀ k∈I, F k=F 0+deriv F 0*k+g k*k^2) ∧ deriv (deriv F) 0=2*g 0 := by have hd := differentiable_of_dd2_bound hI hM₂ hb₂ obtain ⟨g,hg,he⟩ := (second_remainder_lipschitz hI h0 (hd 0 h0) hM₃ hb₃).extend_real have hgm (k : ℝ) (hk : k∈I\{0}) : g k=(dd1 F 0 k-deriv F 0)/k := (he hk).symm have hgd (k : ℝ) (hk : k∈I\{0}) : DifferentiableAt ℝ g k := by have hk0 : k≠0 := hk.2 have hr : DifferentiableAt ℝ (fun k => (dd1 F 0 k-deriv F 0)/k) k := by unfold dd1 exact ((((hd k hk.1).sub_const _).div (differentiableAt_id.sub_const _) (by simpa using hk0)).sub_const _).div differentiableAt_id hk0 apply hr.congr_of_eventuallyEq filter_upwards [(hI.sdiff isClosed_singleton).mem_nhds hk] with y hy exact hgm y hy have hrepr (k : ℝ) (hk : k∈I\{0}) : F k=F 0+deriv F 0*k+g k*k^2 := by rw [hgm k hk] unfold dd1 have hk0 : k≠0 := hk.2 simp only [sub_zero] field_simp ring have hder (k : ℝ) (hk : k∈I\{0}) : deriv F k=deriv F 0+deriv g k*k^2+g k*(2*k) := by have hh := ((hasDerivAt_const k (F 0)).add ((hasDerivAt_id k).const_mul (deriv F 0))).add ((hgd k hk).hasDerivAt.mul ((hasDerivAt_id k).pow 2)) change HasDerivAt (fun y => F 0+deriv F 0*y+g y*y^2) (0+deriv F 0*1+(deriv g k*k^2+g k*(2*k^(2-1)*1))) k at hh norm_num only [Nat.reduceSub,pow_one,mul_one,zero_add] at hh have hhe : HasDerivAt F (deriv F 0+deriv g k*k^2+g k*(2*k)) k := by have ha : deriv F 0+(deriv g k*k^2+g k*(2*k))=deriv F 0+deriv g k*k^2+g k*(2*k) := by ring rw [ha] at hh apply hh.congr_of_eventuallyEq filter_upwards [(hI.sdiff isClosed_singleton).mem_nhds hk] with y hy exact hrepr y hy exact hhe.deriv have hb (k : ℝ) : |deriv g k|≤M₃ := by have hh := (norm_deriv_le_of_lipschitz hg : ‖deriv g k‖≤(⟨M₃,hM₃⟩:ℝ≥0)) change |deriv g k|≤M₃ at hh exact hh have hv : Tendsto (fun k : ℝ => deriv g k*k) (𝓝 0) (𝓝 0) := by have hbound (k : ℝ) : ‖deriv g k*k‖≤M₃*|k| := by rw [Real.norm_eq_abs,abs_mul] exact mul_le_mul_of_nonneg_right (hb k) (abs_nonneg _) have hh : Tendsto (fun k : ℝ => M₃*|k|) (𝓝 0) (𝓝 0) := by have hc := (((continuous_const : Continuous (fun _ : ℝ => M₃)).mul continuous_abs).continuousAt (x:= (0:ℝ))).tendsto change Tendsto (fun k : ℝ => M₃*|k|) (𝓝 0) (𝓝 (M₃*|0|)) at hc simpa only [abs_zero,mul_zero] using hc exact squeeze_zero_norm hbound hh have hsl : Tendsto (slope (deriv F) 0) (𝓝[≠] 0) (𝓝 (2*g 0)) := by have hh : Tendsto (fun k : ℝ => deriv g k*k+2*g k) (𝓝[≠] 0) (𝓝 (0+2*g 0)) := (hv.mono_left nhdsWithin_le_nhds).add ((hg.continuous.continuousAt.tendsto.mono_left nhdsWithin_le_nhds).const_mul 2) simp only [zero_add] at hh apply hh.congr' filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds (hI.mem_nhds h0),self_mem_nhdsWithin] with k hk hk0 rw [slope_def_field,hder k ⟨hk,hk0⟩] have hkne : k≠0 := hk0 simp only [sub_zero] field_simp ring refine ⟨g,hg,?_,(hasDerivAt_iff_tendsto_slope.mpr hsl).deriv⟩ intro k hk by_cases hk0 : k=0 · simp [hk0] · exact hrepr k ⟨hk,hk0⟩ lemma first_derivative_error {F : ℝ → ℝ} {I : Set ℝ} (hI : IsOpen I) (h0 : (0:ℝ)∈I) {M : ℝ} (hM : 0≤M) (hb : ∀ a∈I,∀ b∈I,∀ c∈I,a≠b → a≠c → b≠c → |dd2 F a b c|≤M) {h : ℝ} (hh : h∈I) (hh0 : h≠0) : |dd1 F 0 h-deriv F 0|≤M*|h| := by have hd := differentiable_of_dd2_bound hI hM hb 0 h0 have ht : Tendsto (fun k => |dd1 F 0 h-dd1 F 0 k|) (𝓝[≠] 0) (𝓝 |dd1 F 0 h-deriv F 0|) := by have he : (fun k => dd1 F 0 k)=slope F 0 := funext (fun k => (slope_def_field F 0 k).symm) have hh1 : Tendsto (fun k => dd1 F 0 k) (𝓝[≠] 0) (𝓝 (deriv F 0)) := by rw [he]; exact hd.hasDerivAt.tendsto_slope exact (tendsto_const_nhds.sub hh1).abs have hr : Tendsto (fun k : ℝ => M*|h-k|) (𝓝[≠] 0) (𝓝 (M*|h|)) := by simpa using ((tendsto_const_nhds.sub (tendsto_id.mono_left nhdsWithin_le_nhds)).abs.const_mul M : Tendsto (fun k : ℝ => M*|h-k|) (𝓝[≠] 