import Mathlib import OAI.Probability.JammingConcavity.GaussianStepRight import OAI.Probability.JammingConcavity.UniformPolynomialGrowth namespace OAI /-! Polynomial Comparison. -/ 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 open scoped Topology namespace MicroscopicJamming lemma backward_positive_strict_compact {Q R : ℝ} (hQ : 0 < Q) (hR : 0 < R) {U b d : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ Icc (-R) R)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t)) (ht : ∀ t ∈ Ico 0 Q, ∀ x, HasDerivWithinAt (fun s => U s x) (d t x) (Ici t) t) (hp : ∀ t ∈ Ico 0 Q, ∀ x ∈ Ioo (-R) R, 0 < U t x → 0 < d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x) (hterm : ∀ x ∈ Icc (-R) R, U Q x ≤ 0) (hleft : ∀ t ∈ Icc 0 Q, U t (-R) ≤ 0) (hright : ∀ t ∈ Icc 0 Q, U t R ≤ 0) : ∀ t ∈ Icc 0 Q, ∀ x ∈ Icc (-R) R, U t x ≤ 0 := by have hne : (Icc 0 Q ×ˢ Icc (-R) R).Nonempty := ⟨(0,0),⟨le_rfl,hQ.le⟩,by constructor <;> linarith⟩ obtain ⟨p,hpS,hmax⟩ := (isCompact_Icc.prod isCompact_Icc).exists_isMaxOn hne hc suffices h : U p.1 p.2 ≤ 0 by intro t ht' x hx' exact (hmax (show (t,x) ∈ Icc 0 Q ×ˢ Icc (-R) R from ⟨ht',hx'⟩)).trans h by_contra hn have hpos : 0 < U p.1 p.2 := lt_of_not_ge hn have htQ : p.1 < Q := lt_of_le_of_ne hpS.1.2 (by intro he have hh := hterm p.2 hpS.2 rw [← he] at hh exact (not_le_of_gt hpos) hh) have hxl : -R < p.2 := lt_of_le_of_ne hpS.2.1 (by intro he have hh := hleft p.1 hpS.1 rw [he] at hh exact (not_le_of_gt hpos) hh) have hxr : p.2 < R := lt_of_le_of_ne hpS.2.2 (by intro he have hh := hright p.1 hpS.1 rw [← he] at hh exact (not_le_of_gt hpos) hh) have hxmax : IsMaxOn (U p.1) (Icc (-R) R) p.2 := fun x hx' => hmax (show (p.1,x) ∈ Icc 0 Q ×ˢ Icc (-R) R from ⟨hpS.1,hx'⟩) have hxlocal : IsLocalMax (U p.1) p.2 := hxmax.isLocalMax (Icc_mem_nhds hxl hxr) have hzero := hxlocal.deriv_eq_zero have hsecond := localMax_second_deriv_nonpos (hx p.1 hpS.1) hxlocal have htmax : IsMaxOn (fun t => U t p.2) (Icc p.1 Q) p.1 := fun t ht' => hmax (show (t,p.2) ∈ Icc 0 Q ×ˢ Icc (-R) R from ⟨⟨hpS.1.1.trans ht'.1,ht'.2⟩,hpS.2⟩) have htan : Q-p.1 ∈ posTangentConeAt (Icc p.1 Q) p.1 := sub_mem_posTangentConeAt_of_segment_subset (segment_eq_Icc htQ.le ▸ Subset.rfl) have hd := (ht p.1 ⟨hpS.1.1,htQ⟩ p.2).mono (show Icc p.1 Q ⊆ Ici p.1 from fun _ h => h.1) have hnonpos : (Q-p.1)*d p.1 p.2 ≤ 0 := by simpa only [ContinuousLinearMap.smulRight_apply,ContinuousLinearMap.toSpanSingleton_apply,one_apply_eq_self,smul_eq_mul] using htmax.isLocalMaxOn.hasFDerivWithinAt_nonpos hd htan have hdt : d p.1 p.2 ≤ 0 := by nlinarith [sub_pos.mpr htQ] have hstrict := hp p.1 ⟨hpS.1.1,htQ⟩ p.2 ⟨hxl,hxr⟩ hpos rw [hzero,mul_zero,add_zero] at hstrict linarith def polynomialBarrier (n : ℕ) (ε c Q t x : ℝ) : ℝ := ε*Real.exp (c*(Q-t))*(1+x^2)^(n+2) lemma polynomialBarrier_space (n : ℕ) (ε c Q t x : ℝ) : HasDerivAt (polynomialBarrier n ε c Q t) (ε*Real.exp (c*(Q-t))*(2*(n+2)*x*(1+x^2)^(n+1))) x := by unfold polynomialBarrier convert (((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)).pow (n+2)).const_mul (ε*Real.exp (c*(Q-t))) using 1 <;> first | rfl | (simp only [Nat.cast_add,Nat.cast_ofNat,show n+2-1=n+1 by omega,Pi.add_apply,Pi.pow_apply,id_eq]; ring) lemma polynomialBarrier_space2 (n : ℕ) (ε c Q t x : ℝ) : HasDerivAt (deriv (polynomialBarrier n ε c Q t)) (ε*Real.exp (c*(Q-t))*(2*(n+2)*(1+x^2)^(n+1)+4*(n+2)*(n+1)*x^2*(1+x^2)^n)) x := by have he : deriv (polynomialBarrier n ε c Q t)= fun y => ε*Real.exp (c*(Q-t))*(2*(n+2)*y*(1+y^2)^(n+1)) := funext fun y => (polynomialBarrier_space n ε c Q t y).deriv rw [he] convert (((hasDerivAt_id x).const_mul (2*((n:ℝ)+2))).mul (((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)).pow (n+1))).const_mul (ε*Real.exp (c*(Q-t))) using 1 <;> first | rfl | (simp only [Nat.cast_add,Nat.cast_one,Nat.add_sub_cancel,Pi.add_apply,Pi.pow_apply,id_eq]; ring) lemma polynomialBarrier_time (n : ℕ) (ε c Q t x : ℝ) : HasDerivAt (fun s => polynomialBarrier n ε c Q s x) (-c*polynomialBarrier n ε c Q t x) t := by unfold polynomialBarrier convert ((((hasDerivAt_const t Q).sub (hasDerivAt_id t)).const_mul c).exp.const_mul ε).mul_const ((1+x^2)^(n+2)) using 1 <;> first | rfl | (simp only [Pi.sub_apply,id_eq]; ring) lemma polynomial_transport_neg (n : ℕ) {K b x : ℝ} (hK : 0 ≤ K) (hb : |b| ≤ K*(1+|x|)) : -(2*(n+2)^2+4*K*(n+2)+1)*(1+x^2)^(n+2)+ (1/2:ℝ)*(2*(n+2)*(1+x^2)^(n+1)+4*(n+2)*(n+1)*x^2*(1+x^2)^n)+ b*(2*(n+2)*x*(1+x^2)^(n+1)) < 0 := by have hn : 0 ≤ (n:ℝ) := Nat.cast_nonneg n have hnorm : (1+|x|)*|x| ≤ 2*(1+x^2) := by nlinarith [sq_abs x,sq_nonneg (|x|-1)] have hbx : b*x ≤ 2*K*(1+x^2) := calc b*x ≤ |b*x| := le_abs_self _ _ = |b| * |x| := abs_mul _ _ _ ≤ K*(1+|x|)*|x| := mul_le_mul_of_nonneg_right hb (abs_nonneg _) _ ≤ 2*K*(1+x^2) := by nlinarith [mul_le_mul_of_nonneg_left hnorm hK] have hq : 1+x^2 ≤ (1+x^2)^2 := by nlinarith [sq_nonneg x,sq_nonneg (x^2)] have hxq : x^2 ≤ (1+x^2)^2 := by nlinarith [sq_nonneg (x^2)] have h1 := mul_le_mul_of_nonneg_left hq (show 0 ≤ (n:ℝ)+2 by positivity) have h2 := mul_le_mul_of_nonneg_left hxq (show 0 ≤ 2*((n:ℝ)+2)*(n+1) by positivity) have h3 := mul_le_mul_of_nonneg_left hbx (show 0 ≤ 2*((n:ℝ)+2)*(1+x^2) by positivity) have hneg : -(2*((n:ℝ)+2)^2+4*K*(n+2)+1)*(1+x^2)^2+ (n+2)*(1+x^2)+2*(n+2)*(n+1)*x^2+2*(n+2)*b*x*(1+x^2) < 0 := by nlinarith [sq_nonneg x,sq_pos_of_pos (show 0 < 1+x^2 by positivity)] have hmul := mul_neg_of_pos_of_neg (show 0 < (1+x^2)^n by positivity) hneg convert hmul using 1; first | rfl | (simp only [pow_add,pow_one]; ring) lemma backward_polynomial_bound {Q K C : ℝ} (n : ℕ) (hQ : 0 < Q) (hK : 0 ≤ K) (hC : 0 ≤ C) {U b d : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ univ)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t))) (ht : ∀ t ∈ Ico 0 Q, ∀ x, HasDerivWithinAt (fun s => U s x) (d t x) (Ici t) t) (hb : ∀ t ∈ Icc 0 Q, ∀ x, |b t x| ≤ K*(1+|x|)) (hg : ∀ t ∈ Icc 0 Q, ∀ x, U t x ≤ C*(1+x^2)^n) (hp : ∀ t ∈ Ico 0 Q, ∀ x, 0 < U t x → 0 ≤ d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x) (hterm : ∀ x, U Q x ≤ 0) {ε : ℝ} (hε : 0 < ε) : ∀ t ∈ Icc 0 Q, ∀ x, U t x ≤ polynomialBarrier n ε (2*(n+2)^2+4*K*(n+2)+1) Q t x := by let c := 2*(n+2)^2+4*K*(n+2)+1 have hcpos : 0 ≤ c := by dsimp [c]; positivity let V : ℝ → ℝ → ℝ := fun t x => U t x-polynomialBarrier n ε c Q t x let vd : ℝ → ℝ → ℝ := fun t x => d t x+c*polynomialBarrier n ε c Q t x have hvx (t : ℝ) (ht' : t ∈ Icc 0 Q) (x : ℝ) : HasDerivAt (V t) (deriv (U t) x-ε*Real.exp (c*(Q-t))*(2*(n+2)*x*(1+x^2)^(n+1))) x := ((hx t ht').1 x).hasDerivAt.sub (polynomialBarrier_space n ε c Q t x) have hvxx (t : ℝ) (ht' : t ∈ Icc 0 Q) (x : ℝ) : deriv (deriv (V t)) x=deriv (deriv (U t)) x- ε*Real.exp (c*(Q-t))*(2*(n+2)*(1+x^2)^(n+1)+4*(n+2)*(n+1)*x^2*(1+x^2)^n) := by have he : deriv (V t)=fun y => deriv (U t) y-deriv (polynomialBarrier n ε c Q t) y := by ext y rw [(hvx t ht' y).deriv,(polynomialBarrier_space n ε c Q t y).deriv] rw [he] exact (((hx t ht').2 x).hasDerivAt.sub (polynomialBarrier_space2 n ε c Q t x)).deriv have hvtime (t : ℝ) (ht' : t ∈ Ico 0 Q) (x : ℝ) : HasDerivWithinAt (fun s => V s x) (vd t x) (Ici t) t := by convert (ht t ht' x).sub (polynomialBarrier_time n ε c Q t x).hasDerivWithinAt using 1; first | rfl | (dsimp [V,vd]; ring) have