| 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<x+h by linarith) |
| hf.continuous.continuousOn hf.differentiableOn |
| have he' : deriv f c=(f (x+h)-f x)/h := by simpa using he |
| rw [←he'] |
| have hv := 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 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 |
| |