0) (𝓝 (M*|h-0|))) apply le_of_tendsto_of_tendsto ht hr filter_upwards [Filter.Eventually.filter_mono nhdsWithin_le_nhds (hI.mem_nhds h0),self_mem_nhdsWithin, Filter.Eventually.filter_mono nhdsWithin_le_nhds (isOpen_ne.mem_nhds hh0.symm)] with k hk hk0 hkh rw [dd1_sub_slope F hh0.symm (show (0:ℝ)≠k from (show k≠0 from hk0).symm) hkh.symm,abs_mul] calc |h-k| *|dd2 F 0 h k|≤|h-k| *M := mul_le_mul_of_nonneg_left (hb 0 h0 h hh k hk hh0.symm (show (0:ℝ)≠k from (show k≠0 from hk0).symm) hkh.symm) (abs_nonneg _) _ = M*|h-k| := mul_comm _ _ lemma second_derivative_error {F : ℝ → ℝ} {I : Set ℝ} (hI : IsOpen I) (h0 : (0:ℝ)∈I) {M₂ M₃ : ℝ} (hM₂ : 0≤M₂) (hM₃ : 0≤M₃) (hb₂ : ∀ a∈I,∀ b∈I,∀ c∈I,a≠b → a≠c → b≠c → |dd2 F a b c|≤M₂) (hb₃ : ∀ a∈I,∀ b∈I,∀ c∈I,∀ d∈I, a≠b → a≠c → a≠d → b≠c → b≠d → c≠d → |dd3 F a b c d|≤M₃) {h : ℝ} (hh : h∈I) (hh2 : 2*h∈I) (hh0 : h≠0) : |dd2 F 0 h (2*h)-deriv (deriv F) 0/2|≤5*M₃*|h| := by obtain ⟨g,hg,he,hd⟩ := second_remainder_representation hI h0 hM₂ hM₃ hb₂ hb₃ have hid : dd2 F 0 h (2*h)=2*g (2*h)-g h := by unfold dd2 dd1 rw [he h hh,he (2*h) hh2] field_simp ring rw [hid,hd] have h1 := hg.dist_le_mul h 0 have h2 := hg.dist_le_mul (2*h) 0 change |g h-g 0|≤M₃*|h-0| at h1 change |g (2*h)-g 0|≤M₃*|2*h-0| at h2 simp only [sub_zero,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] at h1 h2 calc |2*g (2*h)-g h-2*g 0/2|=|2*(g (2*h)-g 0)-(g h-g 0)| := by congr 1; ring _ ≤ |2*(g (2*h)-g 0)|+|g h-g 0| := abs_sub _ _ _ = 2*|g (2*h)-g 0|+|g h-g 0| := by rw [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] _ ≤ 5*M₃*|h| := by linarith end Parameter end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming open Higher lemma cylinder_of_poly_error {S : Set ℝ} {f : ℕ → ℝ → ℝ → ℝ} {F : ℝ → ℝ → ℝ} {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0)) {d : ℕ} {C : ℝ} (_hC : 0≤C) (hbd : ∀ n t,t∈S → ∀ x,|f n t x-F t x|≤e n*C*(1+|x|)^d) : CylinderConvergence f F S := by intro R hR rw [Metric.tendstoUniformlyOn_iff] intro ε hε have hc : Tendsto (fun n => e n*C*(1+R)^d) atTop (𝓝 0) := by simpa using (he.mul_const C).mul_const ((1+R)^d) filter_upwards [hc.eventually (gt_mem_nhds hε)] with n hn p hp rw [Real.dist_eq,abs_sub_comm] have hx : |p.2|≤R := abs_le.mpr hp.2 have hepos : 0≤e n*C := by have hh := (abs_nonneg _).trans (hbd n p.1 hp.1 0) simpa using hh exact ((hbd n p.1 hp.1 p.2).trans (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) (by linarith) d) hepos)).trans_lt hn lemma smooth_limit_of_all_jet_convergence {S : Set ℝ} {f : ℕ → ℝ → ℝ → ℝ} {G : ℕ → ℝ → ℝ → ℝ} (hs : ∀ n,JointSpatialSmooth S (f n)) (hv : ∀ k,CylinderConvergence (fun n t => iteratedDeriv k (f n t)) (G k) S) : JointSpatialSmooth S (G 0) ∧ ∀ k t,t∈S → iteratedDeriv k (G 0 t)=G k t := by have hd (k : ℕ) : ∀ t∈S,∀ x,HasDerivAt (G k t) (G (k+1) t x) x := by apply cylinder_limit_hasDerivAt (hv k) (by simpa only [← iteratedDeriv_succ] using hv (k+1)) intro n t ht exact (contDiff_iff_iteratedDeriv.mp ((hs n).1 t ht)).2 k (by simp) have hj : ∀ k t,t∈S → iteratedDeriv k (G 0 t)=G k t := by intro k induction k with | zero => intro t ht; rfl | succ k ih => intro t ht rw [iteratedDeriv_succ,ih t ht] exact funext (fun x => (hd k t ht x).deriv) refine ⟨⟨?_,?