hvp (t : ℝ) (ht' : t ∈ Ico 0 Q) (x : ℝ) (hV : 0 < V t x) : 0 < vd t x+(1/2:ℝ)*deriv (deriv (V t)) x+b t x*deriv (V t) x := by have htcc : t ∈ Icc 0 Q := ⟨ht'.1,ht'.2.le⟩ rw [hvxx t htcc x,(hvx t htcc x).deriv] have hneg := polynomial_transport_neg n hK (hb t htcc x) have hneg' := mul_neg_of_pos_of_neg (mul_pos hε (Real.exp_pos (c*(Q-t)))) hneg have hUp : 0 < U t x := by have : 0 ≤ polynomialBarrier n ε c Q t x := by unfold polynomialBarrier; positivity dsimp [V] at hV linarith have hP := hp t ht' x hUp dsimp [vd,polynomialBarrier,c] at * nlinarith intro t ht' x let R := |x|+C/ε+1 have hdiv : 0 ≤ C/ε := div_nonneg hC hε.le have hR1 : 1 ≤ R := by dsimp [R]; linarith [abs_nonneg x] have hR : 0 < R := by linarith have hxR : x ∈ Icc (-R) R := by dsimp [R]; constructor <;> linarith [le_abs_self x,neg_abs_le x] have hCR : C ≤ ε*(1+R^2) := by have hr : C/ε ≤ 1+R^2 := by dsimp [R] at hR1 ⊢; nlinarith [abs_nonneg x,sq_nonneg R] have hh := (div_le_iff₀ hε).mp hr nlinarith have hCR2 : C ≤ ε*(1+R^2)^2 := hCR.trans (mul_le_mul_of_nonneg_left (by nlinarith [sq_nonneg R,sq_nonneg (R^2)]) hε.le) have hB (s : ℝ) (hs : s ∈ Icc 0 Q) : C*(1+R^2)^n ≤ polynomialBarrier n ε c Q s R := by have he : 1 ≤ Real.exp (c*(Q-s)) := Real.one_le_exp_iff.mpr (mul_nonneg hcpos (sub_nonneg.mpr hs.2)) calc C*(1+R^2)^n ≤ (ε*(1+R^2)^2)*(1+R^2)^n := mul_le_mul_of_nonneg_right hCR2 (by positivity) _ = ε*(1+R^2)^(n+2) := by rw [pow_add]; ring _ ≤ Real.exp (c*(Q-s))*(ε*(1+R^2)^(n+2)) := le_mul_of_one_le_left (by positivity) he _ = _ := by unfold polynomialBarrier; ring have hVc : ContinuousOn (fun p : ℝ × ℝ => V p.1 p.2) (Icc 0 Q ×ˢ Icc (-R) R) := by apply (hc.mono (by intro p hp'; exact ⟨hp'.1,mem_univ _⟩)).sub have hh : Continuous (fun p : ℝ × ℝ => polynomialBarrier n ε c Q p.1 p.2) := by unfold polynomialBarrier fun_prop exact hh.continuousOn have hcomp := backward_positive_strict_compact (b := b) hQ hR hVc (fun s hs y => (hvx s hs y).differentiableAt) hvtime (fun s hs y _ => hvp s hs y) (fun y _ => by dsimp [V] have hn : 0 ≤ polynomialBarrier n ε c Q Q y := by unfold polynomialBarrier; positivity linarith [hterm y]) (fun s hs => by dsimp [V] have hh := hg s hs (-R) have hh' := hB s hs simp only [neg_sq] at hh have he : polynomialBarrier n ε c Q s (-R)=polynomialBarrier n ε c Q s R := by simp [polynomialBarrier] rw [he] linarith) (fun s hs => by dsimp [V]; linarith [hg s hs R,hB s hs]) have hh := hcomp t ht' x hxR dsimp [V] at hh linarith lemma backward_polynomial_comparison {Q K : ℝ} (hQ : 0 < Q) (hK : 0 ≤ K) {U b d : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ univ)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t))) (ht : ∀ t ∈ Ico 0 Q, ∀ x, HasDerivWithinAt (fun s => U s x) (d t x) (Ici t) t) (hb : ∀ t ∈ Icc 0 Q, ∀ x, |b t x| ≤ K*(1+|x|)) (hg : Higher.UniformPolynomialGrowth (Icc 0 Q) U) (hp : ∀ t ∈ Ico 0 Q, ∀ x, 0 < U t x → 0 ≤ d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x) (hterm : ∀ x, U Q x ≤ 0) : ∀ t ∈ Icc 0 Q, ∀ x, U t x ≤ 0 := by obtain ⟨n,C,hC,hg⟩ := hg have hg' (t : ℝ) (ht' : t ∈ Icc 0 Q) (x : ℝ) : U t x ≤ (C*2^n)*(1+x^2)^n := calc U t x ≤ |U t x| := le_abs_self _ _ ≤ C*(1+|x|)^n := hg t ht' x _ ≤ C*(2*(1+x^2))^n := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) (by nlinarith [sq_abs x,sq_nonneg (|x|-1)]) n) hC _ = _ := by rw [mul_pow]; ring intro t ht' x by_contra hn have hpos : 0 < U t x := lt_of_not_ge hn let M := Real.exp ((2*(n+2)^2+4*K*(n+2)+1)*(Q-t))*(1+x^2)^(n+2) have hM : 0 < M := by dsimp [M]; positivity let ε := U t x/(2*M) have hε : 0 < ε := div_pos hpos (by positivity) have hh := backward_polynomial_bound