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110f4f3 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.GeneratingKernel
namespace OAI
noncomputable section
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
open Set
open Set Filter
open scoped Topology ContDiff
namespace WeakMTWTransport
section
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
local instance actionKernelDualNormedAddCommGroup : NormedAddCommGroup (F →L[ℝ] ℝ) :=
ContinuousLinearMap.toNormedAddCommGroup
local instance actionKernelDualNormedSpace : NormedSpace ℝ (F →L[ℝ] ℝ) :=
ContinuousLinearMap.toNormedSpace
local instance actionKernelBilinearNormedAddCommGroup : NormedAddCommGroup (F →L[ℝ] F →L[ℝ] ℝ) :=
ContinuousLinearMap.toNormedAddCommGroup
local instance actionKernelBilinearNormedSpace : NormedSpace ℝ (F →L[ℝ] F →L[ℝ] ℝ) :=
ContinuousLinearMap.toNormedSpace
lemma generating_kernel_velocity_zero {B : E×F → ℝ} {Q V : E → F}
{g : F → F →L[ℝ] F →L[ℝ] ℝ} {R : F → F →L[ℝ] ℝ} {p w : E}
(hB : ContDiffAt ℝ 2 B (p,Q p)) (hmin : IsLocalMin B (p,Q p))
(hQ : DifferentiableAt ℝ Q p) (hV : DifferentiableAt ℝ V p)
(hg : DifferentiableAt ℝ g (Q p)) (hR : DifferentiableAt ℝ R (Q p))
(hpos : ∀ u : F, u ≠ 0 → 0 < g (Q p) u u)
(hstat : ∀ᶠ v in 𝓝 p, ∀ a : E, fderiv ℝ B (v,Q v) (a,0) = 0)
(hcot : ∀ᶠ v in 𝓝 p, ∀ u : F,
fderiv ℝ B (v,Q v) (0,u) = g (Q v) (V v) u + R (Q v) u)
(hker : fderiv ℝ Q p w = 0) : fderiv ℝ V p w = 0 := by
have hz (u : F) : g (Q p) (fderiv ℝ V p w) u = 0 := by
have heq : (fun v => fderiv ℝ B (v,Q v) (0,u)) =ᶠ[𝓝 p]
fun v => g (Q v) (V v) u + R (Q v) u := hcot.mono (fun _ hv => hv u)
have hd := ((((hg.hasFDerivAt.comp p hQ.hasFDerivAt).clm_apply hV.hasFDerivAt).clm_apply
(hasFDerivAt_const u p)).add
((hR.hasFDerivAt.comp p hQ.hasFDerivAt).clm_apply (hasFDerivAt_const u p))).fderiv
change fderiv ℝ (fun v => g (Q v) (V v) u + R (Q v) u) p = _ at hd
have hh := localMin_generating_cotangent_kernel hB hmin hQ hstat hker u
rw [heq.fderiv_eq,hd] at hh
simpa [hker] using hh
by_contra hn
exact (hpos _ hn).ne' (hz _)
lemma generating_graph_position_immersion {B : E×F → ℝ} {Q V : E → F}
{g : F → F →L[ℝ] F →L[ℝ] ℝ} {R : F → F →L[ℝ] ℝ} {p : E}
(hB : ContDiffAt ℝ 2 B (p,Q p)) (hmin : IsLocalMin B (p,Q p))
(hQ : DifferentiableAt ℝ Q p) (hV : DifferentiableAt ℝ V p)
(hg : DifferentiableAt ℝ g (Q p)) (hR : DifferentiableAt ℝ R (Q p))
(hpos : ∀ u : F, u ≠ 0 → 0 < g (Q p) u u)
(hstat : ∀ᶠ v in 𝓝 p, ∀ a : E, fderiv ℝ B (v,Q v) (a,0) = 0)
(hcot : ∀ᶠ v in 𝓝 p, ∀ u : F,
fderiv ℝ B (v,Q v) (0,u) = g (Q v) (V v) u + R (Q v) u)
(himm : Function.Injective (fderiv ℝ (fun v => (Q v,V v)) p)) :
Function.Injective (fderiv ℝ Q p) := by
intro u v huv
have hw : fderiv ℝ Q p (u-v) = 0 := by rw [map_sub,huv,sub_self]
have hv := generating_kernel_velocity_zero hB hmin hQ hV hg hR hpos hstat hcot hw
apply sub_eq_zero.mp
apply himm
rw [(hQ.hasFDerivAt.prodMk hV.hasFDerivAt).fderiv]
simp [hw,hv]
end
lemma weighted_square_le (a b : ℝ) {s t : ℝ} (hs : 0<s) (ht : 0<t) :
(a+b)^2/(s+t) ≤ a^2/s+b^2/t := by
rw [div_add_div _ _ hs.ne' ht.ne']
apply (div_le_div_iff₀ (add_pos hs ht) (mul_pos hs ht)).mpr
nlinarith [sq_nonneg (t*a-s*b)]
lemma three_leg_energy_le {M : Type*} [PseudoMetricSpace M]
(x b q y : M) {L ε r : ℝ} (hL : 0<L) (hε : 0<ε)
(hb : dist x b ≤ L*r) :
dist x y^2/(2*(L+2*ε)) ≤ L*r^2/2 + dist b q^2/(2*ε) + dist q y^2/(2*ε) := by
have htri : dist x y ≤ L*r+(dist b q+dist q y) :=
(dist_triangle x b y).trans (add_le_add hb (dist_triangle b q y))
have hsq : dist x y^2 ≤ (L*r+(dist b q+dist q y))^2 := by
nlinarith [dist_nonneg (x := x) (y := y)]
have h₁ := weighted_square_le (L*r) (dist b q+dist q y) hL (by linarith : 0<2*ε)
have h₂ := weighted_square_le (dist b q) (dist q y) hε hε
have he : (L*r)^2/L = L*r^2 := by field_simp
rw [he] at h₁
have h₂' : (dist b q+dist q y)^2/(2*ε) ≤ dist b q^2/ε+dist q y^2/ε := by
simpa only [two_mul] using h₂
have hh := (div_le_div_of_nonneg_right hsq (show 0≤L+2*ε by positivity)).trans
(h₁.trans (add_le_add le_rfl h₂'))
have hh' := div_le_div_of_nonneg_right hh (show (0:ℝ)≤2 by norm_num)
calc
dist x y^2/(2*(L+2*ε)) = (dist x y^2/(L+2*ε))/2 := by rw [div_div,mul_comm]
_ ≤ (L*r^2+(dist b q^2/ε+dist q y^2/ε))/2 := hh'
_ = _ := by field_simp; ring
end WeakMTWTransport
end
end OAI
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