_⟩,hj⟩ · intro t ht apply contDiff_of_differentiable_iteratedDeriv intro k _ rw [hj k t ht] exact fun x => (hd k t ht x).differentiableAt · intro k apply (cylinder_limit_continuous (f:=fun n t => iteratedDeriv k (f n t)) (F:=G k) (fun n => (hs n).2 k) (hv k)).congr intro p hp exact congrFun (hj k p.1 hp.1) p.2 lemma growth_limit {S : Set ℝ} {f : ℕ → ℝ → ℝ → ℝ} {F : ℝ → ℝ → ℝ} (hf : FamilyGrowth (fun _ : ℕ => True) S f) (hv : ∀ t∈S,∀ x,Tendsto (fun n => f n t x) atTop (𝓝 (F t x))) : ∃ d : ℕ,∃ C : ℝ,0≤C ∧ ∀ t∈S,∀ x,|F t x|≤C*(1+|x|)^d := by obtain ⟨d,C,hC,hb⟩ := hf exact ⟨d,C,hC,fun t ht x => le_of_tendsto (hv t ht x).abs (Eventually.of_forall (fun n => hb n trivial t ht x))⟩ end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming namespace Parameter open Higher def pjet (F : ℝ → ℝ → ℝ → ℝ) (k : ℕ) (t x : ℝ) : ℝ := deriv (fun e => iteratedDeriv k (F e t) x) 0 def pjet2 (F : ℝ → ℝ → ℝ → ℝ) (k : ℕ) (t x : ℝ) : ℝ := deriv (deriv (fun e => iteratedDeriv k (F e t) x)) 0/2 def paramSeq (r : ℝ) (n : ℕ) : ℝ := (r/4)*(1/((n:ℝ)+1)) lemma paramSeq_pos {r : ℝ} (hr : 0 iteratedDeriv k (F e t) x) a b c|≤C*(1+|x|)^d := by obtain ⟨d,C,hC,hb⟩ := (field2_poly hf).bounds k refine ⟨d,C,hC,?_⟩ intro t ht x a ha b hb' c hc hab hac hbc let p : Nodes3 I := ⟨⟨a,b,ha,hb',hab⟩,c,hc,hac,hbc⟩ have hh := hb p trivial t ht x dsimp only at hh rw [field2_jet hf.smooth p k ht x] at hh exact hh lemma RankFamilyData.bound_dd3 {I : Set ℝ} {A B : ℝ → ℝ} {F : ℝ → ℝ → ℝ → ℝ} (hf : RankFamilyData I A B F) (k : ℕ) : ∃ d : ℕ,∃ C : ℝ,0≤C ∧ ∀ t∈Icc (0:ℝ) 1,∀ x, ∀ a∈I,∀ b∈I,∀ c∈I,∀ z∈I,a≠b → a≠c → a≠z → b≠c → b≠z → c≠z → |dd3 (fun e => iteratedDeriv k (F e t) x) a b c z|≤C*(1+|x|)^d := by obtain ⟨d,C,hC,hb⟩ := (field3_poly hf).bounds k refine ⟨d,C,hC,?_⟩ intro t ht x a ha b hb' c hc e he hab hac hae hbc hbe hce let p : Nodes4 I := ⟨⟨⟨a,b,ha,hb',hab⟩,c,hc,hac,hbc⟩,e,he,hae,hbe,hce⟩ have hh := hb p trivial t ht x dsimp only at hh rw [field3_jet hf.smooth p k ht x] at hh exact hh lemma RankFamilyData.first_jet_limit {r : ℝ} (hr : 0 iteratedDeriv k (field1 F 0 (paramSeq r n) t)) (pjet F k) (Icc 0 1) := by obtain ⟨d,C,hC,hb⟩ := hf.bound_dd2 k apply cylinder_of_poly_error (d:=d) (paramSeq_tendsto r) hC intro n t ht x have hj := field1_jet hf.smooth (seqNodes2 hr n) k ht x dsimp only [seqNodes2] at hj rw [hj] have hh := first_derivative_error isOpen_Ioo (show (0:ℝ)∈Ioo (-r) r from ⟨by linarith,hr⟩) (show 0≤C*(1+|x|)^d by positivity) (hb t ht x) (paramSeq_mem hr n) (paramSeq_pos hr n).ne' rw [abs_of_pos (paramSeq_pos hr n)] at hh have he : C*(1+|x|)^d*paramSeq r n=paramSeq r n*C*(1+|x|)^d := by ring exact he ▸ hh lemma RankFamilyData.second_jet_limit {r : ℝ} (hr : 0 iteratedDeriv k (field2 F 0 (paramSeq r n) (2*paramSeq r n) t)) (pjet2 F k) (Icc 0 1) := by obtain ⟨d₂,C₂,hC₂,hb₂⟩ := hf.bound_dd2 k obtain ⟨d₃,C₃,hC₃,hb₃⟩ := hf.bound_dd3 k apply cylinder_of_poly_error (d:=d₃) (paramSeq_tendsto r) (show 0≤5*C₃ by positivity) intro n t ht x have hj := field2_jet hf.smooth (seqNodes3 hr n) k ht x dsimp only [seqNodes3,seqNodes2] at hj rw [hj] have hh := second_derivative_error isOpen_Ioo (show (0:ℝ)∈Ioo (-r) r from ⟨by linarith,hr⟩) (show 0≤C₂*(1+|x|)^d₂ by positivity) (show 0≤C₃*(1+|x|)^d₃ by positivity) (hb₂ t ht x) (hb₃ t ht x) (paramSeq_mem hr n) (twice_paramSeq_mem hr n) (paramSeq_pos hr n).ne' rw [abs_of_pos (paramSeq_pos hr n)] at hh have he : 5*(C₃*(1+|x|)^d₃)*paramSeq r n=paramSeq r n*(5*C₃)*(1+|x|)^d₃ := by ring exact he ▸ hh lemma RankFamilyData.parameter_jets {r : ℝ} (hr : 0 field1_smooth hf.smooth (seqNodes2 hr n)) (hf.first_jet_limit hr) · exact smooth_limit_of_all_jet_convergence (fun n => field2_smooth hf.smooth (seqNodes3 hr n)) (hf.second_jet_limit hr) end Parameter end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming namespace Parameter open Higher def endpointRhs {ι : Type*} (A B : ℝ → ℝ) (F : ℝ → ℝ → ℝ → ℝ) (l r : ι → ℝ) (U H : ι → ℝ → ℝ → ℝ) (i : ι) (t x : ℝ) : ℝ := H i t x-(A t+l i*B t)*iteratedDeriv 2 (U i t) x- ((A t+l i*B t)*t*(iteratedDeriv 1 (F (l i) t) x+iteratedDeriv 1 (F (r i) t) x))*iteratedDeriv 1 (U i t) x lemma RankFamilyData.endpoint_rhs_poly {I : Set ℝ} {A B : ℝ → ℝ} {F : ℝ → ℝ → ℝ → ℝ} (hf : RankFamilyData I A B F) {ι : Type*} (l r : ι → ℝ) (hl : ∀ i,l i∈I) (hr : ∀ i,r i∈I) {U H : ι → ℝ → ℝ → ℝ} (hU : PolynomialJetFamily (fun _ : ι => True) (Icc 0 1) U) (hH : PolynomialJetFamily (fun _ : ι => True) (Icc 0 1) H) : PolynomialJetFamily (fun _ : ι => True) (Icc 0 1) (endpointRhs A B F l r U H) := by obtain ⟨D,hD,ha,hB⟩ := hf.coeff have hd := (hU.jet 2).coeff_mul (fun i _ => hf.diffusion_cont (l i)) hD (fun i _ t ht => by rw [abs_of_nonneg (ha (l i) (hl i) t ht).1]; exact (ha (l i) (hl i) t ht).2) have hlp := hf.poly.comp (R:=fun _ => True) l (fun i _ => hl i) have hrp := hf.poly.comp (R:=fun _ => True) r (fun i _ => hr i) have hb := hf.mul_at l hl ((((hlp.jet 1).add (hrp.jet 1))).mul (hU.jet 1)) convert (hH.sub hd).sub hb using 1 funext i t x dsimp only [endpointRhs] ring def rhs1 {I : Set ℝ} (F : ℝ → ℝ → ℝ → ℝ) (A B : ℝ → ℝ) := endpointRhs A B F (fun p : Nodes2 I => p.a) (fun p => p.b) (fun p => field1 F p.a p.b) (source1 F B) def rhs2 {I : Set ℝ} (F : ℝ → ℝ → ℝ → ℝ) (A B : ℝ → ℝ) := endpointRhs A B F (fun p : Nodes3 I => p.a) (fun p => p.c) (fun p => field2 F p.a p.b p.c) (source2 F A B) lemma rhs1_poly {I : Set ℝ} {A B : ℝ → ℝ} {F : ℝ → ℝ → ℝ → ℝ} (hf : RankFamilyData I A B F) : PolynomialJetFamily (fun _ : Nodes2 I => True) (Icc 0 1) (rhs1 F A B) := hf.endpoint_rhs_poly (fun p : Nodes2 I => p.a) (fun p => p.b) (fun p => p.ha) (fun p => p.hb) (field1_poly hf) (source1_poly hf) lemma rhs2_poly {I : Set ℝ} {A B : ℝ → ℝ} {F : ℝ → ℝ → ℝ → ℝ} (hf : RankFamilyData I A B F) : PolynomialJetFamily (fun _ : Nodes3 I => True) (Icc 0 1) (rhs2 F A B) := hf.endpoint_rhs_poly (fun p : Nodes3 I => p.a) (fun p => p.c) (fun p => p.ha) (fun p => p.hc) (field2_poly hf) (source2_poly hf) end Parameter end MicroscopicJamming open Set Filter MeasureTheory open scoped Topology namespace MicroscopicJamming lemma time_derivative_limit_dominated {f g : ℕ → ℝ → ℝ} {F G : ℝ → ℝ} {C : ℝ} (hfc : ∀ n,ContinuousOn (f n) (Icc 0 1)) (hgc : ∀ n,ContinuousOn (g n) (Icc 0 1)) (hG : ContinuousOn G (Icc 0 1)) (hd : ∀ n t,t∈Ioo (0:ℝ) 1 → HasDerivAt (f n) (g n t) t) (hfv : ∀ t∈Icc (0:ℝ) 1,Tendsto (fun n => f n t) atTop (𝓝 (F t))) (hgv : ∀ t∈Icc (0:ℝ) 1,Tendsto (fun n => g n t) atTop (𝓝 (G t))) (hgb : ∀ n t,t∈Icc (0:ℝ) 1 → |g n t|≤C) : ∀ t∈Ioo (0:ℝ) 1,HasDerivAt F (G t) t := by have hsub (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : uIcc t 1⊆Icc 0 1 := by rw [uIcc_of_le ht.2]; exact Icc_subset_Icc ht.1 le_rfl have hrepr (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : F t=F 1-∫ s in t..1,G s := by have hi : Tendsto (fun n => ∫ s in t..1,g n s) atTop (𝓝 (∫ s in t..1,G s)) := by apply intervalIntegral.tendsto_integral_filter_of_dominated_convergence (fun _ => C) · exact Eventually.of_forall (fun n => (((hgc n).mono (hsub t ht)).aestronglyMeasurable measurableSet_uIcc).mono_set uIoc_subset_uIcc) · exact Eventually.of_forall (fun n => Eventually.of_forall (fun s hs => by simpa only [Real.norm_eq_abs] using hgb n s (hsub t ht (uIoc_subset_uIcc hs)))) · exact intervalIntegrable_const · exact Eventually.of_forall (fun s hs => hgv s (hsub t ht (uIoc_subset_uIcc hs))) have he (n : ℕ) : (∫ s in t..1,g n s)=f n 1-f n t := by apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le ht.2 · exact (hfc n).mono (Icc_subset_Icc ht.1 le_rfl) · exact fun s hs => hd n s ⟨lt_of_le_of_lt ht.1 hs.1,hs.2⟩ · exact ((hgc n).mono (hsub t ht)).intervalIntegrable have hh := tendsto_nhds_unique hi ((hfv 1 ⟨by norm_num,le_rfl⟩).sub (hfv t ht) |>.congr (fun n => (he n).symm)) linarith let E : ℝ → ℝ := fun t => G (projIcc 0 1 (by norm_num) t) have hE : Continuous E := (continuousOn_iff_continuous_domRestrict.mp hG).comp continuous_projIcc have hEe (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : E t=G t := by dsimp only [E]; rw [projIcc_of_mem (by norm_num) ht] intro t ht have hh := (intervalIntegral.integral_hasDerivAt_left (hE.intervalIntegrable t 1) hE.aestronglyMeasurable.stronglyMeasurableAtFilter hE.continuousAt).const_sub (F 1) simp only [neg_neg,hEe t ⟨ht.1.le,ht.2.le⟩] at hh apply hh.congr_of_eventuallyEq filter_upwards [Ioo_mem_nhds ht.1 ht.2] with s hs rw [hrepr s ⟨hs.1.le,hs.2.le⟩] congr 1 apply intervalIntegral.integral_congr intro y hy exact (hEe y (hsub s ⟨hs.1.le,hs.2.le⟩ hy)).symm end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming namespace Parameter open Higher lemma RankFamilyData.base_jet_limit {r : ℝ} (hr : 0 iteratedDeriv k (F (c*paramSeq r n) t) x) atTop (𝓝 (iteratedDeriv k (F 0 t) x)) := by have hc := (MicroscopicJamming.RankFamilyData.parameter_C2 hf isOpen_Ioo ⟨by linarith,hr⟩ k t ht x).1.continuousAt exact hc.tendsto.comp (by simpa only [mul_zero] using (paramSeq_tendsto r).const_mul c) lemma RankFamilyData.offaxis_jet_limit {r : ℝ} (hr : 0 iteratedDeriv k (field1 F (paramSeq r n) (2*paramSeq r n) t) x) atTop (𝓝 (pjet F k t x)) := by have h0 := (hf.first_jet_limit hr k).tendsto_at ht x have h2 := (hf.second_jet_limit hr k).tendsto_at ht x have he (n : ℕ) : iteratedDeriv k (field1 F (paramSeq r n) (2*paramSeq r n) t) x= iteratedDeriv k (field1 F 0 (paramSeq r n) t) x+ (2*paramSeq r n)*iteratedDeriv k (field2 F 0 (paramSeq r n) (2*paramSeq r n) t) x := by have ha := field1_jet hf.smooth (seqNodes3 hr n).bc k ht x have hb := field1_jet hf.smooth (seqNodes2 hr n) k ht x have hc := field2_jet hf.smooth (seqNodes3 hr n) k ht x dsimp only [seqNodes2,seqNodes3,Nodes3.bc] at ha hb hc rw [ha,hb,hc] unfold dd2 have hn := (paramSeq_pos hr n).ne' field_simp ring have hh := h0.add (((paramSeq_tendsto r).const_mul 2).mul h2) simpa only [mul_zero,zero_mul,add_zero,← he] using hh def linearRhs (A B : ℝ → ℝ) (f v : ℝ → ℝ → ℝ) (t x : ℝ) : ℝ := -B t*(iteratedDeriv 2 (f t) x+t*(iteratedDeriv 1 (f t) x)^2)- A t*iteratedDeriv 2 (v t) x-2*A t*t*iteratedDeriv 1 (f t) x*iteratedDeriv 1 (v t) x def quadraticRhs (A B : ℝ → ℝ) (f v g : ℝ → ℝ → ℝ) (t x : ℝ) : ℝ := -A t*t*(iteratedDeriv 1 (v t) x)^2- B t*(iteratedDeriv 2 (v t) x+2*t*iteratedDeriv 1 (f t) x*iteratedDeriv 1 (v t) x)- A t*iteratedDeriv 2 (g t) x-2*A t*t*iteratedDeriv 1 (f t) x*iteratedDeriv 1 (g t) x lemma linearRhs_smooth {A B : ℝ → ℝ} {f v : ℝ → ℝ → ℝ} (hA : ContinuousOn A (Icc 0 1)) (hB : ContinuousOn B (Icc 0 1)) (hf : JointSpatialSmooth (Icc 0 1) f) (hv : JointSpatialSmooth (Icc 0 1) v) : JointSpatialSmooth (Icc 0 1) (linearRhs A B f v) := by have hh := (((hf.jet 2).add (((hf.jet 1).mul (hf.jet 1)).coeff_mul continuousOn_id)).coeff_mul hB.neg).sub ((hv.jet 2).coeff_mul hA) |>.sub ((((hf.jet 1).mul (hv.jet 1)).coeff_mul (((continuousOn_const (c:=(2:ℝ))).mul hA).mul continuousOn_id))) convert hh using 1 funext t x dsimp only [linearRhs,Pi.mul_apply,Pi.neg_apply,id_eq] ring lemma quadraticRhs_smooth {A B : ℝ → ℝ} {f v g : ℝ → ℝ → ℝ} (hA : ContinuousOn A (Icc 0 1)) (hB : ContinuousOn B (Icc 0 1)) (hf : JointSpatialSmooth (Icc 0 1) f) (hv : JointSpatialSmooth (Icc 0 1) v) (hg : JointSpatialSmooth (Icc 0 1) g) : JointSpatialSmooth (Icc 0 1) (quadraticRhs A B f v g) := by have hh := (((hv.jet 1).mul (hv.jet 1)).coeff_mul (hA.neg.mul continuousOn_id)).sub (((hv.jet 2).add (((hf.jet 1).mul (hv.jet 1)).coeff_mul ((continuousOn_const (c:=(2:ℝ))).mul continuousOn_id))).coeff_mul hB) |>.sub ((hg.jet 2).coeff_mul hA) |>.sub ((((hf.jet 1).mul (hg.jet 1)).coeff_mul (((continuousOn_const (c:=(2:ℝ))).mul hA).mul continuousOn_id))) convert hh using 1 funext t x dsimp only [quadraticRhs,Pi.mul_apply,Pi.neg_apply,id_eq] ring lemma