n hQ hK (show 0 ≤ C*2^n by positivity) hc hx ht hb hg' hp hterm hε t ht' x have he : polynomialBarrier n ε (2*(n+2)^2+4*K*(n+2)+1) Q t x=U t x/2 := by dsimp only [polynomialBarrier] rw [mul_assoc] change (U t x/(2*M))*M=U t x/2 field_simp rw [he] at hh linarith end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming lemma backward_reaction_upper {Q K R H : ℝ} (hQ : 0 < Q) (hK : 0 ≤ K) (hR : 0 ≤ R) (hH : 0 ≤ H) {U b d r : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ univ)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t))) (ht : ∀ t ∈ Ico 0 Q, ∀ x, HasDerivWithinAt (fun s => U s x) (d t x) (Ici t) t) (hb : ∀ t ∈ Icc 0 Q, ∀ x, |b t x| ≤ K*(1+|x|)) (hg : Higher.UniformPolynomialGrowth (Icc 0 Q) U) (hr : ∀ t ∈ Ico 0 Q, ∀ x, r t x ≤ R) (hp : ∀ t ∈ Ico 0 Q, ∀ x, d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x+r t x*U t x=0) (hterm : ∀ x, U Q x ≤ H) : ∀ t ∈ Icc 0 Q, ∀ x, U t x ≤ H*Real.exp (R*(Q-t)) := by let E : ℝ → ℝ := fun t => Real.exp (R*(t-Q)) let V : ℝ → ℝ → ℝ := fun t x => E t*U t x-H let vd : ℝ → ℝ → ℝ := fun t x => E t*(d t x+R*U t x) have hE (t : ℝ) : HasDerivAt E (R*E t) t := by convert (((hasDerivAt_id t).sub_const Q).const_mul R).exp using 1 <;> first | rfl | (dsimp [E]; ring) have hEc : Continuous E := continuous_iff_continuousAt.mpr (fun t => (hE t).continuousAt) have hEp (t : ℝ) : 0 < E t := Real.exp_pos _ have hEl (t : ℝ) (ht' : t ∈ Icc 0 Q) : E t ≤ 1 := Real.exp_le_one_iff.mpr (mul_nonpos_of_nonneg_of_nonpos hR (sub_nonpos.mpr ht'.2)) have hvx (t : ℝ) (ht' : t ∈ Icc 0 Q) (x : ℝ) : HasDerivAt (V t) (E t*deriv (U t) x) x := (((hx t ht').1 x).hasDerivAt.const_mul (E t)).sub_const H have hvxx (t : ℝ) (ht' : t ∈ Icc 0 Q) (x : ℝ) : HasDerivAt (deriv (V t)) (E t*deriv (deriv (U t)) x) x := by have he : deriv (V t)=fun y => E t*deriv (U t) y := funext (fun y => (hvx t ht' y).deriv) rw [he] exact ((hx t ht').2 x).hasDerivAt.const_mul (E t) have hvtime (t : ℝ) (ht' : t ∈ Ico 0 Q) (x : ℝ) : HasDerivWithinAt (fun s => V s x) (vd t x) (Ici t) t := by convert ((hE t).hasDerivWithinAt.mul (ht t ht' x)).sub_const H using 1; first | rfl | (dsimp [vd]; ring) have hvg : Higher.UniformPolynomialGrowth (Icc 0 Q) V := by obtain ⟨n,C,hC,hg⟩ := hg refine ⟨n,C+H,add_nonneg hC hH,fun t ht' x => ?_⟩ have he := mul_le_mul_of_nonneg_right (hEl t ht') (abs_nonneg (U t x)) have hp1 : 1 ≤ (1+|x|)^n := one_le_pow₀ (by linarith [abs_nonneg x]) calc |V t x| ≤ |E t*U t x|+|H| := abs_sub _ _ _ = E t*|U t x|+H := by rw [abs_mul,abs_of_pos (hEp t),abs_of_nonneg hH] _ ≤ |U t x|+H := by linarith _ ≤ C*(1+|x|)^n+H*(1+|x|)^n := add_le_add (hg t ht' x) (le_mul_of_one_le_right hH hp1) _ = _ := by ring have hvc : ContinuousOn (fun p : ℝ × ℝ => V p.1 p.2) (Icc 0 Q ×ˢ univ) := ((hEc.comp continuous_fst).continuousOn.mul hc).sub continuousOn_const have hcomp := backward_polynomial_comparison hQ hK hvc (fun t ht' => ⟨fun x => (hvx t ht' x).differentiableAt,fun x => (hvxx t ht' x).differentiableAt⟩) hvtime hb hvg (fun t ht' x hpos => ?_) (fun x => ?_) · intro t ht' x have h := hcomp t ht' x dsimp [V] at h have hexp : E t*Real.exp (R*(Q-t))=1 := by rw [← Real.exp_add]; convert Real.exp_zero using 1; congr 1; ring have hh := mul_le_mul_of_nonneg_right (show E t*U t x ≤ H by linarith) (Real.exp_nonneg (R*(Q-t))) calc U t x = (E t*Real.exp (R*(Q-t)))*U t x := by rw [hexp,one_mul] _ = E t*U t x*Real.exp (R*(Q-t)) := by ring _ ≤ _ := hh · rw [(hvxx t ⟨ht'.1,ht'.2.le⟩ x).deriv,(hvx t ⟨ht'.1,ht'.2.le⟩ x).deriv] have hpp := hp t ht' x have hrp := hr t ht' x have hUp : 0 ≤ E t*U t x := by dsimp [V] at hpos; linarith have hnn := mul_nonneg (sub_nonneg.mpr hrp) hUp dsimp [vd] calc _ = E t*(d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x+r t x*U t x)+ (R-r t x)*(E t*U t x) := by ring _ ≥ 0 := by rw [hpp,mul_zero,zero_add]; exact hnn · dsimp [V,E] simp only [sub_self,mul_zero,Real.exp_zero,one_mul] exact sub_nonpos.mpr (hterm x) lemma backward_reaction_bound {Q K R H : ℝ} (hQ : 0 < Q) (hK : 0 ≤ K) (hR : 0 ≤ R) (hH : 0 ≤ H) {U b d r : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ univ)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t))) (ht : ∀ t ∈ Ico 0 Q, ∀ x, HasDerivWithinAt (fun s => U s x) (d t x) (Ici t) t) (hb : ∀ t ∈ Icc 0 Q, ∀ x, |b t x| ≤ K*(1+|x|)) (hg : Higher.UniformPolynomialGrowth (Icc 0 Q) U) (hr : ∀ t ∈ Ico 0 Q, ∀ x, r t x ≤ R) (hp : ∀ t ∈ Ico 0 Q, ∀ x, d t x+(1/2:ℝ)*deriv (deriv (U t)) x+b t x*deriv (U t) x+r t x*U t x=0) (hterm : ∀ x, |U Q x| ≤ H) : ∀ t ∈ Icc 0 Q, ∀ x, |U t x| ≤ H*Real.exp (R*(Q-t)) := by have hupper := backward_reaction_upper hQ hK hR hH hc hx ht hb hg hr hp (fun x => (le_abs_self _).trans (hterm x)) have hn1 (t : ℝ) (ht' : t ∈ Icc 0 Q) : deriv (fun x => -U t x)=fun x => -deriv (U t) x := by funext x; exact ((hx t ht').1 x).hasDerivAt.neg.deriv have hn2 (t : ℝ) (ht' : t ∈ Icc 0 Q) : deriv (deriv (fun x => -U t x))=fun x => -deriv (deriv (U t)) x := by rw [hn1 t ht']; funext x; exact ((hx t ht').2 x).hasDerivAt.neg.deriv have hneg := backward_reaction_upper (U:=fun t x => -U t x) (d:=fun t x => -d t x) hQ hK hR hH hc.neg (fun t ht' => ⟨(hx t ht').1.neg,by rw [hn1 t ht']; exact (hx t ht').2.neg⟩) (fun t ht' x => (ht t ht' x).neg) hb (by simpa only [neg_one_mul] using hg.const_mul (-1)) hr (fun t ht' x => by rw [hn2 t ⟨ht'.1,ht'.2.le⟩,hn1 t ⟨ht'.1,ht'.2.le⟩]; linarith [hp t ht' x]) (fun x => (neg_le_abs _).trans (hterm x)) intro t ht' x rw [abs_le] exact ⟨by linarith [hneg t ht' x],hupper t ht' x⟩ lemma forward_reaction_bound {Q K R H : ℝ} (hQ : 0 < Q) (hK : 0 ≤ K) (hR : 0 ≤ R) (hH : 0 ≤ H) {U b d r : ℝ → ℝ → ℝ} (hc : ContinuousOn (fun p : ℝ × ℝ => U p.1 p.2) (Icc 0 Q ×ˢ univ)) (hx : ∀ t ∈ Icc 0 Q, Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t))) (ht : ∀ t ∈ Ioc 0 Q, ∀ x, HasDerivAt (fun s => U s x) (d t x) t) (hb : ∀ t ∈ Icc 0 Q, ∀ x, |b t x| ≤ K*(1+|x|)) (hg : Higher.UniformPolynomialGrowth (Icc 0 Q) U) (hr : ∀ t ∈ Ioc 0 Q, ∀ x, r t x ≤ R) (hp : ∀ t ∈ Ioc 0 Q, ∀ x, d t x-(1/2:ℝ)*deriv (deriv (U t)) x-b t x*deriv (U t) x-r t x*U t x=0) (hterm : ∀ x, |U 0 x| ≤ H) : ∀ t ∈ Icc 0 Q, ∀ x, |U t x| ≤ H*Real.exp (R*t) := by have hm (t : ℝ) (ht' : t ∈ Icc 0 Q) : Q-t ∈ Icc 0 Q := ⟨by linarith [ht'.2],by linarith [ht'.1]⟩ have hmo (t : ℝ) (ht' : t ∈ Ico 0 Q) : Q-t ∈ Ioc 0 Q := ⟨by linarith [ht'.2],by linarith [ht'.1]⟩ have hvc : ContinuousOn (fun p : ℝ × ℝ => U (Q-p.1) p.2) (Icc 0 Q ×ˢ univ) := hc.comp (show ContinuousOn (fun p : ℝ × ℝ => (Q-p.1,p.2)) _ by fun_prop) (fun p hp' => ⟨hm p.1 hp'.1,mem_univ _⟩) have hvt (t : ℝ) (ht' : t ∈ Ico 0 Q) (x : ℝ) : HasDerivWithinAt (fun s => U (Q-s) x) (-d (Q-t) x) (Ici t) t := by convert ((ht (Q-t) (hmo t ht') x).comp t ((hasDerivAt_const t Q).sub (hasDerivAt_id t))).hasDerivWithinAt using 1 <;> first | rfl | ring have hcomp := backward_reaction_bound (U:=fun t => U (Q-t)) (b:=fun t => b (Q-t)) (d:=fun t x => -d (Q-t) x) (r:=fun t => r (Q-t)) hQ hK hR hH hvc (fun t ht' => hx (Q-t) (hm t ht')) hvt (fun t ht' => hb (Q-t) (hm t ht')) (hg.comp (fun t ht' => hm t ht')) (fun t ht' => hr (Q-t) (hmo t ht')) (fun t ht' x => by linarith [hp (Q-t) (hmo t ht') x]) (fun x => by simpa using hterm x) intro t ht' x simpa only [sub_sub_cancel] using hcomp (Q-t) (hm t ht') x end