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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | import OAI.Geometry.IsometricImmersion.Curvature.CurvatureLowerTube
import OAI.Geometry.IsometricImmersion.Pulses.PulseDerivativeBounds
import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetNorm
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry
def scalarPulseDerivativeBound (a : ℝ) (ha : 0 < a) (delta : ℝ) (k : ℕ) : ℝ :=
pulseCoreDerivativeBound a ha k * pulseScaleBound 0 delta ^ k
theorem scalarPulseDerivativeBound_nonneg {a : ℝ} (ha : 0 < a) (delta : ℝ) (k : ℕ) :
0 ≤ scalarPulseDerivativeBound a ha delta k :=
mul_nonneg (pulseCoreDerivativeBound_nonneg ha k) (pow_nonneg (pulseScaleBound_pos 0 delta).le k)
theorem pulseScalar_eq_core (a : ℝ) (N : ℕ) (delta tau : ℝ) :
pulseScalar a N delta tau = fun p => (1/tau^N) • pulseCore 0 a delta tau p := by
funext p
simp only [pulseScalar,pulseCore,xiCLM,thetaCLM,temporalCutoff,
add_apply,smul_apply,
ContinuousLinearMap.proj_apply,smul_eq_mul,zero_mul,add_zero]
have ht : tau/delta*p 1 = tau*p 1/delta := by ring
rw [ht]
ring
theorem pulseScalar_derivative_le {a : ℝ} (ha : 0 < a) (N : ℕ) (delta : ℝ)
{tau : ℝ} (ht : 1 ≤ tau) (k : ℕ) (p : Coord) :
‖iteratedFDeriv ℝ k (pulseScalar a N delta tau) p‖ ≤
scalarPulseDerivativeBound a ha delta k * tau^k/tau^N := by
have ht0 : 0 < tau := zero_lt_one.trans_le ht
rw [pulseScalar_eq_core]
rw [iteratedFDeriv_const_smul_apply' (i := k) (a := (1/tau^N : ℝ))
((pulseCore_contDiff 0 a delta tau).contDiffAt.of_le (WithTop.coe_le_coe.mpr le_top)),
norm_smul,Real.norm_of_nonneg (div_nonneg zero_le_one (pow_nonneg ht0.le N))]
calc
_ ≤ (1/tau^N)*(pulseCoreDerivativeBound a ha k*(tau*pulseScaleBound 0 delta)^k) :=
mul_le_mul_of_nonneg_left (pulseCore_derivative_le ha 0 delta ht k p)
(div_nonneg zero_le_one (pow_nonneg ht0.le N))
_ = _ := by rw [mul_pow]; unfold scalarPulseDerivativeBound; ring
def scalarPulseFirstJetBound (a : ℝ) (ha : 0 < a) (delta : ℝ) : ℝ :=
max (scalarPulseDerivativeBound a ha delta 0) (scalarPulseDerivativeBound a ha delta 1)
theorem scalarPulseFirstJetBound_nonneg {a : ℝ} (ha : 0 < a) (delta : ℝ) :
0 ≤ scalarPulseFirstJetBound a ha delta :=
(scalarPulseDerivativeBound_nonneg ha delta 0).trans (le_max_left _ _)
def scalarPulseFirstJetBudget (a : ℝ) (ha : 0 < a) (N : ℕ) (delta tau : ℝ) : ℝ :=
scalarPulseFirstJetBound a ha delta * tau/tau^N
theorem scalarPulseFirstJetBudget_nonneg {a : ℝ} (ha : 0 < a) (N : ℕ) (delta : ℝ)
{tau : ℝ} (ht : 1 ≤ tau) : 0 ≤ scalarPulseFirstJetBudget a ha N delta tau :=
div_nonneg (mul_nonneg (scalarPulseFirstJetBound_nonneg ha delta) (zero_le_one.trans ht))
(pow_nonneg (zero_le_one.trans ht) N)
theorem pulseScalar_first_jet_bounds {a : ℝ} (ha : 0 < a) (N : ℕ) (delta : ℝ)
{tau : ℝ} (ht : 1 ≤ tau) (p : Coord) :
|pulseScalar a N delta tau p| ≤ scalarPulseFirstJetBudget a ha N delta tau ∧
∀ i, |coordPartial i (pulseScalar a N delta tau) p| ≤ scalarPulseFirstJetBudget a ha N delta tau := by
have ht0 : 0 ≤ tau := zero_le_one.trans ht
have hden : 0 ≤ tau^N := pow_nonneg ht0 N
have hB0 : scalarPulseDerivativeBound a ha delta 0 ≤ scalarPulseFirstJetBound a ha delta :=
le_max_left _ _
have hB1 : scalarPulseDerivativeBound a ha delta 1 ≤ scalarPulseFirstJetBound a ha delta :=
le_max_right _ _
constructor
· have h := pulseScalar_derivative_le ha N delta ht 0 p
simp only [norm_iteratedFDeriv_zero,Real.norm_eq_abs,pow_zero,mul_one] at h
exact h.trans (div_le_div_of_nonneg_right
(hB0.trans (le_mul_of_one_le_right (scalarPulseFirstJetBound_nonneg ha delta) ht)) hden)
· intro i
have h := (norm_iteratedCoordPartial_le_jet