RankFamilyData.rhs1_limit {r : ℝ} (hr : 0 rhs1 F A B (seqNodes2 hr n) t x) atTop (𝓝 (linearRhs A B (F 0) (pjet F 0) t x)) := by have hb (k) := hf.base_jet_limit hr k ht x 1 simp only [one_mul] at hb have hv (k) := (hf.first_jet_limit hr k).tendsto_at ht x have hj := (hf.parameter_jets hr).1.2 have hh := (((hb 2).add (((hb 1).pow 2).const_mul t)).const_mul (-B t)).sub ((hv 2).const_mul (A t)) |>.sub ((((tendsto_const_nhds (x:=iteratedDeriv 1 (F 0 t) x)).add (hb 1)).mul (hv 1)).const_mul (A t*t)) convert hh using 1 · funext n; dsimp [rhs1,endpointRhs,source1,seqNodes2]; ring · dsimp only [linearRhs,Pi.mul_apply,Pi.neg_apply,id_eq]; rw [hj 2 t ht,hj 1 t ht]; ring_nf lemma RankFamilyData.rhs2_limit {r : ℝ} (hr : 0 rhs2 F A B (seqNodes3 hr n) t x) atTop (𝓝 (quadraticRhs A B (F 0) (pjet F 0) (pjet2 F 0) t x)) := by have hb (k) := hf.base_jet_limit hr k ht x 1 simp only [one_mul] at hb have hc (k) := hf.base_jet_limit hr k ht x 2 have hv (k) := (hf.first_jet_limit hr k).tendsto_at ht x have hw (k) := hf.offaxis_jet_limit hr k ht x have hg (k) := (hf.second_jet_limit hr k).tendsto_at ht x have hj := hf.parameter_jets hr have hh := (((hv 1).mul (hw 1)).const_mul (-A t*t)).sub (((hw 2).add ((((hb 1).add (hc 1)).mul (hw 1)).const_mul t)).const_mul (B t)) |>.sub ((hg 2).const_mul (A t)) |>.sub ((((tendsto_const_nhds (x:=iteratedDeriv 1 (F 0 t) x)).add (hc 1)).mul (hg 1)).const_mul (A t*t)) convert hh using 1 · funext n; dsimp [rhs2,endpointRhs,source2,seqNodes2,seqNodes3]; ring · dsimp only [quadraticRhs,Pi.mul_apply,Pi.neg_apply,id_eq]; rw [hj.1.2 2 t ht,hj.1.2 1 t ht,hj.2.2 2 t ht,hj.2.2 1 t ht]; ring_nf lemma RankFamilyData.parameter_equations {r : ℝ} (hr : 0 pjet F 0 s x) (linearRhs A B (F 0) (pjet F 0) t x) t) ∧ (∀ t∈Ioo (0:ℝ) 1,∀ x,HasDerivAt (fun s => pjet2 F 0 s x) (quadraticRhs A B (F 0) (pjet F 0) (pjet2 F 0) t x) t) := by have hj := hf.parameter_jets hr have h0 : (0:ℝ)∈Ioo (-r) r := ⟨by linarith,hr⟩ constructor · intro t ht x obtain ⟨d,C,hC,hb⟩ := (rhs1_poly hf).bounds 0 apply time_derivative_limit_dominated (C:=C*(1+|x|)^d) (fun n => ((field1_smooth hf.smooth (seqNodes2 hr n)).2 0).comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) (fun n => ((rhs1_poly hf).smooth (seqNodes2 hr n) trivial).2 0 |>.comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) ((linearRhs_smooth hf.Acont hf.Bcont (hf.smooth 0 h0) hj.1.1).2 0 |>.comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) (fun n s hs => field1_equation hf (seqNodes2 hr n) hs x) (fun s hs => (hf.first_jet_limit hr 0).tendsto_at hs x) (fun s hs => hf.rhs1_limit hr hs x) (fun n s hs => hb (seqNodes2 hr n) trivial s hs x) t ht · intro t ht x obtain ⟨d,C,hC,hb⟩ := (rhs2_poly hf).bounds 0 apply time_derivative_limit_dominated (C:=C*(1+|x|)^d) (fun n => ((field2_smooth hf.smooth (seqNodes3 hr n)).2 0).comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) (fun n => ((rhs2_poly hf).smooth (seqNodes3 hr n) trivial).2 0 |>.comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) ((quadraticRhs_smooth hf.Acont hf.Bcont (hf.smooth 0 h0) hj.1.1 hj.2.1).2 0 |>.comp (continuousOn_id.prodMk continuousOn_const) (fun s hs => ⟨hs,mem_univ _⟩)) (fun n s hs => field2_equation hf (seqNodes3 hr n) hs x) (fun s hs => (hf.second_jet_limit hr 0).tendsto_at hs x) (fun s hs => hf.rhs2_limit hr hs x) (fun n s hs => hb (seqNodes3 hr n) trivial s hs x) t ht end Parameter end MicroscopicJamming open Set namespace MicroscopicJamming namespace Hessian structure Jet (u ux uxx ut : ℝ → ℝ → ℝ) : Prop where x : ∀ t∈Icc (0:ℝ) 1,∀ z,HasDerivAt (u t) (ux t z) z xx : ∀ t∈Icc (0:ℝ) 1,∀ z,HasDerivAt (ux t) (uxx t z) z time : ∀ t∈Ico (0:ℝ) 1,∀ z,HasDerivWithinAt (fun s => u s z) (ut t z) (Ici t) t