MicroscopicJamming open Set Filter open scoped Topology namespace MicroscopicJamming namespace Higher lemma gaussianRowOperator_third_bound {u : ℝ → ℝ} {A B C κ Q a L D K H : ℝ} (hu : Twice.RowTwiceTerminal u A B C κ Q) (hs : SmoothPolynomial u) (ha : 0 ≤ a) (ha1 : a ≤ 1) (hL : 0 < L) (hsub : a*κ*L < 1) (hD : 0 ≤ D) (hK : 0 ≤ K) (hH : 0 ≤ H) (hfirst : ∀ T ∈ Icc 0 L, ∀ x, |deriv (gaussianRowOperator a T u) x| ≤ D*(1+|x|)) (hsecond : ∀ T ∈ Icc 0 L, ∀ x, deriv (deriv (gaussianRowOperator a T u)) x ≤ K) (hthird : ∀ x, |iteratedDeriv 3 u x| ≤ H) : ∀ T ∈ Icc 0 L, ∀ x, |iteratedDeriv 3 (gaussianRowOperator a T u) x| ≤ H*Real.exp (3*K*T) := by let F : ℝ → ℝ → ℝ := fun t => gaussianRowOperator a t u let U : ℝ → ℝ → ℝ := fun t => iteratedDeriv 3 (F t) let b : ℝ → ℝ → ℝ := fun t x => a*iteratedDeriv 1 (F t) x let r : ℝ → ℝ → ℝ := fun t x => 3*a*iteratedDeriv 2 (F t) x let d : ℝ → ℝ → ℝ := fun t x => (1/2:ℝ)*iteratedDeriv 5 (F t) x+ a*iteratedDeriv 1 (F t) x*iteratedDeriv 4 (F t) x+3*a*iteratedDeriv 2 (F t) x*iteratedDeriv 3 (F t) x have hsp (t : ℝ) (ht : t ∈ Icc 0 L) : SmoothPolynomial (F t) := gaussianRowOperator_smoothPolynomial hu hs ha ht.1 ((mul_le_mul_of_nonneg_left ht.2 (mul_nonneg ha hu.kappa_nonneg)).trans_lt hsub) have hU1 (t : ℝ) : deriv (U t)=iteratedDeriv 4 (F t) := (iteratedDeriv_succ (n:=3)).symm have hU2 (t : ℝ) : deriv (deriv (U t))=iteratedDeriv 5 (F t) := by rw [hU1 t]; exact (iteratedDeriv_succ (n:=4)).symm have hUx (t : ℝ) (ht : t ∈ Icc 0 L) : Differentiable ℝ (U t) ∧ Differentiable ℝ (deriv (U t)) := by refine ⟨(hsp t ht).differentiable 3,?_⟩ rw [hU1 t] exact (hsp t ht).differentiable 4 have hUb (t : ℝ) (ht : t ∈ Icc 0 L) (x : ℝ) : |b t x| ≤ D*(1+|x|) := by change |a*iteratedDeriv 1 (gaussianRowOperator a t u) x| ≤ _ rw [abs_mul,abs_of_nonneg ha,iteratedDeriv_one] exact (mul_le_mul_of_nonneg_left (hfirst t ht x) ha).trans (mul_le_of_le_one_left (by positivity) ha1) have hUr (t : ℝ) (ht : t ∈ Ioc 0 L) (x : ℝ) : r t x ≤ 3*K := by have hsec := hsecond t ⟨ht.1.le,ht.2⟩ x have hh := mul_le_mul_of_nonneg_left hsec ha have hh' := mul_le_of_le_one_left hK ha1 dsimp [r,F] simp only [iteratedDeriv_succ,iteratedDeriv_zero] nlinarith have hUp (t : ℝ) (ht : t ∈ Ioc 0 L) (x : ℝ) : d t x-(1/2:ℝ)*deriv (deriv (U t)) x-b t x*deriv (U t) x-r t x*U t x=0 := by rw [hU2 t,hU1 t] dsimp [d,b,r,U]; ring have he : F 0=u := funext (fun x => gaussianRowOperator_zero a u x) apply forward_reaction_bound hL hD (show 0 ≤ 3*K by positivity) hH (gaussianRowOperator_third_continuous hs ha (fun x => (hu.bounds x).2.1)).continuousOn hUx (fun t ht x => gaussianRowOperator_third_equation hs ha (fun x => (hu.bounds x).2.1) ht.1 x) hUb (gaussianRowOperator_third_growth hu hs ha hL.le hsub) hUr hUp intro x change |iteratedDeriv 3 (F 0) x| ≤ H rw [he] exact hthird x end Higher end MicroscopicJamming open Set namespace MicroscopicJamming namespace Higher lemma gaussianRowComposition_third_bound {u : ℝ → ℝ} {A B C κ Q H : ℝ} (hQ : 0 < Q) (hu : RowAnalyticTerminal u A B C κ Q) (hH : 0 ≤ H) (hthird : ∀ x, |iteratedDeriv 3 u x| ≤ H) : ∀ rs : List (ℝ × ℝ), (∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → gaussianStepTime rs ≤ Q → ∀ x, |iteratedDeriv 3 (gaussianRowComposition rs u) x| ≤ H*Real.exp (3*(C+κ/(1-κ*Q))*gaussianStepTime rs) := by let K := C+κ/(1-κ*Q) have hden : 0 < 1-κ*Q := by linarith [hu.2.2.2.2.1] have hK : 0 ≤ K := add_nonneg hu.2.2.1 (div_nonneg hu.2.2.2.1 hden.le) obtain ⟨D,hD,hfirst⟩ := gaussian_step_slope A B C κ Q hQ hu.2.1 hu.2.2.1 hu.2.2.2.1 hu.2.2.2.2.1 intro rs induction rs with | nil => intro _ _ x; simpa [gaussianRowComposition,gaussianStepTime] using hthird x | cons r rs ih => intro hrs hsum have hr := hrs r (by simp) have ht : ∀ v ∈ rs, 0 ≤ v.1 ∧ v.1 ≤ 1 ∧ 0 ≤ v.2 := fun v hv => hrs v (by simp [hv]) have htime : gaussianStepTime (r::rs)=r.2+gaussianStepTime rs := by simp [gaussianStepTime] rw [htime] at hsum ⊢ have htq : gaussianStepTime rs ≤ Q := by linarith [hr.2.2] have htw := gaussianRowComposition_twice hQ hu ht htq have hsp := gaussianRowComposition_smoothPolynomial hQ hu rs ht htq have hinitial : ∀ x, |iteratedDeriv 3 (gaussianRowComposition rs u) x| ≤ H*Real.exp (3*K*gaussianStepTime rs) := ih ht htq by_cases hz : r.2=0 · have he : gaussianRowComposition (r::rs) u=gaussianRowComposition rs u := by funext x change gaussianRowOperator r.1 r.2 (gaussianRowComposition rs u) x=_ rw [hz,gaussianRowOperator_zero] rw [he,hz,zero_add] exact hinitial · have hT : 0 < r.2 := lt_of_le_of_ne hr.2.2 (Ne.symm hz) have hvalid (T : ℝ) (hT' : T ∈ Icc 0 r.2) : ∀ v ∈ (r.1,T)::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 ⟨hr.1,hr.2.1,hT'.1⟩ · exact ht v h have htot (T : ℝ) (hT' : T ∈ Icc 0 r.2) : gaussianStepTime ((r.1,T)::rs) ≤ Q := by have hh : T+gaussianStepTime rs ≤ Q := by linarith [hT'.2] simpa [gaussianStepTime] using hh have hsec (T : ℝ) (hT' : T ∈ Icc 0 r.2) (x : ℝ) : deriv (deriv (gaussianRowOperator r.1 T (gaussianRowComposition rs u))) x ≤ K := by have hh := ((gaussian_step_bounds u A B C κ Q hQ hu _ (hvalid T hT') (htot T hT')).2.2 x).2.2 have hcap : κ/(1-κ*gaussianStepTime ((r.1,T)::rs)) ≤ κ/(1-κ*Q) := div_le_div_of_nonneg_left hu.2.2.2.1 hden (by nlinarith [mul_le_mul_of_nonneg_left (htot T hT') hu.2.2.2.1]) exact hh.trans (by dsimp [K]; linarith [hu.2.2.1]) have hh := gaussianRowOperator_third_bound htw hsp hr.1 hr.2.1 hT (gaussianRowComposition_block_subcritical hQ hu ht hr.1 hr.2.1 hr.2.2 hsum) hD hK (mul_nonneg hH (Real.exp_nonneg _)) (fun T hT' => hfirst u hu ((r.1,T)::rs) (hvalid T hT') (htot T hT')) hsec hinitial intro x have hbound := hh r.2 ⟨hT.le,le_rfl⟩ x have he : (H*Real.exp (3*K*gaussianStepTime rs))*Real.exp (3*K*r.2)= H*Real.exp (3*K*(r.2+gaussianStepTime rs)) := by rw [mul_assoc,← Real.exp_add] congr 2 ring rw [he] at hbound exact hbound end Higher theorem gaussian_step_third : GaussianStepThirdStatement := by intro u A B C κ Q hQ hu obtain ⟨H,hthird⟩ := hu.2.2.2.2.2.1 3 (by norm_num) have hH : 0 ≤ H := (abs_nonneg _).trans (hthird 0) let K := C+κ/(1-κ*Q) have hden : 0 < 1-κ*Q := by linarith [hu.2.2.2.2.1] have hK : 0 ≤ K := add_nonneg hu.2.2.1 (div_nonneg hu.2.2.2.1 hden.le) refine ⟨H*Real.exp (3*K*Q),mul_nonneg hH (Real.exp_nonneg _),?_⟩ intro rs hrs hsum t ht htT obtain ⟨hv,hvT⟩ := gaussianStepRemainder_valid hrs ht rw [max_eq_left (sub_nonneg.mpr htT)] at hvT have hvQ : gaussianStepTime (gaussianStepRemainder rs t) ≤ Q := by rw [hvT]; linarith have hs := Higher.gaussianRowComposition_smoothPolynomial hQ hu _ hv hvQ have hb := Higher.gaussianRowComposition_third_bound hQ hu hH hthird _ hv hvQ refine ⟨?_,fun x => ?_⟩ · change Differentiable ℝ (deriv (deriv (gaussianRowComposition (gaussianStepRemainder rs t) u))) simpa only [iteratedDeriv_succ,iteratedDeriv_zero] using hs.differentiable 2 · have hh := (hb x).trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hvQ (by positivity : 0 ≤ 3*K))) hH) change |deriv (deriv (deriv (gaussianRowComposition (gaussianStepRemainder rs t) u))) x| ≤ _ simpa only [iteratedDeriv_succ,iteratedDeriv_zero,K] using hh end MicroscopicJamming namespace MicroscopicJamming def GaussianStepGradientTimeStatement : Prop := ∀ A B C κ Q : ℝ, 0 < Q → 0 ≤ A → 0 ≤ C → 0 ≤ κ → κ*Q < 1 → ∃ D K : ℝ, 0 ≤ D ∧ 0 ≤ K ∧ ∀ u : ℝ → ℝ, RowAnalyticTerminal u A B C κ Q → ∀ rs : List (ℝ × ℝ), (∀ r ∈ rs, 0 ≤ r.1 ∧ r.1 ≤ 1 ∧ 0 ≤ r.2) → gaussianStepTime rs ≤ Q → ∀ s t : ℝ, 0 ≤ s → s ≤ gaussianStepTime rs → 0 ≤ t → t ≤ gaussianStepTime rs → ∀ x h : ℝ, 0 < h → h ≤ 1 → |deriv (gaussianStepPath rs u t) x-deriv (gaussianStepPath rs u s) x| ≤ 2*D*(1+(|x|+1)^2)*(|t-s|/h)+2*K*h end MicroscopicJamming open Set namespace MicroscopicJamming lemma deriv_secant_error {f : ℝ → ℝ} {K x h : ℝ} (hf : Differentiable ℝ f) (hf' : Differentiable ℝ (deriv f)) (hK : 0 ≤ K) (hb : ∀ z, |deriv (deriv f) z| ≤ K) (hh : 0 < h) : |deriv f x-(f (x+h)-f x)/h| ≤ K*h := by obtain ⟨c,hc,he⟩ := exists_deriv_eq_slope f (show x 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 hv apply hv.trans apply mul_le_mul_of_nonneg_left _ hK rw [abs_of_neg (by linarith [hc.1] : x-c<0)] linarith [hc.2] theorem gaussian_step_gradient_time : GaussianStepGradientTimeStatement := by intro A B C κ Q hQ hA hC hk hsub obtain ⟨D,L,hD,hL,hpath⟩ := gaussian_step_path 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 ⟨D,K,hD,hK,?_⟩ intro u hu rs hrs hsum s t hs hsT ht htT x h hh hh1 obtain ⟨_,_,htime,hspace⟩ := hpath u hu rs hrs hsum have hcurv (v : ℝ) (hv : 0 ≤ v) (hvT : v ≤ gaussianStepTime rs) (z : ℝ) : |deriv (deriv (gaussianStepPath rs u v)) z| ≤ K := by have hz := ((hspace v hv hvT).2.2 z).2 have hsmall : gaussianStepTime rs-v ≤ Q := by linarith have hden' : 0 < 1-κ*(gaussianStepTime rs-v) := by nlinarith [mul_le_mul_of_nonneg_left hsmall hk] have hcap : κ/(1-κ*(gaussianStepTime rs-v)) ≤ κ/(1-κ*Q) := div_le_div_of_nonneg_left hk hden (by nlinarith [mul_le_mul_of_nonneg_left hsmall hk]) rw [abs_le] dsimp [K] constructor <;> linarith [div_nonneg hk hden.le,hz.1,hz.2] have heT := deriv_secant_error (hspace t ht htT).1 (hspace t ht htT).2.1 hK (hcurv t ht htT) (x:=x) hh have heS := deriv_secant_error (hspace s hs hsT).1 (hspace s hs hsT).2.1 hK (hcurv s hs hsT) (x:=x) hh let E := D*(1+(|x|+1)^2)*|t-s| have hb (y : ℝ) (hy : |y| ≤ |x|+1) : |gaussianStepPath rs u t y-gaussianStepPath rs u s y| ≤ E := by have hy2 : y^2 ≤ (|x|+1)^2 := by have hh := (sq_le_sq₀ (abs_nonneg y) (by positivity : 0 ≤ |x|+1)).mpr hy simpa only [sq_abs] using hh exact (htime s t hs ht y).trans (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (by linarith) hD) (abs_nonneg _)) have hx1 : |x+h| ≤ |x|+1 := by have hh' := abs_add_le x h rw [abs_of_pos hh] at hh' linarith have hx0 : |x| ≤ |x|+1 := by linarith have hv1 := hb (x+h) hx1 have hv0 := hb x hx0 have hslope : |(gaussianStepPath rs u t (x+h)-gaussianStepPath rs u t x)/h- (gaussianStepPath rs u s (x+h)-gaussianStepPath rs u s x)/h| ≤ 2*E/h := by rw [← sub_div,abs_div,abs_of_pos hh] apply div_le_div_of_nonneg_right _ hh.le have he : (gaussianStepPath rs u t (x+h)-gaussianStepPath rs u t x)- (gaussianStepPath rs u s (x+h)-gaussianStepPath rs u s x)= (gaussianStepPath rs u t (x+h)-gaussianStepPath rs u s (x+h))- (gaussianStepPath rs u t x-gaussianStepPath rs u s x) := by ring rw [he] exact (abs_sub _ _).trans (by linarith) have ht1 := abs_sub_le (deriv (gaussianStepPath rs u t) x) ((gaussianStepPath rs u t (x+h)-gaussianStepPath rs u t x)/h) (deriv (gaussianStepPath rs u s) x) have ht2 := abs_sub_le ((gaussianStepPath rs u t (x+h)-gaussianStepPath rs u t x)/h) ((gaussianStepPath rs u s (x+h)-gaussianStepPath rs u s x)/h) (deriv (gaussianStepPath rs u s) x) rw [abs_sub_comm (deriv (gaussianStepPath rs u s) x)] at heS have he : 2*E/h=2*D*(1+(|x|+1)^2)*(|t-s|/h) := by dsimp [E]; ring rw [he] at hslope linarith end MicroscopicJamming end end OAI