(pulseScalar_contDiff a N delta tau).contDiffOn isOpen_univ [i] (mem_univ p)).trans
(pulseScalar_derivative_le ha N delta ht 1 p)
simp only [iteratedCoordPartial,Real.norm_eq_abs,pow_one] at h
exact h.trans (div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right hB1 ht0) hden)
theorem thetaPulse_first_increment_bound {a : ℝ} (ha : 0 < a) (N : ℕ) (delta : ℝ)
{tau : ℝ} (ht : 1 ≤ tau) (p : Coord) :
‖thetaCurvatureFirstIncrement (pulseScalar a N delta tau) p‖ ≤
scalarPulseFirstJetBudget a ha N delta tau :=
thetaCurvatureFirstIncrement_norm_le _ p (scalarPulseFirstJetBudget_nonneg ha N delta ht)
(pulseScalar_first_jet_bounds ha N delta ht p).1 (pulseScalar_first_jet_bounds ha N delta ht p).2
theorem scalarPulseFirstJetBudget_tendsto_zero {a : ℝ} (ha : 0 < a) (N : ℕ)
(hN : 1 < N) (delta : ℝ) :
Tendsto (scalarPulseFirstJetBudget a ha N delta) atTop (𝓝 0) := by
have hn : N-1 ≠ 0 := by omega
have hi := (tendsto_inv_atTop_zero : Tendsto (fun t : ℝ => t⁻¹) atTop (𝓝 0)).pow (N-1)
have h : Tendsto (fun t : ℝ => scalarPulseFirstJetBound a ha delta*(t⁻¹)^(N-1)) atTop (𝓝 0) := by
simpa only [zero_pow hn,mul_zero] using hi.const_mul (scalarPulseFirstJetBound a ha delta)
apply h.congr'
filter_upwards [eventually_gt_atTop (0 : ℝ)] with t ht
have hN' : N = 1+(N-1) := by omega
have hr : t/t^N = (t⁻¹)^(N-1) := by
conv_lhs => rw [hN',pow_add,pow_one]
rw [inv_pow]
field_simp [ht.ne']
simp only [scalarPulseFirstJetBudget,mul_div_assoc,hr]
theorem exists_uniform_actual_pulse_lower_bound (B : ℝ) {d : ℝ} (hd : 0 < d)
{a : ℝ} (ha : 0 < a) (N : ℕ) (hN : 1 < N) :
∃ C : ℝ, 0 ≤ C ∧ ∀ delta : ℝ, ∃ T : ℝ, 1 ≤ T ∧
∀ tau : ℝ, T ≤ tau → ∀ (g : MetricField) (p : Coord),
‖actualCurvatureFirstInput g p‖ ≤ B → d ≤ (g p).det →
|thetaPulseLowerDifference g (pulseScalar a N delta tau) p| ≤
C*scalarPulseFirstJetBudget a ha N delta tau := by
obtain ⟨C,hC,hbound⟩ := exists_uniform_thetaPulseLowerDifference_bound B hd
refine ⟨C,hC,?_⟩
intro delta
let beta := max B 0+1
have hbeta : 0 < beta := by dsimp only [beta]; linarith [le_max_right B 0]
let e := min 1 (d/(2*beta))
have he : 0 < e := lt_min zero_lt_one (div_pos hd (mul_pos (by norm_num) hbeta))
have hev : ∀ᶠ tau : ℝ in atTop, scalarPulseFirstJetBudget a ha N delta tau < e :=
(scalarPulseFirstJetBudget_tendsto_zero ha N hN delta).eventually (eventually_lt_nhds he)
obtain ⟨T,hT⟩ := eventually_atTop.mp hev
refine ⟨max T 1,le_max_right _ _,?_⟩
intro tau ht g p hg hdet
have ht1 : 1 ≤ tau := (le_max_right _ _).trans ht
have htT : T ≤ tau := (le_max_left _ _).trans ht
have hsmall1 : scalarPulseFirstJetBudget a ha N delta tau ≤ 1 :=
(hT tau htT).le.trans (min_le_left _ _)
have hsmallD : scalarPulseFirstJetBudget a ha N delta tau ≤ d/(2*beta) :=
(hT tau htT).le.trans (min_le_right _ _)
have hg00 : |g p 0 0| ≤ beta := by
have h0 : |g p 0 0| ≤ ‖actualCurvatureFirstInput g p‖ := by
simpa only [Real.norm_eq_abs] using
((norm_le_pi_norm (g p 0) 0).trans (norm_le_pi_norm (g p) 0)).trans
(norm_fst_le (actualCurvatureFirstInput g p))
exact (h0.trans hg).trans (by dsimp only [beta]; linarith [le_max_left B 0])
have hproduct : |g p 0 0| * |pulseScalar a N delta tau p| ≤ d/2 := by
calc
_ ≤ beta*scalarPulseFirstJetBudget a ha N delta tau :=
mul_le_mul hg00 (pulseScalar_first_jet_bounds ha N delta ht1 p).1
(abs_nonneg _) hbeta.le
_ ≤ beta*(d/(2*beta)) := mul_le_mul_of_nonneg_left hsmallD hbeta.le
_ = d/2 := by field_simp [hbeta.ne']
have hinc := thetaPulse_first_increment_bound ha N delta ht1 p
exact (hbound g (pulseScalar a N delta tau) p hg hdet (hinc.trans hsmall1) hproduct).trans
(mul_le_mul_of_nonneg_left hinc hC)
end SmoothLocal.Pulse
end
end OAI
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