lemma Jet.spatial_deriv {u ux uxx ut : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : deriv (u t)=ux t := funext (fun z => (h.x t ht z).deriv) lemma Jet.spatial_deriv2 {u ux uxx ut : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : deriv (deriv (u t))=uxx t := by rw [h.spatial_deriv ht]; exact funext (fun z => (h.xx t ht z).deriv) lemma Jet.add {u ux uxx ut v vx vxx vt : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) (k : Jet v vx vxx vt) : Jet (fun t z => u t z+v t z) (fun t z => ux t z+vx t z) (fun t z => uxx t z+vxx t z) (fun t z => ut t z+vt t z) := ⟨fun t ht z => (h.x t ht z).add (k.x t ht z),fun t ht z => (h.xx t ht z).add (k.xx t ht z), fun t ht z => (h.time t ht z).add (k.time t ht z)⟩ lemma Jet.neg {u ux uxx ut : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) : Jet (fun t z => -u t z) (fun t z => -ux t z) (fun t z => -uxx t z) (fun t z => -ut t z) := ⟨fun t ht z => (h.x t ht z).neg,fun t ht z => (h.xx t ht z).neg,fun t ht z => (h.time t ht z).neg⟩ lemma Jet.mul {u ux uxx ut v vx vxx vt : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) (k : Jet v vx vxx vt) : Jet (fun t z => u t z*v t z) (fun t z => ux t z*v t z+u t z*vx t z) (fun t z => uxx t z*v t z+2*ux t z*vx t z+u t z*vxx t z) (fun t z => ut t z*v t z+u t z*vt t z) := by refine ⟨fun t ht z => (h.x t ht z).mul (k.x t ht z),fun t ht z => ?_, fun t ht z => (h.time t ht z).mul (k.time t ht z)⟩ exact (((h.xx t ht z).mul (k.x t ht z)).add ((h.x t ht z).mul (k.xx t ht z))).congr_deriv (by ring) lemma Jet.const (c : ℝ) : Jet (fun _ _ => c) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 0) := ⟨fun _ _ z => hasDerivAt_const z c,fun _ _ z => hasDerivAt_const z 0, fun t _ _ => (hasDerivAt_const t c).hasDerivWithinAt⟩ lemma Jet.rank : Jet (fun t _ : ℝ => t) (fun _ _ => 0) (fun _ _ => 0) (fun _ _ => 1) := ⟨fun t _ z => hasDerivAt_const z t,fun _ _ z => hasDerivAt_const z 0, fun t _ _ => (hasDerivAt_id t).hasDerivWithinAt⟩ lemma Jet.coeff {a da : ℝ → ℝ} (ha : ∀ t∈Ico (0:ℝ) 1,HasDerivWithinAt a (da t) (Ici t) t) : Jet (fun t _ => a t) (fun _ _ => 0) (fun _ _ => 0) (fun t _ => da t) := ⟨fun t _ z => hasDerivAt_const z (a t),fun _ _ z => hasDerivAt_const z 0,fun t ht _ => ha t ht⟩ lemma Jet.inv {u ux uxx ut : ℝ → ℝ → ℝ} (h : Jet u ux uxx ut) (hn : ∀ t∈Icc (0:ℝ) 1,∀ z,u t z≠0) : Jet (fun t z => (u t z)⁻¹) (fun t z => -ux t z/(u t z)^2) (fun t z => 2*(ux t z)^2/(u t z)^3-uxx t z/(u t z)^2) (fun t z => -ut t z/(u t z)^2) := by refine ⟨fun t ht z => (h.x t ht z).inv (hn t ht z),fun t ht z => ?_, fun t ht z => (h.time t ht z).inv (hn t ⟨ht.1,ht.2.le⟩ z)⟩ exact ((h.xx t ht z).neg.div ((h.x t ht z).pow 2) (pow_ne_zero 2 (hn t ht z))).congr_deriv (by simp only [Pi.pow_apply,Pi.neg_apply]; field_simp [hn t ht z]; ring) end Hessian end MicroscopicJamming open Set Filter open scoped Topology BigOperators namespace MicroscopicJamming namespace Parameter open Higher lemma RankFamilyData.parameter_growth {r : ℝ} (hr : 0 iteratedDeriv k (pjet F 0 t))) ∧ (∀ k,UniformPolynomialGrowth (Icc 0 1) (fun t => iteratedDeriv k (pjet2 F 0 t))) := by have hj := hf.parameter_jets hr constructor · intro k obtain ⟨d,C,hC,hb⟩ := growth_limit (((field1_poly hf).bounds k).comp (seqNodes2 hr) (fun _ _ => trivial)) (fun t ht x => (hf.first_jet_limit hr k).tendsto_at ht x) exact ⟨d,C,hC,fun t ht x => by dsimp only; rw [hj.1.2 k t ht]; exact hb t ht x⟩ · intro k obtain ⟨d,C,hC,hb⟩ := growth_limit (((field2_poly hf).bounds k).comp (seqNodes3 hr) (fun _ _ => trivial)) (fun t ht x => (hf.second_jet_limit hr k).tendsto_at ht x) exact ⟨d,C,hC,fun t ht x => by dsimp only; rw [hj.2.2 k t ht]; exact hb t ht x⟩ lemma RankFamilyData.parameter_terminal {r : ℝ} (hr : 0 iteratedDeriv