| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.GeneratingKernel |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
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| section |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
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| 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 |
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| 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 |
| |
