File size: 3,011 Bytes
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 | import OAI.Geometry.IsometricImmersion.Comparison.ComparisonEnergyScaling
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
theorem pulseStrip_subset_comparisonSlab
{a delta tau speed : ℝ} (_ha : 0 ≤ a) (_hd : 0 ≤ delta) (_ht : 0 < tau)
(hs : 0 ≤ speed) (htravel : speed*(2*delta/tau) ≤ 3*a) :
pulseStrip a delta tau ⊆ shrinkingSlab (-(4*a)) (4*a) (-(delta/tau)) (delta/tau) speed := by
intro p hp
have hpT : p 1 ∈ Icc (-(delta/tau)) (delta/tau) := hp.2
have hm : speed*(p 1-(-(delta/tau))) ≤ speed*(2*delta/tau) := by
apply mul_le_mul_of_nonneg_left _ hs
calc
_ ≤ delta/tau+delta/tau := by linarith [hpT.2]
_ = _ := by ring
refine ⟨hpT,?_⟩
change -(4*a)+speed*(p 1-(-(delta/tau))) ≤ p 0 ∧
p 0 ≤ 4*a-speed*(p 1-(-(delta/tau)))
constructor <;> linarith [hp.1.1,hp.1.2]
theorem comparisonSlab_subset_dataStrip
{a delta tau speed : ℝ} (ha : 0 ≤ a) (hs : 0 ≤ speed) :
shrinkingSlab (-(4*a)) (4*a) (-(delta/tau)) (delta/tau) speed ⊆
pulseStrip (8*a) delta tau := by
intro p hp
have hinc : 0 ≤ speed*(p 1-(-(delta/tau))) := mul_nonneg hs (sub_nonneg.mpr hp.1.1)
refine ⟨?_,hp.1⟩
change -(8*a) ≤ p 0 ∧ p 0 ≤ 8*a
change p 1 ∈ Icc (-(delta/tau)) (delta/tau) ∧
(-(4*a)+speed*(p 1-(-(delta/tau))) ≤ p 0 ∧
p 0 ≤ 4*a-speed*(p 1-(-(delta/tau)))) at hp
constructor <;> linarith [hp.2.1,hp.2.2]
theorem comparison_pulse_travel_bound
{a delta tau speed : ℝ} (ha : 0 < a) (ht : 0 < tau)
(hlarge : 2*speed*delta/(3*a) ≤ tau) : speed*(2*delta/tau) ≤ 3*a := by
rw [← mul_div_assoc]
apply (div_le_iff₀ ht).mpr
convert (div_le_iff₀ (by positivity : 0 < 3*a)).mp hlarge using 1 <;> ring
theorem comparison_pulse_geometry_eventually
{a speed : ℝ} (ha : 0 < a) (hs : 0 ≤ speed) (delta : ℝ) (hd : 0 ≤ delta) :
∀ᶠ tau : ℕ in atTop, 1 ≤ (tau : ℝ) ∧
speed*(2*delta/(tau : ℝ)) ≤ 3*a ∧
pulseStrip a delta (tau : ℝ) ⊆
shrinkingSlab (-(4*a)) (4*a) (-(delta/(tau : ℝ))) (delta/(tau : ℝ)) speed ∧
shrinkingSlab (-(4*a)) (4*a) (-(delta/(tau : ℝ))) (delta/(tau : ℝ)) speed ⊆
pulseStrip (8*a) delta (tau : ℝ) := by
have hlarge : ∀ᶠ tau : ℕ in atTop, max 1 (2*speed*delta/(3*a)) ≤ (tau : ℝ) :=
(tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually
(eventually_ge_atTop _)
filter_upwards [hlarge] with tau htau
have ht1 : 1 ≤ (tau : ℝ) := (le_max_left _ _).trans htau
have ht : (0 : ℝ) < tau := zero_lt_one.trans_le ht1
have htravel := comparison_pulse_travel_bound ha ht ((le_max_right _ _).trans htau)
exact ⟨ht1,htravel,pulseStrip_subset_comparisonSlab ha.le hd ht hs htravel,
comparisonSlab_subset_dataStrip ha.le hs⟩
end SmoothLocal.Pulse
end
end OAI
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