n (pjet F 0 s) x) (iteratedDeriv n (linearRhs A B (F 0) (pjet F 0) t) x) (Ici t) t) ∧ (∀ n,∀ t∈Ico (0:ℝ) 1,∀ x,HasDerivWithinAt (fun s => iteratedDeriv n (pjet2 F 0 s) x) (iteratedDeriv n (quadraticRhs A B (F 0) (pjet F 0) (pjet2 F 0) t) x) (Ici t) t) := by have hj := hf.parameter_jets hr have hd := hf.parameter_equations hr have h0 : (0:ℝ)∈Ioo (-r) r := ⟨by linarith,hr⟩ have hG := linearRhs_smooth hf.Acont hf.Bcont (hf.smooth 0 h0) hj.1.1 have hH := quadraticRhs_smooth hf.Acont hf.Bcont (hf.smooth 0 h0) hj.1.1 hj.2.1 exact ⟨evolution_all_jets (by norm_num) (hj.1.1.2 0) hG.2 hj.1.1.1 hG.1 hd.1, evolution_all_jets (by norm_num) (hj.2.1.2 0) hH.2 hj.2.1.1 hH.1 hd.2⟩ end Parameter end MicroscopicJamming open Set open scoped BigOperators namespace MicroscopicJamming namespace Hessian open Higher Parameter lemma smooth_jet {f : ℝ → ℝ} (hf : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) f) (n : ℕ) : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (iteratedDeriv n f) := by apply contDiff_of_differentiable_iteratedDeriv intro k _ simpa only [iteratedDeriv_iteratedDeriv] using (contDiff_iff_iteratedDeriv.mp hf).2 (k+n) (by simp) def ftime (A t : ℝ) (f : ℕ → ℝ) (n : ℕ) : ℝ := -A*(f (n+2)+t*∑ i∈Finset.range (n+1),(n.choose i:ℝ)*f (i+1)*f (n-i+1)) def vtime (A B t : ℝ) (f v : ℕ → ℝ) (n : ℕ) : ℝ := -B*(f (n+2)+t*∑ i∈Finset.range (n+1),(n.choose i:ℝ)*f (i+1)*f (n-i+1))- A*v (n+2)-2*A*t*∑ i∈Finset.range (n+1),(n.choose i:ℝ)*f (i+1)*v (n-i+1) def gtime (A B t : ℝ) (f v g : ℕ → ℝ) : ℝ := -A*t*(v 1)^2-B*(v 2+2*t*f 1*v 1)-A*g 2-2*A*t*f 1*g 1 lemma iterated_rhs (A t : ℝ) {f : ℝ → ℝ} (hf : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) f) (n : ℕ) (x : ℝ) : iteratedDeriv n (fun y => -A*(iteratedDeriv 2 f y+t*(iteratedDeriv 1 f y)^2)) x= ftime A t (fun k => iteratedDeriv k f x) n := by have hc (k : ℕ) : ContDiff ℝ (n:WithTop ℕ∞) (iteratedDeriv k f) := smooth_jet hf k |>.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤)) rw [iteratedDeriv_const_mul_field] rw [iteratedDeriv_fun_add (hc 2).contDiffAt (contDiff_const.mul ((hc 1).pow 2)).contDiffAt] rw [iteratedDeriv_const_mul_field] simp only [pow_two] rw [iteratedDeriv_fun_mul (hc 1).contDiffAt (hc 1).contDiffAt] simp only [iteratedDeriv_iteratedDeriv,ftime] lemma iterated_linearRhs {A B : ℝ → ℝ} {f v : ℝ → ℝ → ℝ} {t : ℝ} (hf : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (f t)) (hv : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (v t)) (n : ℕ) (x : ℝ) : iteratedDeriv n (linearRhs A B f v t) x= vtime (A t) (B t) t (fun k => iteratedDeriv k (f t) x) (fun k => iteratedDeriv k (v t) x) n := by have hc (k : ℕ) : ContDiff ℝ (n:WithTop ℕ∞) (iteratedDeriv k (f t)) := smooth_jet hf k |>.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤)) have hd (k : ℕ) : ContDiff ℝ (n:WithTop ℕ∞) (iteratedDeriv k (v t)) := smooth_jet hv k |>.of_le (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤)) have hbase := (contDiff_const (c:= -B t)).mul ((hc 2).add ((contDiff_const (c:=t)).mul ((hc 1).pow 2))) have hU := (contDiff_const (c:=A t)).mul (hd 2) have hV := ((contDiff_const (c:=2*A t*t)).mul (hc 1)).mul (hd 1) unfold linearRhs rw [iteratedDeriv_fun_sub (hbase.sub hU).contDiffAt hV.contDiffAt, iteratedDeriv_fun_sub hbase.contDiffAt hU.contDiffAt] rw [iterated_rhs (B t) t hf n x,iteratedDeriv_const_mul_field] have he : (fun y => 2*A t*t*iteratedDeriv 1 (f t) y*iteratedDeriv 1 (v t) y)= fun y => (2*A t*t)*(iteratedDeriv 1 (f t) y*iteratedDeriv 1 (v t) y) := by funext y; ring rw [he,iteratedDeriv_const_mul_field,iteratedDeriv_fun_mul (hc 1).contDiffAt (hd 1).contDiffAt] simp only [ftime,vtime,iteratedDeriv_iteratedDeriv] end Hessian end MicroscopicJamming end end OAI